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5 answers

Use the following employee’s conditions below in answering the questions.a. A medical technologist who has been in the job for

Question:

Use the following employee’s conditions below in answering the questions.a. A medical technologist who has been in the job for 10 years and works independently with excellent performance.b. A medical technologist new to his/her job.c. A medical technologist who has worked self-reliantly but was not working well nor is productive.d. A medical technologist working in the Clinical Chemistry section for already a month and is still learning. 1. Using the Managerial Grid, what type of leadership style would you use for each of the situations below. Support your answer.

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2 . Six resistors Ri=7.00 0, R2-8.00 $, R3=13.0 0, R4-10.0 &, R;=10.0$, and Rs=7.00 \$ are mounted as shown in the figure The voltage of the battery Is V 22 Volts What is the power dissipated in the Rz resistor? Watts...

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# Dirac action and conventions

I have a (possibly) fundamental question, which is driving me crazy.

Notation

When considering the Dirac action (say reading Peskin's book), one have

$\int dV\;\bar{\psi}\left(\imath\not\partial-m\right)\psi,$

or (following Weinberg)

$\int dV\;\bar{\psi}\left(-\not\partial-m\right)\psi.$

Where Dirac conjugation is defined according to the metric signature.

Question

Is there a reason why the kinetic term in those actions have that particular sign choice?

Are these actions: $\int dV\;\bar{\psi}\left(-\imath\not\partial-m\right)\psi$ and $\int dV\;\bar{\psi}\left(\not\partial-m\right)\psi$ ill-possed?

Personal Thoughts

• Elements of Clifford algebra are well defined up to a sign, therefore, I would say the second set of actions is Ok.

• I'm afraid that the different sign would spoil the positivity of energy. However, I'm not sure how to prove that statement.

Any help or thoughts are welcome! Thank you.

-
This is the most annoying part of modern QFT--- the lack of good standardized Dirac notation and results. This can't be solved by committee, because it was standardized in the 1950s, but in the 1970s, when we needed higher dimensions, it was realized the standardized thing was suboptimal, and the dust hasn't settled yet. I would go with Weinberg, but I don't have a good answer for the optimal conventions, and there are about half a dozen reasonable ones floating around (Euclidean/Lorentzian Weyl/Dirac Majorana/C-matrix are each discrete choices, also complex-coordinates/real-coordinates). – Ron Maimon Oct 24 '12 at 17:15
What do you mean @Dox by Elements of Clifford algebra are well defined up to a sign? that $\gamma^\mu$ and $-\gamma^\mu$ yield the same? you are actually changing by $i$. Didn't you write $\partial\!\!\! /$ instead of $\mathrm{i}\partial\!\!\! /$? – c.p. Oct 24 '12 at 19:08
@JorgeCampos: You are right, but I mean one a convention is picked (say signature of metric), changing $\gamma^\mu\to(\gamma^\mu)'= -\gamma^\mu$ span the same Clifford algebra. – Dox Oct 24 '12 at 19:53

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# Solution

Determine if the following is an identity: $$\frac{1+\tan t}{\tan t} = \cot t + \sec^2 t - \tan^2 t$$

Let us start by trying to simplify the left side, $$\begin{array}{rclc} \frac{1+\tan t}{\tan t} &=& \frac{\displaystyle{1+\frac{\sin t}{\cos t}}}{\displaystyle{\frac{\sin t}{\cos t}}} & \quad \scriptsize{\textrm{rewriting in terms of } \sin \theta \textrm{ and } \cos \theta}\\\\ &=& \frac{\displaystyle{1+\frac{\sin t}{\cos t}}}{\displaystyle{\frac{\sin t}{\cos t}}} \cdot \frac{\cos t}{\cos t} & \overset{\normalsize{\textrm{to simplify the complex fraction}}}{\scriptsize{\textrm{we multiply by a well-chosen value of one}}}\\\\ &=& \frac{\cos t + \sin t}{\sin t} \end{array}$$

With it unclear how to further simplify the left side, we turn to simplifying the right side -- attempting to match the last expression above:

$$\begin{array}{rclc} \cot t + \sec^2 t - \tan^2 t &=& \frac{\cos t}{\sin t} + \frac{1}{\cos^2 t} - \frac{\sin^2 t}{\cos^2 t} & \quad \scriptsize{\textrm{rewriting in terms of } \sin \theta \textrm{ and } \cos \theta}\\\\ &=& \frac{\cos t}{\sin t} + \frac{1 - \sin^2}{\cos^2 t} & \overset{\normalsize{\textrm{note the left-most term matches part of } \frac{\cos t + \sin t}{\sin t}}}{\scriptsize{\textrm{so we only combine the other two terms...}}}\\\\ &=& \frac{\cos t}{\sin t} + \frac{\cos^2 t}{\cos^2 t} & \scriptsize{\textrm{recalling} \sin^2 \theta + \cos^2 \theta = 1}\\\\ &=& \frac{\cos t}{\sin t} +1 & \overset{\normalsize{\textrm{now we combine fractions as the side we are}}}{\scriptsize{\textrm{attempting to match is a fraction, not a sum}}}\\\\ &=& \frac{\cos t}{\sin t} +\frac{\sin t}{\sin t}\\\\ &=& \frac{\cos t + \sin t}{\sin t} \end{array}$$

Since this just happens to match the simplified version of the left side, we now know that the equation above is an identity.


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## Stream: Geographic locality

### Topic: Pittsburgh, PA

#### Koundinya Vajjha (Apr 12 2020 at 13:15):

Pittsburgh represent!
The Pittsburgh area has a very active Lean community, centered around Carnegie Mellon University and the neighboring University of Pittsburgh.

We have weekly Lean meetings every Tuesday at 4.30pm in Thackeray Hall at Pitt. (At least we used to, until the university closed down.)

#### Jeremy Avigad (Apr 12 2020 at 13:45):

We use the cmu-lean mailing list on Google groups for the weekly meeting announcements. It is a private mailing list but anyone is welcome to join. You should be able to find it on Google groups and click a button to send a request, but in any case you can email me at avigad@cmu.edu and I will send an invite.

The meetings are informal. Sometimes we just talk shop, but I generally try to line up a volunteer to present something they are working on or a paper they are reading, or lead a discussion. The topics are not restricted to Lean; participants are generally interested in all aspects of formal methods and their applications to mathematics.

We have had a few meetings by Zoom since the lockdown began. We'll take a break this week, but we're still planning to meet sporadically, as people volunteer to present. I have nearly all the slides done for a talk I was going to give at a phil math conference last week (https://princetonphilmath.wordpress.com/), so I'll present those a week from Tuesday (April 21) unless we get a better offer.

#### Tim Daly (Apr 14 2020 at 20:06):

Tim Daly

Out near the KPIT airport, just behind the Raccoon State Park.

#### William Sealy Gosset (Oct 06 2020 at 11:37):

I am in Bethel Park. Once this whole pandemic thing has resolved itself, I hope to become more involved with the Lean community at CMU and Pitt. My background is in financial engineering, and as a result my mathematics background is more on the applied side.

Last updated: Jul 29 2021 at 21:10 UTC


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# Can theories with axioms beyond arithmetic make false promises about integer existence?

In this paper, author Nik Weaver warns that there could be questions of $$\Sigma_1$$-validity of $${\mathrm{ZFC}}$$ set theory. As I understand it, he suggests that the axioms of a set theory might be strong enough to decide PA-undecidable statements about Turing machines, and moreover decide them wrongly. Wrongly in the sense that e.g. $${\mathrm{ZFC}}$$ might prove for a particular machine $$t$$ there to "exist" a $$n\in {\mathbb N}$$ such that $$t$$ halts at step $$n$$ (i.e. $${\mathrm{ZFC}}$$ proves $$\exists n.T(n)$$ for some statement $$T$$ about a particular coding up of a machine $$t$$), but "in reality" the machine actually can't do that. This wouldn't be a problem w.r.t. consistency, since it affects a statement where Peano arithmetic ($${\mathrm{PA}}$$) axioms were too weak to make a call on it.

See here for some relevant definitions and this related SE question. The relation between $${\mathrm{PA}}$$ and $${\mathrm{ZFC}}$$ is subtle but e.g. from reading the answer on the mentioned questions, one would think $${\mathrm{PA}}$$ got us covered for any given Turing machine w.r.t. $$\sigma_1$$. So I'm trying to understand where a loophole is that Weaver points out. To clarify, I'm not interested in an inconsistent $${\mathrm{PA}}$$ situation or an inconsistent $${\mathrm{ZFC}}$$ situation, but I won't presume any soundness properties of the systems. And for the question, I'm also only interested in machines that could (edit: assuming we could wait arbitrary long and provide the memory capacity) be put on the quest to search for a natural and fail to ever halt - despite $${\mathrm{ZFC}}$$ claims.

I know that there are conservativity results of $${\mathrm{PA}}$$ over $${\mathrm{HA}}$$ on, I think, that level of the hierarchy, and similarly with second order-$${\mathrm{PA}}$$ v. second order-$${\mathrm{HA}}$$, as well as $${\mathrm{ZF}}$$ over $${\mathrm{IZF}}$$. I.e. we know that for $$\exists n.(f(n)=0)$$-statements of $${\mathrm{HA}}$$, won't see anything new in $${\mathrm{PA}}$$ ($${\mathrm{PA}}$$ can't fail the world of HA on that level, so to speak). But do we know anything more about $${\mathrm{ZFC}}$$ speaking about PA statements? I'm not sure if results I just listed do in fact clarify the $$\Sigma_1$$-validity issue on their own. Further (since I read someone bring it up in this context) I wonder whether or how $${\mathrm{ZF}}$$ models of itself or of $${\mathrm{PA}}$$ could clarify what's possible and what's inconsistent. Is this issue discussed somewhere on that level of the hierarchy, where validity issues in connection with "the real world" would still be "hands on" in this sense (i.e. not stuff about Turing jumps somewhere removed from recursive machines)? Assuming $${\mathrm{PA}}$$ is right about the world, can $${\mathrm{ZFC}}$$ be wrong about the world in this "wrong promise/wait forever" kind of way? I do understand that "wrong promise" might be orthogonal to a hard "wrong" notion in proof theory.

• I've heard that iteration of a process which at step $n$ writes an integer in "recursive base $2n$" (where the exponents also get translated into recursive base $2n$), change all bases to $2n+1$, and then subtracts one, always terminating is sufficient to show consistency of PA, and therefore can't be proved by PA. But ZFC does prove termination of this calculation with any starting input, essentially as a result of the fact that $\varepsilon_0 = \omega^{\omega^{\cdots}}$ is well-ordered. Though I guess that's probably $\Pi_2$ not $\Sigma_1$. – Daniel Schepler Sep 19 at 0:03
• @DanielSchepler Sounds interesting. It's not a bad apple case, though, is it? Weaver, in chapter 2 starting at around page 6, 7 in the above text, speaks about (quote) "if ZFC gives us bad information about which Turing machines halt", where, for this to be persuasive, I suppose there needs to be a means of judging "bad". If your ordinal-type consistency statement can't be dismissed by other means (the machine would indeed just halt), then this would just give another fair theorem and not a false promise of sorts. Or is a real-world-halting that PA doesn't predict automatically bad? – Nikolaj-K Sep 19 at 0:14
• @Nikolaj-K "is a real-world-halting that PA doesn't predict automatically bad?" You have to distinguish between single halting instances and universally halting programs. The former are pretty boring here: $\mathsf{PA}$ is $\Sigma^0_1$-complete (indeed this is gigantic overill; see e.g. the discussion here), so there are no individual instances of halting computations which $\mathsf{PA}$ or $\mathsf{ZFC}$ miss. However, the statement that such-and-such program halts on all inputs is significantly more complicated - it is $\Pi^0_2$ in general. – Noah Schweber Sep 19 at 2:04
• (As Daniel said above.) By Godel, no "reasonable" theory (namely: no consistent computably axiomatizable theory interpreting $\mathsf{Q}$) is $\Pi^0_1$-complete, so a fortiori there will be true $\Pi^0_2$ statements which (say) $\mathsf{ZFC}$ misses. As to the question you ask, it is indeed possible that $\mathsf{PA}$ is fully sound but $\mathsf{ZFC}$ is inconsistent, or that $\mathsf{PA}$ is fully sound and $\mathsf{ZFC}$ is consistent but that $\mathsf{ZFC}$ is not $\Sigma_1$-sound, or so on. – Noah Schweber Sep 19 at 2:06
• @Nikolaj-K I find your question as written a bit unclear, hence my comments as opposed to an answer. If my comments addressed your question I'll be happy to turn them into an answer, let me know - if not, in what way do they not? – Noah Schweber Sep 19 at 2:10

Any "appropriate" theory $$T$$ - like $$\mathsf{PA}$$ or $$\mathsf{ZFC}$$ - is strong enough to verify all true halting facts. That is, such theories are $$\Sigma_1$$-complete. Conversely, per Godel no such theory is $$\Pi_1$$-complete. In particular, if $$T$$ is "appropriate" in the relevant sense then so is $$T+\neg\mathsf{Con}(T)$$; since this latter theory is $$\Sigma_1$$-complete, consistent, and proves a false $$\Sigma_1$$-sentence, it (and a fortiori $$T$$ itself) must not be $$\Pi_1$$-complete.
So a consistent theory of the type we're looking at cannot afford to make any false $$\Pi_1$$ assertions, but it could conceivably make a false $$\Sigma_1$$ assertion. This distinction between $$\Pi_1$$ and $$\Sigma_1$$ explains why there is no tension between Weaver's observation and the "sufficiency" of $$\mathsf{PA}$$ (and indeed much less) for $$\Sigma_1$$ propositions.
Finally, $$\mathsf{ZFC}$$ could indeed be consistent but $$\Sigma_1$$-unsound even if $$\mathsf{PA}$$ is fully sound - at least, working within a reasonably weak background theory. Specifically, assuming $$\mathsf{ZFC+Con(ZFC)}$$ is consistent in the first place then so is $$\mathsf{ZFC+Con(ZFC)+\neg Sound_{\Sigma_1}(ZFC)+Sound(PA)}$$. (The "$$+\mathsf{Sound(PA)}$$"-bit is redundant since $$\mathsf{ZFC}$$ already proves that $$\mathsf{PA}$$ is fully sound, I'm just including it for explicitness.) So basically we cannot use the correctness of $$\mathsf{PA}$$ to justify even the $$\Sigma_1$$-correctness of $$\mathsf{ZFC}$$ without baking the latter assumption into our base theory at the outset. However, this happens long before we hit $$\mathsf{ZFC}$$: the same holds for $$\Pi^1_1$$-$$\mathsf{CA_0}$$, if memory serves, which is a tiny fragment of $$\mathsf{Z_2}$$ which is itself a tiny tiny tiny fragment of $$\mathsf{ZFC}$$. So leaping all the way to $$\mathsf{ZFC}$$ here is massive overkill, and I think makes this phenomenon appear more mysterious than it actually is.


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# Deriving distribution from conditional distribution

Hi guys I am having problems deriving $$P(X = k)$$ if $$P(X = k|X+Y = n)$$ = $${n}\choose{k}$$ $$\times$$ $$2^{-n}$$

X and Y are i.i.d. random variables with values in $$\mathbb{N_0}$$.

After playing a bit with the formula and using independency of the variables I get to:

$$P(X = k)$$ = $${n}\choose{k}\times$$ $$2^{-n}\times$$ $$\frac{P(X+Y = n)}{P(Y = n-k)}$$

and I could theoretically rewrite $$P(X+Y = n)$$ = $$\sum_{k = 0}^{n}P(X = k) P(Y = n-k)$$, although I am not sure whether the indexing is correct and it does not really help me to get further.

I know from an exercise that at some point I should get to an induction, but I just do not know how. Thank you for any help.

• Hint: the expression you got can be written as $k! \,P(X=k) \,(n-k)! \,P(Y=n-k) = 2^n\, n!\, P(X+Y=n)$ and with a little imagination that can be rewritten $\dfrac{P(X=k)}{\frac{c^k}{k!}d} \dfrac{P(Y=n-k)}{\frac{c^{n-k}}{(n-k)!}d} = \dfrac{P(X+Y=n)}{\frac{(2c)^n}{n!}d^2}$ for suitable $c,d$ – Henry May 15 at 16:32
• Sorry, but could you clarify it a bit more? – Jakub May 15 at 17:32
• Do you know any discrete distributions whose probability mass function involves a factorial in the denominator and a power term in the numerator? – Henry May 15 at 19:22

## 1 Answer

Let $$p_n=P(X=n)$$. Consider $$P(X=n|X+Y=n)=\frac{P(X=n)P(Y=0)}{P(X+Y=n)}=2^{-n},$$ which together with $$P(X+Y=n)=\sum_{k=0}^n p_kp_{n-k}$$ implies $$p_np_0=2^{-n}\sum_{k=0}^n p_kp_{n-k}.\tag1$$ Using $$(1)$$, and induction, you can prove that for all $$n\ge 0$$, that $$p_n=p_0\cdot \frac{(p_1/p_0)^n}{n!}.\tag{2}$$ Summing $$(2)$$ over all $$n\ge 0$$, you get $$\sum_n p_n=1=p_0\sum_n \frac{(p_1/p_0)^n}{n!}=p_0e^{p_1/p_0}\implies p_1=-p_0\log p_0. \tag3$$ Therefore, combining $$(3)$$ and $$(2)$$, $$p_n=p_0\frac{(-\log p_0)^n}{n!}.$$ Therefore, the distribution of $$X$$ is determined up to a free parameter, $$p_0$$. Note that $$X$$ has a Poisson$$(\lambda)$$ distribution with $$\lambda=-\log p_0$$.

In case you have trouble proving $$(2)$$, note that the base cases $$n=0,1$$ are immediate, and for all $$n\ge 2$$, the inductive step is as follows (hidden behind spoilers in case you want to find it yourself):

\begin{align}p_np_0&=2^{-n}\cdot 2p_0p_n+2^{-n}\sum_{k=1}^{n-1} p_kp_{n-k}\\&=2^{-n+1}p_0p_n+2^{-n}\sum_{k=1}^{n-1} p_0^2 \frac{(p_1/p_0)^k}{k!}\frac{(p_1/p_0)^{n-k}}{(n-k)!}\\&=2^{-n+1}p_0p_n+2^{-n}\frac{p_0^2(p_1/p_0)^n}{n!}\sum_{k=1}^{n-1}\binom{n}k\\&=2^{-n+1}p_0p_n+2^{-n}\frac{p_0^2(p_1/p_0)^n}{n!}\cdot (2^n-2), \end{align}which simplifies to $$(2)$$.


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# Manually define clusters in Seurat and determine marker genes

I want to define two clusters of cells in my dataset and find marker genes that are specific to one and the other. Is there a way to do this in Seurat? Say, if I produce two subsets by the SubsetData function, is there a way to feed them into some other function that would calculate marker genes? If not, what other packages would you recommend for doing that?

If you look here:

https://satijalab.org/seurat/seurat_clustering_tutorial_part2.html

I just need a way to define ident myself, the number of levels (2) and assign numbers to each cell (0, 1), and then run DE between 0 and 1 clusters which is obvious how to do afterwards.

• A related issue was answered on github. Just as an alternative reference to the excellent answers below) – Tapper Mar 14 '19 at 17:31

## 2 Answers

I think you are looking to FindAllMarkers function from Seurat. As you said, you just have to define your ident, that have to have the structure of a table (cell names as names and cluster as value):

pident=as.factor(clusters)
names(pident)=cellNames
object1@ident=pident

And then run the FindAllMarkers function:

FindAllMarkers(object1, min.pct = 0.25, min.diff.pct = 0.25)

You can specify several parameters in this function (type of DE to perform, thresholds of expression, etc).

Seurat has functions for adding metadata and setting identities. Get unique cell names:

cell.labels <- seuratobject@ident

Replace column and its name with your cluster labels (e.g.), then:

seuratobject <- AddMetaData(seuratobject, metadata=cell.labels)
seuratobject <- SetAllIdent(seuratobject, id='yourclusterlabels')

Because you want to contrast two clusters against each other, I suggest using FindMarkers() as opposed to FindAllMarkers():

FindMarkers(object, ident.1, ident.2)

It can also compare combinations of clusters.


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# How do you find the exact value of cos 33pi/4?

Then teach the underlying concepts
Don't copy without citing sources
preview
?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

7
Alan P. Share
Dec 7, 2015

$\cos \left(\frac{33 \pi}{4}\right) = \frac{\sqrt{2}}{2}$

#### Explanation:

$\frac{33 \pi}{4}$ is equivalent to $\frac{32 \pi}{4} + \frac{\pi}{4} = 2 \cdot \left(2 \pi\right) + \frac{\pi}{4} = \frac{\pi}{4} \textcolor{w h i t e}{\text{XX}}$(since $2 \pi = 0$, a complete circle)

$\cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \textcolor{w h i t e}{\text{XXX}}$(this is one of the standard angles)

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# present value of annuity formula

THE CERTIFICATION NAMES ARE THE TRADEMARKS OF THEIR RESPECTIVE OWNERS. When I try to reproduce the periodic steps of this formula in Excel, the principal is exhausted in period 33. and similar publications. It is denoted by P. Step 2: Next, figure out the interest rate on the basis of the ongoing market rates and it will be used to discount each periodic payment to the present day. The present value of an annuity due is a calculation that estimates the value of an investment that would begin right away based on future payments. ALL RIGHTS RESERVED. Present Value of Annuity Due is calculated using the formula given below, PV of Annuity Due = PMT * [(1 – (1 / (1 + r) ^ n))/ r] * (1 + r). How will it affect the final sum of the money invested? 2) The rate does not change This is especially true with the dependability of fixed interest rates. Present Value of Annuity Formula (Table of Contents). The annuity would have a 4% annual interest rate. Rather than calculating each payment individually and then adding them all up, however, you can use the following formula, which will tell you how much money you'd have in the end: FVOrdinary Annuity=C×[(1+i)n−1i]where:C=cash flow per periodi=interest raten=number of payments\begin{aligned} &\text{FV}_{\text{Ordinary~Annuity}} = \text{C} \times \left [\frac { (1 + i) ^ n - 1 }{ i } \right] \\ &\textbf{where:} \\ &\text{C} = \text{cash flow per period} \\ &i = \text{interest rate} \\ &n = \text{number of payments} \\ \end{aligned}FVOrdinary Annuity=C×[i(1+i)n−1]where:C=cash flow per periodi=interest raten=number of payments. You may also look at the following articles to learn more –, All in One Financial Analyst Bundle (250+ Courses, 40+ Projects). Here, we use the same numbers, as in our previous examples: FVAnnuity Due=$1,000×[(1+0.05)5−10.05]×(1+0.05)=$1,000×5.53×1.05=5,801.91\begin{aligned} \text{FV}_{\text{Annuity Due}} &= \1,000 \times \left [ \frac{ (1 + 0.05)^5 - 1}{ 0.05 } \right ] \times (1 + 0.05) \\ &= \$1,000 \times 5.53 \times 1.05 \\ &= \$5,801.91 \\ \end{aligned}FVAnnuity Due=$1,000×[0.05(1+0.05)5−1]×(1+0.05)=$1,000×5.53×1.05=$5,801.91. Present Value of an Annuity Due Calculator You can use the present value of an annuity due calculator below to work out the cash value of your immediate investment by entering the required numbers. The present value of an annuity is the value of money you would invest now an annuity, directly affected by the interest and payments the annuity would make in the future. The present value of an annuity is determined by using the following variables in the calculation. Therefore, the present value of the annuity is$14,430. All payments in an annuity due would be paid at the beginning of every pay period. Pro members can track their course progress and get access to exclusive downloads, quizzes and more! If you are making the payments, then an ordinary annuity is better if the option is available to you. Calculate the price of the bond. By using the geometric specifics. In the example shown, the formula in C9 is: = PV(C5, C6, C4,0,0) Consequently, an annuity due will always be of greater value than an ordinary annuity (assuming everything else is equal). Present Value of an Annuity Due Formula PV = C \times \bigg[ \dfrac{1 - (1 + r)^{-n}}{r} \bigg] \times (1 + i) C = cash flow per period; r = interest rate; n = number of periods; It’s important to remember that in finding the annuity due, the payments must begin immediately. You can calculate the present or future value for an ordinary annuity or an annuity due using the following formulas. Study Finance is an educational platform to help you learn fundamental finance, accounting, and business concepts. The reason the values are higher is that payments made at the beginning of the period have more time to earn interest. To account for payments occurring at the beginning of each period, it requires a slight modification to the formula used to calculate the future value of an ordinary annuity and results in higher values, as shown below. tabel pv annuity due.pdf - Formula PV =(1 i x(1 1(1 i)n i Present Value Annuity Due Tables n\/i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 to be used in the formula. Contact us at: Below is how much you would have at the end of the five-year period. It is important to pay particular attention to the rate as you are calculating this equation. By closing this banner, scrolling this page, clicking a link or continuing to browse otherwise, you agree to our Privacy Policy, Download Present Value of Annuity Due Formula Excel Template, Black Friday Offer - All in One Financial Analyst Bundle (250+ Courses, 40+ Projects) Learn More, You can download this Present Value of Annuity Due Formula Excel Template here –, 250+ Online Courses | 1000+ Hours | Verifiable Certificates | Lifetime Access, Present Value of Annuity Due Formula Excel Template, Finance for Non Finance Managers Course (7 Courses), Investment Banking Course(117 Courses, 25+ Projects), Financial Modeling Course (3 Courses, 14 Projects), Examples of Interest Formula with Excel Template, Annuity Due Formula (Examples with Excel Template), Finance for Non Finance Managers Training Course, PV of Annuity Due = $1,000 * [(1 – (1 / (1 + 5%)^3)) / 5%] * (1 + 5%), PV of Annuity Due =$100,000 * [(1 – (1 / (1 + 5%)^8)) / 5%] * (1 + 5%), PV of Annuity Due = $1,000 * [(1 – (1 / (1 + 13.2%)^12)) / 13.2%] * (1 + 13.2%), PV of Annuity Due =$500 * [(1 – (1 / (1 + 12%)^12)) / 12%] * (1 + 12%). Company ABC Private limited wants to purchase machinery in installment purchase system method and it will the third party an amount of $100,000 at the starting of each year for the next 8 years. eval(ez_write_tag([[300,250],'studyfinance_com-medrectangle-3','ezslot_1',108,'0','0'])); To understand the present value of an annuity due, you should first have a solid understanding of what an annuity is and the two types. In fact, it is predominantly used by accountants, actuaries and insurance personnel to calculate the present value of structured future cash flows. The canceled out throughout the equation by doing this. © 2020 - EDUCBA. © 2020 - EDUCBA. A deferred annuity pays the initial payment at a later time. Here we discuss how to calculate Present Value of Annuity along with practical examples. The formula for the present value of an annuity due identifies 3 variables: the cash value of payments, the interest rate, and the number of payments. The type of interest rate that you use in the calculation should match the number of payments you are using in your equation. He can choose between an annuity of$50,000 paid annually at the end of each year for 25 years or a \$1,000,000 lump sum. Mathematically, it is represented as, PVA Due = P * [1 – (1 + r/n)-t*n] * [ (1 + r/n) / (r/n)] The user should use information provided by any tools or material at his The present value of annuity formula determines the value of a series of future periodic payments at a given time. Annuity payments can be sent out or required at different frequencies. It is also useful in the decision – whether a lump sum payment is better than a series of future payments based on the discount rate. This site was designed for educational purposes. These recurring or ongoing payments are technically referred to as "annuities" (not to be confused with the financial product called an annuity, though the two are related). Calculate the present value of the future cash inflow if the relevant discounting rate based on the ongoing market rate is 5% while the payment is received: Present Value of Ordinary Annuity is calculated using the formula given below, PVA Ordinary = P * [1 – (1 + r/n)-t*n] / (r/n). Also, you will often see the interest rate referred to as a discount rate when discussing the present value of an annuity due. Are Variable Annuities Subject to Required Minimum Distributions? The interest rate is 13.2%. PVOrdinary Annuity=C×[1−(1+i)−ni]\begin{aligned} &\text{PV}_{\text{Ordinary~Annuity}} = \text{C} \times \left [ \frac { 1 - (1 + i) ^ { -n }}{ i } \right ] \\ \end{aligned}PVOrdinary Annuity=C×[i1−(1+i)−n].

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For any system of linear algebraic equations described in the form represented in (4), a solution exists if only if b is a linear combination of nonzero column vectors. To solve such a system graphically, we will graph both lines on the same set of axis and look for the point of intersection. However speed is an important factor for solving linear … In this lesson, you will study systems of linear and quadratic equations.This type of system can have one solution, two solutions, or no solutions. Parallel Lines And The Coordinate Plane Graphing Linear. Thus, if the rank r, of A solving a system of equations. Indeed, you probably spent much of your time in high school learning to manipulate and plot solutions to equations of the form Ax + By = C or y = mx + b. Theorem 3.1 The system of two equations in n unknowns over a field F . ENCE 203 Œ CHAPTER 5a. Materials include course notes, lecture video clips, JavaScript Mathlets, a quiz with solutions, practice problems with solutions, a problem solving video, and problem sets with solutions. 1.3.1. So there are in nitely many solutions. Ixl Solve A System Of Equations By Graphing Word Problems. Many answers. One of the last examples on Systems of Linear Equations was this one: Example: Solve. Notes – Systems of Linear Equations ... ***To solve a system of equations by graphing simply graph both equations on the same coordinate plane and find where they intersect. For example, a linear equation in three variables represents a plane in three-dimensional space. 2X – Y = 6 3X + 5Y = 22 4. Graphing Systems Of Linear Equations Worksheet Pdf At. A system of linear equations can have either one solution, no solutions, or inﬁnitely many solutions. ©2 r2C0 K1C22 RKNuftXa 8 MSyo Jf3t cwJadrqe 7 XLOLkCt. Systems of Linear Equations . Or like y + 0.5x − 3.5 = 0 and more. We learned how to solve linear equations here in the Systems of Linear Equations and Word Problems Section. A system of equations AX = B is called a homogeneous system if B = O. NOTE: You might need to multiply an ENTIRE equation before you can add the two equations to eliminate a variable. This might introduce extra solutions. The method has been used to derive applied models in diverse topics like ecology, chemistry, heating and cooling, kinetics, mechanics and electricity. Various methods have been evolved to solve the linear equations but there is no best method yet proposed for solving system of linear equations[1]. In this paper linear equations are discussed in detail along with elimination method. We can use either Substitution or Elimination, depending on what’s easier. Ex: x + y = 1, 2x + y = 5-2-Create your own worksheets like this one with Infinite Algebra 2. 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Two systems of linear equations are said to be equivalent if they have equal solution sets. Solving a System of Linear Equations in Three Variables Steps for Solving Step 1: Pick two of the equations in your system and use elimination to get rid of one of the variables. Bolster skills in various methods of solving systems of linear equations with this practice worksheet for grade 8. Typically we consider B= 2Rm 1 ’Rm, a column vector. Variable Names Each compartment is labelled with a variable X. Download System of Linear Equations solved questions PDF, from the link mentioned below. 2 ' Assakkaf Slide No. The method. Guassian elimination and Guass Jordan schemes are carried out to solve the linear system of equation. In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same set of variables. Solve the following system. 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Write a system of equations, state whether it is best to solve by substitution or elimination and state your answer. Geometrically, solving a system of linear equations in two (or three) unknowns is equivalent to determining whether or not a family of lines (or planes) has a common point of intersection. Systems of Linear Equations 0.1 De nitions Recall that if A2Rm n and B2Rm p, then the augmented matrix [AjB] 2Rm n+p is the matrix [AB], that is the matrix whose rst ncolumns are the columns of A, and whose last p columns are the columns of B. Step 3: The results from steps one and two will each be an equation in two variables. If B ≠ O, it is called a non-homogeneous system of equations. Advanced. x5yz11 3z12 2x4y2z8 +−=− = +−= All of the following operations yield a system which is equivalent to the original. This array of free pdf worksheets is meticulously designed for 8th grade and high school students. Systems of large number of equations require the use of numerical methods and computers. A Linear Equation is an equation for a line. The systems of linear equations are a classic section of numerical methods which was already known BC. equations is termed as system of linear equations. 25) Write a system of equations with the solution (4, −3). SAMPLE APPLICATION OF DIFFERENTIAL EQUATIONS 3 Sometimes in attempting to solve a de, we might perform an irreversible step. Solving simple 2x2 systems using elementary row operations. e.g., 2x + 5y = 0 3x – 2y = 0 is a homogeneous system of linear equations whereas the system of equations given by e.g., 2x + 3y = 5 x + y = 2 is a non-homogeneous system of linear equations. are opposites, if we add the two equations together, we can eliminate the y-variable. https://www.patreon.com/ProfessorLeonardWhat a System of Linear Equations represents and how to find a solution. (Equivalent systems have the same solution.) A linear system may behave in any one of 3 possible ways: The system has no solution, a single unique solution or infinitely many solutions. In this situation, the lines are parallel, as we can see from the graph. Free trial available at KutaSoftware.com 1.2. Systems of Non-Linear Equations (Note that solving trig non-linear equations can be found here). Solve the following system. Graphing a. Graph one equation b. Graph the other equation on the same plane. 2 Ł … Graphing Standard Form Worksheets Algebra . Hide Ads About Ads. System of linear equations From Wikipedia, the free encyclopedia In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the … a z 9AmltlU Or Gi 5gUh vtIs k Hrfe bs OeWrGvie KdP.r A UMxa3d0e 3 owYigt lh 9 aIWnafYi RnSi YtMe8 lAnlngNe8brYaM M1Y.b Worksheet by Kuta Software LLC A compartment diagram consists of the following components. A linear system in three variables determines a collection of planes. Solving Systems of Equations - Any Method | #Worksheet 1 . CCSS: HSA-REI.6. Sometimes we need solve systems of non-linear equations, such as those we see in conics. (Note that with non-linear equations, there will most likely be more than one intersection; ... Systems of Equations Calculator Screens: Notes $$\displaystyle \begin{array}{l}y=-x+4\\y=-x-2\end{array}$$ Notice that the slope of these two equations is the same, but the $$y$$-intercepts are different. system of linear diﬀerential equations. x +y ≤6 Inequality 1 2x ºy >4 Inequality 2 A of a system of linear inequalities is an ordered pair that is a solution of each inequality in the system. They are often based on same set of variables. Examples: 6!−5!=21 2!+5!=−5 Since 5! This page is only going to make sense when you know a little about Systems of Linear Equations and Matrices, so please go and learn about those if you don't know them already! Consider the following simple 2x2 systemof linear equations a11 x1 + a12x2 = b1 (7) a21 x1 + a22 x2 = b2 We can write this in matrix form as Ax= b A = a11 a12 a21 a22 ,x= x1 x2 ,b= b1 b2 . and −5! Refer to Figure 2. The Example. 4x + 3y = 10 2x + y = 4 SOLUTION If we multiply the bottom equation by −3, the coefficients of y in the resulting equation and the top equation will be opposites. New Solving And Graphing Inequalities Cheat Sheet. 3X – 7Y = 4 6X + 14Y = 1 5. A linear equation is not always in the form y = 3.5 − 0.5x, It can also be like y = 0.5(7 − x) Or like y + 0.5x = 3.5. 156 Chapter 3 Systems of Linear Equations and Inequalities Graphing and Solving Systems of Linear Inequalities GRAPHING A SYSTEM OF INEQUALITIES The following is a in two variables. When solving a system containing two linear equations there will be one ordered pair (x,y) that will work in both equations. In the last section, we used the Gauss-Jordan method to solve systems that had exactly one solution. EXAMPLE 1.1.1 Solve the equation 2x+ 3y= 6: Solution. Solve the following system. Show Ads. Students can easily access and download chapter wise solutions in PDF format for free. A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. c. Find the point, or points, or intersection. For example, + − = − + = − − + − = is a system of three equations in the three variables x, y, z.A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. Systems Of Linear Equations. Step 2: Pick a different two equations and eliminate the same variable. The intersection point is the solution. Every system of equations has either one solution, no solution, or infinitely many solutions. _____ 3.


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# Mathematical term e

The number e is a mathematical constant, approximately equal to 271828, which appears in many different settings throughout mathematics it was discovered by the swiss mathematician jacob bernoulli while studying compound interest. In mathematics, a factorial is a function applied to natural numbers greater than zero the symbol for the factorial function is an exclamation mark after a number, like this: 2 the usual definition of n factorial is n = n (n - 1) (n - 2) 2 1, where 'n' is a. Terms to be added together were simply juxtaposed, while subtraction required the special symbol $\wedge$ equality was denoted by the letter $\iota$ (from the greek \$\iota\sigma which makes particularly extensive use of mathematical symbols mathematical logic and classification of. Mathematical terms and definitions: are gr bner covers in dynamic geometry related to gr bner basis in algebra. If the series terms do not go to zero in the limit then there is no way the series can converge since this would violate the theorem one of those tricks that you see occasionally where you get a contradictory result because of a hard to spot math/logic error. This is a glossary of math definitions for common and important mathematics terms used in arithmetic, geometry, and statistics.

A mathematical process of addition, subtraction, multiplication or division axes: two perpendicular number lines that are used to locate points in a coordinate plane by convention a term used to indicate that two or more quantities are of the same value equation: a. This page attempts to show some of the ambiguities in defining some of the mathematical terms that might be encountered at the high school level. In word, you can insert mathematical symbols into equations or text by using the equation tools. 1 de nition and properties of the exp function 11 de nition of the exp function number e de nition 1 the number e is de ned by lne = 1 ie, the unique number at which lnx = 1 remark let l(x) = lnx and e(x) = ex for x rational then. Edge any side of a polyhedron's faces elevations two-dimensional views of three-dimensional figures given from the top, front, or sides elevations usually include measurements and a scale.

Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Operations & algebraic thinking print this page kindergarten generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers but should show the mathematics in the problem. The story of mathematics - glossary of mathematical terms decimal number: a real number which expresses fractions on the base 10 standard numbering system using place value, eg 37 100 = 037. More generally, the definition of sine (and other trigonometric functions) similarly, python defines mathsin(x) within the built-in math module complex sine functions are also available within the cmath module, eg cmathsin(z. Math explained in easy language, plus puzzles, games, quizzes, videos and worksheets for k-12 kids, teachers and parents.

## Mathematical term e

What is e e is a numerical constant that is equal to 271828 for more information on e, visit the the math forum at mathforumorg reference: the mathforumorg : provide feedback sponsors & contributors terms & conditions about the site.

• Glossary of terms that have been discussed or mentioned on these pages letter a.
• Mathwords home page mathwords: terms and formulas from beginning algebra to calculus an interactive math dictionary with enoughmath words, math terms.
• I have a math expression which has what appears to be a function called exp() and i don't know what it means it's on this site right here here is the link to the picture of the expression itself exp means exponential function in other words.
• I tried searching wikipedia for e and maths but found the e mathematical constant with a scary looking graph my guess is e stand for exponential or something and -5 is the power it is raised to and the time is a really small number.

Comprehensive math vocabulary lists are based on the common core state math standards and organized by k-12 grade level. Math 120 the logistic function elementary functions examples & exercises in the past weeks and polynomial functions as mathematical models in many different contexts another type of function, called the logistic function, occurs often in describing the term e-bx becomes smaller and. Get an answer for 'what do the letters r, q, n, and z mean in math' and find homework help for other math questions at enotes. Barcodes,inc home articles the complete mathematical terms dictionary the complete mathematical terms dictionary understanding math concepts is critical in our world today. Why is betterexplained different i enjoyed math until a poorly-taught class nearly destroyed that passion bring large numbers down to everyday terms fencepost problem counting spans or points number systems when the.

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# How to calculate probability of particle survival for particle filter?

I'm trying to figure out a way that I can calculate the probability that a particle will survive the re-sampling step in the particle filter algorithm.

For the simple case of multinomial re-sampling, I can assume that we can model the process like a Binomial distribution if we only care about one sample/particle.

So if the particle has a weight of w that is also the probability that it will get selected in a step of the re-sampling. So we use 1 - P(k, p, n) where P is the Binomial distribution, k is 0 (we did not select the particle in all our tries), p is equal to w and n is equal to M, the number of particles.

What is the case though in the systematic re-sampling, where the probability of a particle being selected is proportional but not equal to its weight?

• Multinomial and systematic re-sampling
– Bar
Dec 4, 2013 at 9:14
• Deleted: My comment was unnecessary, as you clearly provided the information I was asking for in the question...my mistake. Dec 4, 2013 at 16:02

Let $w_1 \dots w_n$ be the weights of $n$ particles, $p_i \triangleq \frac{w_i}{\sum\limits_{j=1}^{n}w_j}, \sum\limits_{j=1}^{n} p_j = 1$, then as you posted, the probability of the $i$th particle surviving in the resampling procedure for multinomial resampling is: $$P(Survival_i) = 1 - (1 - p_i)^n$$

In systematic resampling, one concatenate $p_1 \dots p_n$ as a ring of interval [0,1). First select $\tilde{u} \sim U[0,1)$, and take $n$ points $u_k = \frac{k-1 + \tilde{u}}{n}, k = 0, \dots n-1$ equally spaced $\frac{1}{n}$ apart from each other on that ring, and take the $n$ samples which have partitions $p_i$ covering the $n$ points.

Therefore, if $p_i \geq \frac{1}{n}$, the probability of survival for the $i$th particle is 1. If $p_i < \frac{1}{n}$, the probability of survival depends on $u_k$. Without loss of generality, let's always assume $i=1, p_1 < \frac{1}{n}$, and consider the probability of the first particle not being selected (you can always re-index the particles to see this is valid for all $i$).

In order for the first particle not to be selected, there need to be two equally spaced points $\{u_j, u_{j+1}, | j = 0 \dots n-1\}$ such that $u_j < 0$ and $u_{j+1} > p_1$. The probability of this event happening for $j=0$ is equal to $\frac{1}{n}-p_1$ (consider the ring interval $[1-\frac{1}{n}+p_1, 1)$). The probability of the first particle not to be selected, is a combination of $n$ exclusive events $j=0, \dots n-1$. Therefore the probability of the first particle not selected is $n\times (\frac{1}{n}-p_1) = 1-n p_1$.

Same derivation for other particles $i=1\dots n$.

In conclusion, $$P(Survival_i) = 1, \text{if } p_i >= \frac{1}{n}\\ P(Survival_i) = np_i, \text{if } p_i < \frac{1}{n}$$

• @nil could you please cite a scientific paper that calculates the survival likelihoods of particles using the formula you described above? Jan 28, 2018 at 20:29

Survival rate for the case of multinomial resampling and the case of $$w \geq \frac{1}{n}$$ has been covered well by the accepted answer.

However, I didn't find the case of $$w < \frac{1}{n}$$ intuitive enough for myself, so I will share my own intuitive understanding about it, even though it might not be as formal. Forgive me for the lack of pictures.

How I visualize systematic resampling - Casino roulette: First, I visualize systematic resampling similarly to a casino roulette over which we spread our cumulative distribution, starting from 0 and draw samples from the start of each roulette field, after spinning the roulette by the randomly sampled value $$r_0 \in [0,1/n]$$. The variable $$n$$ here is the number of particles.

A bit more formally: we split the cumulative distribution into $$n$$ bins of size $$1/n$$ each, with the first bin starting from $$0$$. Then we shift the position of those bins by the sampled random number $$r_0$$, sampling a weight from the beginning of each bin.

Survival (or no survival) of a weight $$w$$: If $$0 \leq w \leq \frac{1}{n}$$, then the probability of $$w$$ surviving depends on the value of the random variable $$r_0$$. In order for $$w$$ to not survive, we need to generate an $$r_0$$ so that $$w$$ falls between two of our equally spaced indices that define our bins after shifting the position of the bins by $$r_0$$. In terms of casino roulettes, we should spin the roulette so that after the spin $$w$$ falls within a single roulette field and not between them.

We can visualize this $$r_0$$ offset sampling procedure as having a single $$1/n$$ sized bin from which we sample the starting position of all bins after the spin (each roulette field) at the same time, with $$w$$ not surviving if we sample an offset that results in no starting position being covered by $$w$$ for any of the $$n$$ bins.

That is equivalent to the probability of $$r_0$$ having a value in $$[0, 1/n]$$ but not falling on the part of the space covered by $$w$$ in any of the $$1/n$$ sized bins ($$w$$ can actually be present in 2 bins at most). More formally:

$$$$prob(\bar{w}) = \frac{1/n - w}{1/n} = 1 - nw$$$$

where $$prob(\bar{w})$$ is the probability of $$w$$ not surviving. Hence, the probability of surviving is equal to $$nw$$.


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## Thursday, March 4, 2021

### Liuquan Wang (Wuhan University) March 4, 2021 - 3:55 PM (IST)

The next talk is by Liuquan Wang of Wuhan University.

Talk announcement

Title: Parity of coefficients of mock theta functions

Speaker: Liuquan Wang (Wuhan University, PRC)

When: March 4, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Zoom:  Please write to sfandnt@gmail.com for a link or watch here:

Abstract: We study the parity of coefficients of classical mock theta functions. Suppose $g$ is a formal power series with integer coefficients, and let $c(g;n)$ be the coefficient of $q^n$ in its series expansion. We say that $g$ is of parity type $(a,1-a)$ if $c(g;n)$ takes even values with probability $a$ for $n\geq 0$. We show that among the 44 classical mock theta functions, 21 of them are of parity type $(1,0)$. We further conjecture that 19 mock theta functions are of parity type $(\frac{1}{2},\frac{1}{2})$ and 4 functions are of parity type $(\frac{3}{4},\frac{1}{4})$. We also give characterizations of $n$ such that $c(g;n)$ is odd for the mock theta functions of parity type $(1,0)$.


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## Educators

Problem 1

If $\frac{d y}{d x}=\frac{7 x^{2}}{y^{3}}$ and $y(3)=2,$ find an equation for $y$ in terms of $x$.

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Problem 2

If $\frac{d y}{d x}=5 x^{2} y$ and $y(0)=6,$ find an equation for $y$ in terms of $x$.

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Problem 3

If $\frac{d y}{d x}=\frac{e^{x}}{y^{2}}$ and $y(0)=1,$ find an equation for $y$ in terms of $x$.

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Problem 4

If $\frac{d y}{d x}=\frac{y^{2}}{x^{3}}$ and $y(1)=2,$ find an equation for $y$ in terms of $x$.

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Problem 5

If $\frac{d y}{d x}=\frac{\sin x}{\cos y}$ and $y(0)=\frac{3 \pi}{2},$ find an equation for $y$ in terms of $x$.

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Problem 6

A colony of bacteria grows exponentially and the colony's population is $4,000$ at time $t=0$ and $6,500$ at time $t=3 .$ How big is the population at time $t=10 ?$

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Problem 7

A rock is thrown upward with an initial velocity, $v(t),$ of 18 $\mathrm{m} / \mathrm{s}$ from a height, $h(t),$ of 45 $\mathrm{m}$ If the acceleration of the rock is a constant $-9 \mathrm{m} / \mathrm{s}^{2},$ find the height of the rock at time $t=4 .$

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Problem 8

A radioactive element decays exponentially in proportion to its mass. One-half of its original amount remains after $5,750$ years. If $10,000$ grams of the element are present initially, how much will be left after $1,000$ years?

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Problem 9

Sketch the slope field for $\frac{d y}{d x}=2 x$.

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Problem 10

Sketch the slope field for $\frac{d y}{d x}=-\frac{x}{y}$.

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text:
## Intermediate Algebra (6th Edition)

$x-2\sqrt{x+1}+2$
Using $(a+b)^2=a^2+2ab+b^2$ or the square of a binomial, the given expression, $\left( \sqrt{x+1}-1 \right)^2 ,$ is equivalent to \begin{array}{l}\require{cancel} (\sqrt{x+1})^2+2(\sqrt{x+1})(-1)+(-1)^2 \\\\= x+1-2\sqrt{x+1}+1 \\\\= x-2\sqrt{x+1}+2 .\end{array}


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# 8.12: Congruent Figures

Difficulty Level: At Grade Created by: CK-12
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Chelsie wants to put a checkerboard pattern on her locker for her birthday. She has to make the pattern on poster board then tape it to her locker so that she does not permanently damage her locker. She decides that the easiest technique is to get construction paper and cut out enough squares to make her pattern. How can she tell if the squares that she cuts out are congruent?

In this concept, you will learn about congruent figures.

### Congruent Figures

Congruent figures have exactly the same size and shape. They have congruent sides and congruent angles

Here are some pairs of congruent figures.

Compare the figures in each pair. They are exactly the same. If you’re not sure, imagine that you could cut out one figure and place it on top of the other. If they match exactly, they are congruent.

Test for congruency by comparing each side and angle of two figures. Each side and angle of one figure corresponds to a side or angle in the other. These are called corresponding parts. For instance, the top point of one triangle corresponds to the top point of the other triangle in a congruent pair.

It is not always easy to see the corresponding parts of two figures. One figure may be rotated differently so that the corresponding parts appear to be in different places. If you’re not sure, trace one figure and place it on top of the other to see if you can make them match.

Is the pair of figures below congruent?

First, look at the shape of each figure.

The shapes look similar.

Next, determine if there are corresponding parts.

The given angles are not corresponding because they have different values.

Then, determine whether the shapes are corresponding.

No.

The answer is that the pair of figures are not congruent.

You can use information about one figure in a pair of congruent figures to find the measure of a corresponding angle or side in the other figure. Let’s see how this works. Take a look at the congruent figures below.

These two parallelograms are congruent. Find the corresponding parts.

First, trace one parallelogram and place it on top of the other. Rotate it until the parts correspond.

You can see that side AB\begin{align*}AB\end{align*} corresponds to side PQ\begin{align*}PQ\end{align*} because they are congruent. Therefore,

ABPQ.\begin{align*}AB \cong PQ.\end{align*}

List all of the congruent sides.

ABBCADDCPQQRPSSR\begin{align*}AB &\cong PQ\\ BC &\cong QR\\ AD &\cong PS\\ DC &\cong SR\end{align*}

List the congruent angles.

ABPQDSCR\begin{align*}\angle A &\cong \angle P && \angle D \cong \angle S\\ \angle B &\cong \angle Q && \angle C \cong \angle R\end{align*}

Now let's find the length of side AB\begin{align*}AB\end{align*}.

AB\begin{align*}AB\end{align*} is congruent to PQ\begin{align*}PQ\end{align*}, so if you find the length of PQ\begin{align*}PQ\end{align*} then it will be the same for AB.PQ\begin{align*}AB. PQ\end{align*} is 7 centimeters. Therefore AB\begin{align*}AB\end{align*} must also be 7 centimeters long.

Now let’s look at the angles. Let's find the measure of C\begin{align*}\angle C\end{align*}. The angle corresponds to R\begin{align*}\angle R\end{align*}, but you do not know the measure of R\begin{align*}\angle R\end{align*} either. However, you do know the measures of two of the angles in the first parallelogram: 70\begin{align*}70^\circ\end{align*} and 110\begin{align*}110^\circ\end{align*}. You do not know the measure of B\begin{align*}\angle B\end{align*}, but this time you do know the measure of its corresponding angle, Q\begin{align*} \angle Q\end{align*}. These two angles are congruent, so you know that B\begin{align*}\angle B\end{align*} must measure 70\begin{align*}70^\circ\end{align*}. Now you know three of the angles in the first figure, so you can subtract to find the measure of C\begin{align*}\angle C\end{align*}.

360(70+110+70)360250110=C=C=C\begin{align*}360 - (70 + 110 + 70) &= \angle C\\ 360 - 250 &= \angle C\\ 110^\circ &= \angle C\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Chelsie and her checkerboard themed locker.

She has cut out the shapes, how can she tell if the squares that she cut out are congruent?

First, measure the lengths of the sides.

All of the sides are equal.

Next, measure the angles.

All of the angles should be equal.

Then, draw a conclusion.

The shapes are congruent.

The answer is that the shapes are congruent.

#### Example 2

Is the pair below congruent?

First, look at the shape of each figure.

The shapes look similar but one shape is smaller than the other.

Next, determine if there are corresponding parts.

There appears to be corresponding angles, but the sides that are in the same position are not congruent.

Then, determine if the shapes are corresponding.

No.

The answer is that the pair of figures are not congruent.

#### Example 3

Is the pair of figures congruent?

First, look at the shape of each figure.

The shapes look similar.

Next, determine if there are corresponding parts.

There appears to be corresponding angles, but not all of the sides that are in the same position are congruent.

Then, determine if the shapes are corresponding.

No.

The answer is that the pair of figures are not congruent.

#### Example 4

What is the measure of M\begin{align*}\angle M\end{align*}?

First, identify the angle that corresponds with angle L\begin{align*}L\end{align*}.

Angle L\begin{align*}L\end{align*} = Angle E\begin{align*}E\end{align*} = 95o\begin{align*}95^o\end{align*}

Next, understand that the sum of three interior angles is 180o\begin{align*}180^o\end{align*} and write an equation.

95+35+x=180\begin{align*}95 + 35 + x = 180\end{align*}

Then, solve for the missing angle measure.

130+x=180\begin{align*}130 + x = 180\end{align*}

x=50\begin{align*}x = 50\end{align*}

The answer is that the measure of angleM\begin{align*}M\end{align*}  is 50\begin{align*}50^\circ\end{align*}.

#### Example 5

Triangle DEF\begin{align*}DEF\end{align*} is congruent with triangle KLM\begin{align*}KLM\end{align*}. If angle D=42o\begin{align*}D = 42^o\end{align*}, what is the measure of angle K\begin{align*}K\end{align*}?

First, determine if angle K\begin{align*}K\end{align*} is congruent to angle D\begin{align*}D\end{align*}.

Yes, it is.

Next, remember the relationship between congruent parts.

Congruent parts are equal.

Then, write the measure of angle K\begin{align*}K\end{align*}.

42o\begin{align*}42^o\end{align*}

The answer is that angle K=42o\begin{align*}K = 42^o\end{align*}.

### Review

Name the corresponding parts to those given below.

1. R\begin{align*}\angle R\end{align*}
2. MN\begin{align*}MN\end{align*}
3. O\begin{align*}\angle O\end{align*}

Use the relationships between congruent figures to find the measure of g\begin{align*}g\end{align*}. Show your work.

Use the relationships between congruent figures to find the measure of T\begin{align*}\angle T\end{align*}. Show your work.

Answer each of the following questions.

1. Triangles ABC\begin{align*}ABC\end{align*} and DEF\begin{align*}DEF\end{align*} are congruent. If the measure of angle A\begin{align*}A\end{align*} is \begin{align*}58^{\circ}\end{align*}, what is the measure of angle \begin{align*}D\end{align*} if it corresponds to angle \begin{align*}A\end{align*}?
2. True or false. Congruent figures are exactly the same in every way.

Identify the given triangles as visually congruent or not.

12.

Answer each of the following questions.

1. Triangles \begin{align*}ABC\end{align*} and \begin{align*}DEF\end{align*} are congruent. Does this mean that their angle measures are the same? Why?
2. Define Congruent.
3. True or false. If two figures are congruent, then they have the same length sides but not the same angle measures.
4. Triangles \begin{align*}ABC\end{align*} and \begin{align*}DEF\end{align*} are congruent. If the measure of angle \begin{align*}A\end{align*} is \begin{align*}58^{\circ}\end{align*}, what is the measure of angle \begin{align*}D\end{align*} if it corresponds to angle \begin{align*}A\end{align*}?

To see the Review answers, open this PDF file and look for section 8.12.

### Notes/Highlights Having trouble? Report an issue.

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### Vocabulary Language: English

Angle

A geometric figure formed by two rays that connect at a single point or vertex.

Congruent

Congruent figures are identical in size, shape and measure.

Trapezoid

A trapezoid is a quadrilateral with exactly one pair of parallel opposite sides.

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.

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# Open packaging question for the readers

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Just a question about software packaging, specifically for mailed products. How important is packaging to you if I was to mail you a CD?

For example, let's take four scenarios for a piece of software that I mail to you.

1. Full color CD in DVD-style clamshell case with full-color insert with screenshots on the back. two-page mini-manual inside, shipped in a fitted cardboard mailer. Total cost:

Media (inkjet printable CD-R) - $0.25 DVD Case -$0.29
Insert and mini-manual printing - $0.25 Cardboard DVD Mailer -$0.43
Postage - $1.50 (first class or media mail) Total cost:$2.72

2. Full color printed CD in plastic jewel case with full-color square insert. Setup instructions printed on the jewel case insert or CD. Mailed in a small cardboard jewel-case box. Total cost:

Media (inkjet printable CD-R) - $0.25 CD Jewel Case -$0.24
Insert printing - $0.10 Cardboard mailer -$0.26
Postage - $1.25 (first class) Total cost:$2.10

3. Full color printed CD packed and mailed in a scratch-resistant cardboard mailer Installation instructions printed on the CD.

Media (inkjet printable CD-R) - $0.25 Mailer -$0.59
Postage - $0.60 (first class) Total cost:$1.44

4. Full color printed CD packed and mailed in folded orgami CD case printed on stiff paper. Installation instructions printed on the paper and/or the CD.

Media (inkjet printable CD-R) - $0.25 Insert printing (stiff paper) -$0.15
Postage - $0.60 (first class) Total cost:$1.00

The question is, how important would it be for you to receive the "deluxe" packaging over the no-frills one? Would you be willing to take no-frills packaging if you could save $1 or$2 on the product?

Or, as an alternate question, what price-point would you expect to pay for the above items? For example, would you feel ripped off if you paid $20 for packaging choice 3 or 4? How about if you paid$5? At what price-point would you expect to get each packaging choice?

I'll start the discussion by posting my own worthless opinion first. Respond with your own worthless opions.

My own worthless opinion.

Packaging choice 4 doesn't bother me as long as it's software that doesn't require a manual and it arrives safely. Choice 3 might prevent having to re-mail discs that are scratched in transit, so it might be worth the extra 44 cents. I don't need cases, as I store all my software in a file-box in slim jewel cases, so if I get software in a larger case (thick jewel case or DVD case), I always end up throwing out the packaga anyway. If I could save a few bucks for cheap packaging (long as it's safe), I'd take it.

I'd go for deluxe packaging over price for prices like this... I don't like being sent "traditional" CD jewel cases as they tend to be easily cracked/broken by careless posties.

I personally prefer the DVD cases over CD jewel cases.

In truth, the deciding factor for me is overall price (in my case, i've often declined buying things because of extortionate international shipping fees) and the speed of delivery. CD-WOW are dirt-cheap, but can take *ages* to deliver a CD - I've waited 2-3 weeks from the time they officially despatched it.

Jack

Personaly I get the cheap feeling from something that comes in minimalist packaging and display. The dollar or so difference in the costs of goods is not worth the possible feeling loss of a customer in my opinion. I would go with option 1 myself for that little extra polished touch. Also I have had very good luck with Jiffy Padded mailers and DVD cases. Using Stamps.com is nice too.

If I paid 5$for it, I would expect to 'get what I paid for' to me$5 isn't all that much. If I paid 30$however (much like peole do when they purchase our physical games) I would expect somthing with quality. Personaly, I don't mind spending more on a quality product, but more importantly a quality product on the outside, helps sway my purchase. I would say the big issue for you comes in terms of labor, sure 3 bucks per cd production isnt all that much (can easily be passed on to the customer) but it can be labor intensive; I am very glad we out-sourced our production, because I wouldn't have time to think let alone make anothe game. It depends on your target market I would say, assuming this is for duck-tiles, you probably don't need to go crazy with the packaging, a good looking full-color CD should be fine. Also, If you have your copy of Morning's Wrath handy, for reference, the CD is done via heat-transfer printing, it's automated machine and does burning and printing, might be somthing to look into if you are going to have high voulume. for ink-jet labels I would go with photo ones, i found memorex photo-labels were the best quality and produced good looking results. #### Share this comment ##### Link to comment I wouldn't expect to see a$5 product in a DVD case. Because then I'd likely get the impression that all the money was put into packaging instead of development and the producer was trying to make you think you got more product than they can actually deliver. However, a well designed piece of software can usually turn around those expectations, but I really can't remember the last time I saw a well designed piece of software for $5. Likewise I wouldn't expect to see a$20+ product in a paper sleeve. Then I'd have the impression that the producer doesn't really care anything about delivery or that they went for the cheapest packaging option in order to keep more of the $20. The packaging should be appropriate to the price point. A cheap package or demo version is OK to ship in a paper envelope. But if I paid a reasonable amount of money, I'd expect a keep case (jewel case minimum) and at least minimal printed instructions. #### Share this comment ##### Link to comment Personally I prefer This type of case. They are extremly sturdy, small, easy to mail and protect the CDs well. For an inexpensive product I would combine option 3 or 4 with this type of case. They start at about$0.22/ea (carton of 200) from the site above and go down as you buy more.

_Evan

If I were to pay $20+ for software, I expect it to ship in the package #1. Anything less and I get the feeling that the software I paid for is also a cheap piece of crap. For things under$20 though package #2 or #4. If I'm paying only 5, 10, or 15 bucks for it, I expect that I'm only paying for the software and no fancyness.

Personally, I wouldn't expect a physical media at all with a cheaply priced game. I could see adding option 3 or 4 to a standard download as a value-add, but not as a selling point in itself. However, if I'm paying near retail prices for your game, I would expect something more like option 1. And if you're charging me \$30 per game, the extra buck or two for the DVD case seems relatively minor.

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Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$

Let $$f(a)=\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx,$$ where $\operatorname{sech}(z)=\frac2{e^z+e^{-z}}$ is the hyperbolic secant.

Here are values of $f(a)$ at some particular points: $$f(0)=\pi,\hspace{.15in}f(1)=2,\hspace{.15in}f(2)=\left(\sqrt2-1\right)\,\pi,\hspace{.15in}f\left(\frac34\right)=\left(4\sqrt{2+\sqrt2}-\frac{20}3\right)\,\pi.$$ Athough I do not yet have a proof ready, it seems that for every $a\in\mathbb{Q},\ f(a)=\alpha+\beta\,\pi$, where $\alpha$ and $\beta$ are algebraic numbers.

I wonder, if it is possible to express $f\left(\sqrt2\right)$ in a closed form?

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A chain of substitutions yields the equivalent form $$f(a/b)=16b\int_0^1\frac{u^{a+b-1}du}{(u^{2a}+1)(u^{2b}+1)}.$$ Although my complex analysis is rusty, when I try partial fractions (via residue theorem) on this I unfortunately get tons of trigonometric expressions that I don't want to handle. At least it appears $f({\Bbb Q})\subseteq {\Bbb Q}^{\rm rab}\oplus\pi{\Bbb Q}^{\rm rab}$ as conjectured anyway. (By ${\Bbb Q}^{\rm rab}$ I mean the maximal real abelian extension of $\Bbb Q$, or equivalently $\Bbb Q$ with all values in $\cos(\pi\Bbb Q)$ adjoined.) – anon Sep 12 '13 at 17:17
@anon: That doesn't look correct to me. – Eric Naslund Mar 29 '15 at 22:19
@EricNaslund Interesting how it took you a year and a half to correct that. – Akiva Weinberger Mar 30 '15 at 1:32
@columbus8myhw: I had never looked at this question before. – Eric Naslund Mar 30 '15 at 1:35
@EricNaslund True enough. – Akiva Weinberger Mar 30 '15 at 1:36

In what follows we obtain the exact formula $$f\left(\frac{c}{d}\right)=\int_{-\infty}^\infty \text{sech}(x)\text{sech}\left(\frac{c}{d} x\right)dx$$ $$= 2d\pi\left(\frac{1}{c}\sum_{k=1}^{c}\frac{\left(1-\frac{2k-1}{2c}\right)}{\sin\left((c+d)\frac{2k-1}{2c}\pi\right)}+\frac{1}{d}\sum_{k=1}^{d}\frac{\left(1-\frac{2k-1}{2d}\right)}{\sin\left((c+d)\frac{2k-1}{2d}\pi\right)}\right),$$ where $c/d$ is a rational number in lowest terms $>0$ not equal to $1$ with at least one of $c,d$ even. The method uses the Chebyshev polynomials. (If $c$,$d$ are both odd, the same method will work, however we have to change the term corresponding to the double pole at $0$.) This proves that for rational $a\neq 1$ the integral is always of the form $\pi\alpha$ where $\alpha$ is algebraic, and it allows us to easily evaluate $f\left(c/d\right)$ for many values of $c,d$, (values for which Wolfram Alpha fails to give anything) such as $$f\left(\frac{4}{5}\right)=\pi\left(1-\sqrt{5\left(26+5\sqrt{2}-8\sqrt{10+5\sqrt{2}}\right)}\right),$$ $$f\left(\frac{4}{7}\right)=\pi\left(2\sec\left(\frac{\pi}{7}\right)+2\csc\left(\frac{\pi}{8}\right)-\frac{7}{2}\sec\left(\frac{\pi}{8}\right)+2\csc\left(\frac{3\pi}{14}\right)-2\csc\left(\frac{\pi}{14}\right)-1\right),$$ $$f\left(\frac{5}{6}\right)=\pi\left(-\frac{6}{5}+\frac{24}{5}\sqrt{5}+2\sqrt{2}+4\sqrt{6}\right),$$ $$f(8)=\pi-\frac{\pi}{2}\sqrt{2\left(6-\sqrt{10+7\sqrt{2}}\right)}.$$ Moreover, the nature of this identity strongly suggests that there is no closed form for irrational $a$. It is interesting to note that the value is strongly tied to the roots of the Chebyshev polynomials, and the degree of the algebraic field in which $\alpha$ lives depends directly on $d$ and $c$.

Proof: Notice that $$\int_{-\infty}^{\infty}\frac{1}{e^{x}+e^{-x}}\frac{1}{e^{ax}+e^{-ax}}dx=\int_{-\infty}^{\infty}\frac{4}{e^{(a+1)x}+e^{-(a+1)x}+e^{(a-1)x}+e^{-(a-1)x}}dx$$ $$=\int_{-\infty}^{\infty}\frac{2}{\cosh((a+1)x)+\cosh((a-1)x)}dx.$$

Thus, if $a=\frac{c}{d}$ is a rational number greater than $0$ and not equal to $1$, then $f\left(\frac{c}{d}\right)$ equals $$\int_{-\infty}^{\infty}\frac{2}{\cosh\left(\frac{c+d}{d}x\right)+\cosh\left(\frac{c-d}{d}x\right)}dx=2d\int_{-\infty}^{\infty}\frac{1}{\cosh\left(\left(c+d\right)x\right)+\cosh\left(\left(d-c\right)x\right)}dx.$$ Let $T_{n}$ denote the $n^{th}$ Chebyshev polynomial of the first kind. Then $$f\left(\frac{c}{d}\right)=2d\int_{-\infty}^{\infty}\frac{1}{T_{c+d}\left(\cosh(x)\right)+T_{d-c}\left(\cosh(x)\right)}dx,$$ and so by the product formula for Chebyshev polynomials of the first kind, this is $$f\left(\frac{c}{d}\right)=d\int_{-\infty}^{\infty}\frac{1}{T_{d}\left(\cosh(x)\right)T_{c}\left(\cosh(x)\right)}dx.$$ Letting $\cosh(x)=u$, we have that $$f\left(\frac{c}{d}\right)=2d\int_{1}^{\infty}\frac{1}{T_{d}\left(x\right)T_{c}\left(x\right)}\frac{1}{\sqrt{x^{2}-1}}dx.$$

Now, since $$T_{n}(x)=2^{n-1}\prod_{k=1}^{n}\left(x-\cos\left(\frac{2k-1}{2n}\pi\right)\right),$$ when $c,d$ are relatively prime the polynomial $T_{c}(x)T_{d}(x)$ will have $d+c$ distinct roots as long as $c,d$ are not both odd, and all of these roots lie in the interval $[-1,1]$. Thus we may write $$\frac{1}{T_{c}(x)T_{d}(x)}=\sum_{i=1}^{d+c}\frac{\beta_{i}}{x-\alpha_{i}}$$ where $\beta_{i}=\text{Res}_{x=\alpha_{i}}\frac{1}{T_{c}(x)T_{d}(x)}$ is an algebraic number. Now, since $$\int_{1}^{\infty}\frac{1}{(x-\alpha_{i})\sqrt{x^{2}-1}}dx=\frac{\pi-\cos^{-1}(\alpha_{i})}{\sqrt{1-\alpha_{i}^{2}}}=\frac{\pi-\pi q_{i}}{\sqrt{1-\alpha_{i}^{2}}}$$ for $q_i$ satisfying $\alpha_i=\cos(\pi q_i)$, we see that $$f\left(\frac{c}{d}\right)=2d\sum_{i=1}^{d+c}\frac{\beta_{i}\pi\left(1-q_{i}\right)}{\sqrt{1-\cos^{2}\left(q_{i}\pi\right)}}=2d\sum_{i=1}^{d+c}\frac{\beta_{i}\pi\left(1-q_{i}\right)}{\sin\left(q_{i}\pi\right)}.$$ To find exact form for $\beta_{i}$ notice that $$\beta_{i}=\frac{1}{T'_{c}(\alpha_{i})T_{d}(\alpha_{i})+T_{c}(\alpha_{i})T_{d}'(\alpha_{i})}=\frac{1}{cU_{c-1}(\alpha_{i})T_{d}(\alpha_{i})+dT_{c}(\alpha_{i})U_{d-1}(\alpha_{i})}$$ where $U_{n}(x)$ is a Chebyshev Polynomial of the second kind. Since $U_{n-1}(\cos(\theta))=\frac{\sin(n\theta)}{\sin(\theta)}$, $T_{n}(\cos(\theta))=\cos(n\theta)$, we see that $$\beta_{i}=\frac{\sin(q_{i}\pi)}{c\sin(cq_{i}\pi)\cos(dq_{i}\pi)+d\cos(cq_{i}\pi)\sin(dq_{i}\pi)}.$$ Notice that when $q_{i}=\frac{2k-1}{2d}$, $\sin(cq_{i}\pi)\cos(dq_{i}\pi)=0$, and so we may change its coefficient from $c$ to $d$ and use the addition identity for $\sin$ to obtain $$\beta_{i}=\frac{\sin(q_{i}\pi)}{d\sin\left((c+d)\frac{2k-1}{2d}\pi\right)}.$$ Similarly when $q_{i}=\frac{2k-1}{2c}$ we see that $$\beta_{i}=\frac{\sin(q_{i}\pi)}{c\sin\left((c+d)\frac{2k-1}{2d}\pi\right)}.$$ Thus we obtain the final form $$f\left(\frac{c}{d}\right)=2d\pi\left(\frac{1}{c}\sum_{k=1}^{c}\frac{\left(1-\frac{2k-1}{2c}\right)}{\sin\left((c+d)\frac{2k-1}{2c}\pi\right)}+\frac{1}{d}\sum_{k=1}^{d}\frac{\left(1-\frac{2k-1}{2d}\right)}{\sin\left((c+d)\frac{2k-1}{2d}\pi\right)}\right),$$ as desired.

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Does $\frac cd$ need to be in lowest terms? Also, how do you explain $I(1)=2$, given by OP? – Akiva Weinberger Mar 30 '15 at 1:36
@columbus8myhw: Yes, the rational number $c/d$ should be in lowest terms. I have added that. It cannot equal $1$, as the proof would breakdown in that case. I added the slightly stronger condition that in lowest terms, at least one of $c,d$ must be even. This is because I did not want to deal with the partial fraction expansion when there is a double root at $x=0$. It should not take much more work extend this result to all rational numbers, but the form may be slightly different when both $c,d$ are odd. – Eric Naslund Mar 30 '15 at 1:47
Lastly, if you are skeptical, I have added several exact computations, and you may verify numerically that these are correct. – Eric Naslund Mar 30 '15 at 1:48

${\large\mbox{We just need to evaluate}\ {\rm f}\left(a\right)\ \mbox{when}\ a \in \left\lbrack 0, 1\right\rbrack}$ since $${\rm f}\left(-a\right) = {\rm f}\left(a\right) \quad\mbox{and}\quad {\rm f}\left(1 \over a\right) = \left\vert a\right\vert\,{\rm f}\left(a\right)$$

\begin{align} {\rm f}\left(1 \over a\right) &= \int_{-\infty}^\infty {\rm sech}\left(x\right)\,{\rm sech}\left(x \over a\right)\,{\rm d}x = a\int_{-\infty}^\infty {\rm sech}\left(a\,{x \over a}\right){\rm sech}\left(x \over a\right) \,{{\rm d}x \over a} \\[3mm]&= \left\vert a\right\vert\int_{-\infty}^\infty {\rm sech}\left(ax\right)\,{\rm sech}\left(x\right)\,{\rm d}x = \left\vert a\right\vert\,{\rm f}\left(a\right) \end{align}

For example \begin{align} {\rm f}\left(1 \over 2\right) &= 2\,{\rm f}\left(2\right) = 2\left(\sqrt{2\,} - 1\right)\pi \\[3mm] {\rm f}\left(4 \over 3\right) &= {3 \over 4}\,{\rm f}\left(3 \over 4\right) = \left(3\sqrt{2 +\sqrt{\vphantom{\large A}2\,}\,} - 5\right)\,\pi \\[3mm] {\rm f}\left(a\right) & = {1 \over \left\vert a\right\vert}\,{\rm f}\left(1 \over a\right) \approx {1 \over \left\vert a\right\vert}\,{\rm f}\left(0\right) = {\pi \over \left\vert a\right\vert}\,, \quad \left\vert a \right\vert \gg 1 \end{align}

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You can have this form of solution

$$\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx =\frac{2}{a}\sum _{k=0}^{\infty } \left( -1 \right)^{k} \left( \psi \left( \,{\frac {3\,a+2\,k+1}{4a}} \right) -\psi \left( {\frac {2\,k+1+a} {4a}} \right) \right),$$

where $\psi(x)$ is the digamma function. Note that, $a=0$ is a special case.

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Note that $af(a)=f(1/a)$. For similar integrals see Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.76,498-No.541(1975), 49-50. It is amusing to note that $$\int_{-\infty}^{\infty}\operatorname{sech}(x) \operatorname{sech}[ax(x+i\pi)]\,\mathrm dx=\pi \operatorname{sech}(\pi^2 a/4)$$ but I doubt $f(a)$ has a closed form expression.

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can you, please, specify where one can find that paper. Because it is a specific one. – Caran-d'Ache Sep 19 '13 at 5:37
@Caran-d'Ache, I believe that the paper is "Evaluation of a class of definite integrals" by M.L. Glasser. It claims to examine a class of integrals which would include $\int_{-\infty}^\infty \operatorname{sech} x \operatorname{sech} a x \, dx$ as a special case, but then goes on to evaluate a different special case (where the integrand equals $e^{-\alpha x} \operatorname{sech} x \operatorname{sech}(x + a) \operatorname{sech}(x + b)$ with $a \not= b \not= 0$ and $\operatorname{Re} \alpha| < 3$) which is not at all applicable to the one we are interested in. – Kyle Dec 9 '13 at 4:41


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# Motion of a Plate

• October 5th 2007, 05:40 PM
edeffect
Motion of a Plate
Hey All

Im trying to solve this problem, but I keep getting the wrong answer? Im using the relative motion formulas and $V=(Ang Vel)*r$ and the $i$ and $j$ components of the problem. Could some explain there process of how they would solve this problem so I can learn how I should do this in future and see where I have gone wrong.

Cheers

Edward Fitzgerald
• October 6th 2007, 03:53 AM
Opalg
I would write $\theta$ for the angle between AB and the horizontal (given as 56 degrees in the diagram), and similarly $\phi,\;\psi$ for the angles that BD and DE make with the horizontal. Also, write x for the length of DE. Then the horizontal and vertical distances from A to E remain constant as the system moves. Therefore

$55\cos\theta+135\cos\phi+x\cos\psi = \text{const.},$
$55\sin\theta-135\sin\phi+x\sin\psi = \text{const.}$

Differentiate these equations with respect to time:

$55\sin\theta.\dot{\theta}+135\sin\phi.\dot{\phi}+x \sin\psi.\dot{\psi} = 0,$
$55\cos\theta.\dot{\theta}-135\cos\phi.\dot{\phi}+x\cos\psi.\dot{\psi} = 0.$

Now plug in the values $\theta=56,\; \dot{\theta}=2.1,\;\phi=35,\;\psi=32.$ You can eliminate $x\dot{\psi}$ between the two equations, and you're left with an equation for $\dot{\phi}$ (which is the angular velocity of the plate, in a clockwise direction).
• October 7th 2007, 06:33 AM
edeffect
Thanks Heaps Opalg! I been using the working you provided to solve the question, but I continue to get the wrong answer? I get -0.0939 rad/s for an answer. I would think the answer would be positive and possibly larger, but Im not 100% sure. Could you please help I really wont to master this process!

Edward
• October 7th 2007, 08:35 AM
Opalg
Quote:

Originally Posted by edeffect
Thanks Heaps Opalg! I been using the working you provided to solve the question, but I continue to get the wrong answer? I get -0.0939 rad/s for an answer. I would think the answer would be positive and possibly larger, but Im not 100% sure. Could you please help I really wont to master this process!

Edward

The numbers I get are

$95.7538 + 77.4328\dot{\phi} + 0.5299x\dot{\psi} = 0,$
$64.5868 - 110.5855\dot{\phi} + 0.8480x\dot{\psi} = 0.$

Multiply the first equation by 0.8480 and the second one by 0.5299, and subtract, to get

$\dot{\phi} = \frac{34.2245 - 81.2038}{65.6667 + 58.5993} \approx -0.378.$

Arithmetic is not my strong point, so I don't guarantee those numbers. At least we both agree that the answer is negative. Remember that $\phi$ is measured in such a way that $\dot{\phi}$ is positive when the plate is rotating in a clockwise direction. It seems to me quite plausible from the picture that $\phi$ should be decreasing as $\theta$ increases.


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Magnetic Susceptibility of Down Quarks

We can find analytic solutions for some parameters in terms of electron, muon and proton characteristics such as the mass $m$ and the magnetic moment $\mu$. This is done by writing out $\mu$ in terms of magnetic susceptibilities $\chi_{\it{m}}$ and then solving the resulting equations to obtain

\begin{align} \chi_{\it{m}}(\sf{d}) = \chi_{\it{m}}(\sf{b}) - \chi_{\it{m}}(\sf{t}) + \left( \frac{\mathrm{2} \pi \, \mu_{\sf{N}} }{\it{ h e}} \right) \, \mu(\sf{p^{+}}) \, \it{m}(\sf{p^{+}}) \end{align}

where $\mu_{\sf{N}}$ is the and $\pi$, $h$ and $e$ are constants.