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# Homework Help: Moment produced by a fluid of varying density

1. Feb 21, 2014

### wahaj

1. The problem statement, all variables and given/known data
Find F, the force acting at the top of the gate.
The gate is 2m high and 2m into the page. It is hinged at the bottom (red dot in the diagram)
Density of fluid varies linearly. 1000 kg/m3 at the top and 1600 kg/m3 at the bottom (depth of 4m)

2. Relevant equations

$$M_s=-\int \int \vec{r} \times \hat{n} \ P dA$$

3. The attempt at a solution
My origin is at the red dot. z is positive upwards and x is positive towards the left into the fluids
$$\vec{r} \times \hat{n} = z \hat{k} \times \hat{i} = z \hat{j}$$
I'm going to ignore the vector part and also the negative sign of this since I only need the magnitude.
$$P= -\rho g z + C \ (negative\ because\ g\ is\ negative) \\ P = 0\ when\ z = 4 \\ 0= -4 \rho g + C \\ C = 4 \rho g \\ P = \rho g (4 - z) \\ \rho = 1000 \ when\ z = 4 \ \ \ \rho = 1600 \ when\ z = 0 \\ \rho = mz +b \\ 1000 = 4m +b \ \ \ \ \ \ \ \ \ \ \ 1600 = 0m+b \\ \rho = 1600 - 150z \\ dA = dz dy , \ limits: \ [0,2],[0,2] \\ M = g \int_0^2 \int_0^2 z(1600 - 150z)(4 - z) \ dzdy \\ M = 147.804 kN \\ 147.804kN = 2F \\ F = 73.902 kN$$
The actual answer is 63.1 kN. What am I doing wrong?

#### Attached Files:

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Last edited: Feb 21, 2014
2. Feb 21, 2014

### SteamKing

Staff Emeritus
Is there supposed to be a diagram attached?

3. Feb 21, 2014

### wahaj

I did attach the diagram. I don't know where it went. Let me try again
Edit: There we go

4. Feb 21, 2014

### SteamKing

Staff Emeritus
Your calculations check. I can't see how F = 63.1 kN.

5. Feb 21, 2014

### wahaj

That's the answer my professor gave. If I am right that means the professor must have given the answer for a different question by mistake. I will ask him about this when classes start again. Thanks for checking my work, I've been losing my mind trying to find the mistake in my work.

format: plain


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## Michael H. Albert ; Marie-Louise Lackner ; Martin Lackner ; Vincent Vatter - The Complexity of Pattern Matching for $321$-Avoiding and Skew-Merged Permutations

dmtcs:1308 - Discrete Mathematics & Theoretical Computer Science, December 21, 2016, Vol. 18 no. 2, Permutation Patterns 2015 - https://doi.org/10.46298/dmtcs.1308
The Complexity of Pattern Matching for $321$-Avoiding and Skew-Merged Permutations

Authors: Michael H. Albert ; Marie-Louise Lackner ; Martin Lackner ; Vincent Vatter

The Permutation Pattern Matching problem, asking whether a pattern permutation $\pi$ is contained in a permutation $\tau$, is known to be NP-complete. In this paper we present two polynomial time algorithms for special cases. The first algorithm is applicable if both $\pi$ and $\tau$ are $321$-avoiding; the second is applicable if $\pi$ and $\tau$ are skew-merged. Both algorithms have a runtime of $O(kn)$, where $k$ is the length of $\pi$ and $n$ the length of $\tau$.

Volume: Vol. 18 no. 2, Permutation Patterns 2015
Section: Permutation Patterns
Published on: December 21, 2016
Accepted on: December 21, 2016
Submitted on: December 21, 2016
Keywords: Mathematics - Combinatorics,Computer Science - Data Structures and Algorithms,05A05, 68Q25


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# An elevator moving up and a bolt falling down

1. Aug 28, 2007

### chaotixmonjuish

A bolt comes loose from the bottom of an elevator that is moving upward at a speed of 6.38 m/s. The bolt reaches the bottom of the elevator shaft in 3.85 s. How high up was the elevator when the bolt came loose?

I integrated 6.38 to get the position equation and plugged in 3.85. However, I'm not really sue that's right at all. Could someone show me what type of equation I should be using?

2. Aug 28, 2007

### chaotixmonjuish

Should I get the height of the elevator at 3.85 seconds. Then use 9.8m/s and create a position equation for the falling bolt.

3. Aug 28, 2007

$$s = ut + \frac{1}{2}at^2$$


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# Relation between Weil Conjecture and Langlands Program

Recently I read Gelbart's An Elementary Introduction To The Langlands Program, which explained the origin of the program, and this question came to me. For an elliptic curve over finite field, the associated L-function will coincide with the L-function associated to a certain automorphic cusp form (in Silverman's "Advanced topics in the arithmetic of elliptic curves" there is an alternative statement using Grössencharakter).

Since both Weil conjecture and Langlands program consist rationality and functional equation and by the preceding result (BTW, some approaches are similar), are there some inner relations between the two conjectures?

• There is a misunderstanding here. The $L$-function of an elliptic curve over a finite field is very simple analytically, namely of the form $(1-\alpha p^{-s})(1-\beta p^{-s})$. (Depending on the definition you use, this is either trivial or a simple fact.) Hasse-Weil $L$-functions are similarly simple in this regard, namely they are rational functions of $p^{-s}$. (This fact is due to Dwork and lies much deeper of course.) The $L$-function of a cusp form, on the other hand, is an infinite product over all the primes. (Continued in next comment.) – GH from MO Nov 5 '15 at 19:25
• In a sense a Hasse-Weil $L$-function of a variety over the $p$-element field is much like the reciprocal of a single Euler factor (namely the one at $p$) of an automorphic $L$-function. In short, your second sentence is in error: we do not associate cusp forms to elliptic curves over $\mathbb{F}_p$. Instead, we associate cusp forms to elliptic curves over $\mathbb{Q}$. – GH from MO Nov 5 '15 at 19:25

The Weil conjectures asks for rationality, functional equation, Riemann hypothesis, and degree relating to the Hasse-Weil zeta function of a nonsingular projective variety $X$ over a finite field $k$, which is a product of L-functions. Since you are reading Gelbart, as Artin you can think of these as L-functions associated to the Galois representation of Gal$(\bar k/k)$ acting on each degree of the (etale) cohomology of $X$ respectively.
The Langlands program does not ask for rationality, but to each automorphic representation it does ask for a functional equation and meromorphic continuation for the associated automorphic $L$-function. (This also depends on the representation of the $L$-group.) It is generally believed that these $L$-functions also satisfy the Riemann hypothesis, but this I do not think is explicitly a part of the program.
What you do get is that the functoriality conjecture for $H=\{1\}$ and $G=GL_n$ (p.208 of Gelbart) is what some call Langlands reciprocity: to every Galois representation is associated an automorphic representation, so if you like, you may think of the Hasse-Weil zeta function as (conjecturally) a product of automorphic $L$-functions instead, though I am not sure what one gains from this.
• "I am not sure what one gains from this": The $L$-function of an irreducible cuspidal representation of $GL_n$ is automatically analytic on the entire complex plane and satisfies the usual functional equation, hence what you call Langlands reciprocity would imply that Artin $L$-functions also have these properties. – GH from MO Nov 5 '15 at 19:13
• Do you mean automorphic $L$-functions over a function field? Because over a number field a product of automorphic $L$-functions is never a rational function. – GH from MO Nov 6 '15 at 1:44


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Napier's e

Original message:

rational approximation of an irrational number
by vivek gandhi (posted 1/9/10)

students find it really curious, and more intriguing when i tell them that (1+(1/n))^n in the limit of n tending to infinity is an irrational number 'e'. Here n is a positive integer. They find it really difficult to believe that a rational expression on the left hand side tends to an irrational number. How do I convince them ?


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Which one has the largest area:
a circle with a diameter of 10 inches
a rectangle that measures 12 inches by 11 inches
a circle with a radius of 6 inches
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2. rectangle that measures 12 inches by 11 inches
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Measurement of space inside a polygon.
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Which statement shows how to compute the area of a circle with a radius of 4 feet?
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4. $pi$ times 16 squared
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Find the area of the triangle.

Label the triangle
AB = 15 meters
height = 8 meters
AC = 13 meters

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Grade 7 Area
1. Label this rectangle (length 9 cm width 5 cm)
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The length of a line drawn A to C is 8. What is the area of the circle?
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4. 210 Square centimeters
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The area of the circle is $64pi$. Which expression should be used to find the area of the shaded regions?
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Article Contents

# A framework for the development of implicit solvers for incompressible flow problems

• This survey paper reviews some recent developments in the design of robust solution methods for the Navier--Stokes equations modelling incompressible fluid flow. There are two building blocks in our solution strategy. First, an implicit time integrator that uses a stabilized trapezoid rule with an explicit Adams--Bashforth method for error control, and second, a robust Krylov subspace solver for the spatially discretized system. Numerical experiments are presented that illustrate the effectiveness of our generic approach. It is further shown that the basic solution strategy can be readily extended to more complicated models, including unsteady flow problems with coupled physics and steady flow problems that are nondeterministic in the sense that they have uncertain input data.
Mathematics Subject Classification: Primary: 65M22, 76D05, 76D05; Secondary: 76M10.

 Citation:

•  [1] Uri M. Ascher, Steven J. Ruuth and Brian T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797-823.doi: 10.1137/0732037. [2] Alexei Bespalov, Catherine E. Powell and David Silvester, A priori error analysis of stochastic Galerkin mixed approximations of elliptic PDEs with random data., MIMS Eprint 2011.91, http://eprints.ma.man.ac.uk/1696/. [3] Jonathan Boyle, Milan Mihajlović and Jennifer Scott, HSL_MI20: an efficient {AMG} preconditioner for finite element problems in 3D, Internat. J. Numer. Methods Engrg, 82 (2010), 64-98. [4] H. Damanik, J. Hron, A. Ouazzi and S. Turek, A monolithic FEM-multigrid solver for non-isothermal incompressible flow on general meshes, J. Comput. Phys., 228 (2009), 3869-3881.doi: 10.1016/j.jcp.2009.02.024. [5] Philip G. Drazin, "Introduction to Hydrodynamic Stability,'' Cambridge University Press, Cambridge, 2002. [6] Howard Elman, Milan Mihajlović and David Silvester, Fast iterative solvers for buoyancy driven flow problems, J. Comput. Phys., 230 (2011), 3900-3914.doi: 10.1016/j.jcp.2011.02.014. [7] Howard C. Elman, Alison Ramage and David J. Silvester, Algorithm 866: IFISS, a MATLAB toolbox for modelling incompressible flow, ACM Trans. Math. Softw., 33 (2007), 2-14. [8] Howard C. Elman, David J. Silvester and Andrew J. Wathen, Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations, Numer. Math., 90 (2002), 665-688.doi: 10.1007/s002110100300. [9] Howard C. Elman, David J. Silvester and Andrew J. Wathen, "Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics,'' Oxford University Press, New York, 2005. [10] Howard C. Elman and Ray S. Tuminaro, Boundary conditions in approximate commutator preconditioners for the Navier-Stokes equations, Electron. Trans. Numer. Anal., 35 (2009), 257-280. [11] Oliver G. Ernst, Catherine E. Powell, David J. Silvester and Elisabeth Ullmann, Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data, SIAM J. Sci. Comput., 31 (2009), 1424-1447.doi: 10.1137/070705817. [12] P. M. Gresho, D. K. Gartling, J. R. Torczynski, K. A. Cliffe, K. H. Winters, T. J. Garratt, A. Spence and J. W. Goodrich, Is the steady viscous incompressible two-dimensional flow over a backward-facing step at Re=800 stable?, Internat. J. Numer. Methods Fluids, 17 (1993), 501-541.doi: 10.1002/fld.1650170605. [13] Philip M. Gresho, David F. Griffiths and David J. Silvester, Adaptive time-stepping for incompressible flow; Part I: Scalar advection-diffusion, SIAM J. Sci. Comput., 30 (2008), 2018-2054.doi: 10.1137/070688018. [14] P. M. Gresho and R. L. Sani, "Incompressible Flow and the Finite Element Method: Volume 2: Isothermal Laminar Flow,'' John Wiley, Chichester, 1998. [15] David A. Kay, Philip M. Gresho, David F. Griffiths and David J. Silvester, Adaptive time-stepping for incompressible flow; Part II: Navier-Stokes equations, SIAM J. Sci. Comput., 32 (2010), 111-128.doi: 10.1137/080728032. [16] David Kay, Daniel Loghin and Andrew Wathen, A preconditioner for the steady-state Navier-Stokes equations, SIAM J. Sci. Comput., 24 (2002), 237-256.doi: 10.1137/S106482759935808X. [17] William Layton, "Introduction to the Numerical Analysis of Incompressible Viscous Flow,'' SIAM, Philadelphia, 2008. xx+213 pp. ISBN: 978-0-898716-57-3. [18] O. P. Le Maître and O. M. Knio, "Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics,'' Springer, New York, 2010. xvi+536 pp. ISBN: 978-90-481-3519-6. [19] Catherine E. Powell and Howard C. Elman, Block-diagonal preconditioning for spectral stochastic finite-element systems, IMA J. Numer. Anal., 29 (2009), 350-375.doi: 10.1093/imanum/drn014. [20] Catherine E. Powell and David Silvester, Preconditioning steady-state Navier-Stokes equations with random data, MIMS Eprint 2012.35, http://eprints.ma.man.ac.uk/1792/. [21] David Silvester, Howard Elman, David Kay and Andrew Wathen, Efficient preconditioning of the linearised Navier-Stokes equations for incompressible flow, J. Comput. Appl. Math., 128 (2001), 261-279.doi: 10.1016/S0377-0427(00)00515-X. [22] David Silvester, Howard Elman and Alison Ramage, "Incompressible Flow and Iterative Solver Software (IFISS),'' Version 3.2, 2012. Available from http://www.manchester.ac.uk/ifiss/. [23] J. C. Simo and F. Armero, Unconditional stability and long-term behaviour of transient algorithms for the incompressible Navier-Stokes and Euler equations, Comput. Methods Appl. Mech. Engrg., 111 (1994), 111-154.doi: 10.1016/0045-7825(94)90042-6. [24] Dongbin Xiu, "Numerical Methods for Stochastic Computations: A Spectral Method Approach,'' Princeton University Press, Princeton, NJ, 2010. xiv+127 pp. ISBN: 978-0-691-14212-8.
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Calculating expected residual service times

This is a queueing theory-related question. Suppose we have two types of arrivals, call them A and B, who arrive according to a Poison proces with rates $\lambda_A = 1/20$ per second and $\lambda_B = 1/30$ per second, respectively. These arrivals arrive at a single server. Type A arrivals have a service time of exactly $B_A= 8$ seconds, type B arrivals have a service time of exactly $B_B = 12$ seconds. I now want to calculate the expected residual time of an arbitrary job. The service time and residual time are related through $$E(R) = \frac{E(B^2)}{2E(B)}$$

Since the expected service time of an arbitrary job can be calculated through (using $P(\text{type A job}) = \frac{\lambda_A}{\lambda_A + \lambda_B} = \frac{3}{5}$ and $P(\text{type B job}) = \frac{\lambda_B}{\lambda_A + \lambda_B} =\frac{2}{5}$)$$E(B) = E(B|\text{type A job})P(\text{type A job}) + E(B|\text{type B job})P(\text{type B job}) = 8\cdot\frac{3}{5} + 12\frac{2}{5} = \frac{48}{5}$$ I thought I'd do the same for the expected residal time, through $$E(R) = E(R|\text{type A job})P(\text{type A job}) + E(R|\text{type B job})P(\text{type B job}) = \frac{8^2}{2\cdot 8}\frac{3}{5} + \frac{12^2}{2\cdot 12}\frac{2}{5} = \frac{24}{5}$$ However, a solution sheet gives as correct answer $$E(R) = \frac{1}{2}(\frac{8^2}{2\cdot 8} +\frac{12^2}{2\cdot 12})$$

I don't quite see why this is the case. I'm sure I'm missing something obvious but I would welcome any advice.

NB This is not homework. Also, the reader I use can be found at http://www.win.tue.nl/~iadan/queueing.pdf

-

Applying the definitions you recall, one gets $$\mathrm E(B)=8\cdot\frac35+12\cdot\frac25,\qquad\mathrm E(B^2)=8^2\cdot\frac35+12^2\cdot\frac25,$$ hence $$\mathrm E(R)=\frac12\cdot\frac{\mathrm E(B^2)}{\mathrm E(B)}=\frac12\cdot\frac{8^2\cdot3+12^2\cdot2}{8\cdot3+12\cdot2}=5.$$


text:
Simplify a logarithm function

$$2\log\sqrt[4]{10}-\ln e^{-7}+\log_9\sqrt 3$$

I want to simplify this function. I believe that $\,2\log\sqrt[4]{10}\,$ can become $\,\log\sqrt{10}\,$ but now I'm stuck.

Is it possible that $\ln e^{-7}\,$ can be just $\,-7\,$?

• Not only is it possible: it actually is that. – DonAntonio Mar 29 '13 at 14:17

$$\log\sqrt[4]{10}= \frac14\log10$$ $$\ln e^{-7}=-7\ln e = -7\cdot1$$ $$\log_9\sqrt{3} = \frac12\log_9 3=\frac12\log_9\sqrt{9}=\frac12\cdot\frac12=\frac14$$

Probably "$\log10$" is intended to mean $\log_{10}10$, so that is equal to $1$.

I disapprove of using "$\log$" with no base to mean $\log_{10}$ when there's no special context saying that's the right base to use.

• The main reason to disapprove is that nowadays one usually means $\log = \ln$, whereas until some years ago, when slide rules and tables of logarithms were still widespread, usually one meant $\log=\log_{10}$. By the way, as I still use these tools myself (for fun, I confess, a kind of masochistic fun), I prefer the $\log=\log_{10}$ convention :-) – Jean-Claude Arbaut Mar 29 '13 at 14:29
• In some fields, maybe biology, chemistry, astronomy, etc., base-10 logarithms are still used, and they write "log" with no base for those. In some other fields (mathematics, statistics, computer science) "log" with no base means base-$e$. It's context-dependent, and at least that fact should be mentioned. – Michael Hardy Mar 30 '13 at 1:38

You are on the right path. Also, note that if $\log_9 (\sqrt{3}) = x$, then $9^x = \sqrt{3}$.

And if $\log = \log_{10}$, then you may be able to simplify $\log_{10} \sqrt{10}$ as well. In that case, you'd be looking for a number $y$ satisfying $10^y = \sqrt{10}$...

$$2\log\sqrt[4]{10}-\ln e^{-7}+\log_9\sqrt 3=\frac{2}{4}\log{10}-(-7)+\frac{1}{2}\log_93=\frac{1}{2}+7+\frac{1}{4}\ldots$$

Properties used:

$$\log_aa^n=n\;\;,\;\;\log_ax^n=n\log_ax$$

and, of course, the very definition of logarithm and the assumed fact that you surely meant $\,\log=\log_{10}\,$

Formulas regarding log:

Basically $log_aM=N$ means that $a^N=M$

Now the problem is easy,Ihope you'll be able to solve it yourself.


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## Observations on the PARCC sample Geometry exam

Part 1: Observations on the PARCC sample Algebra I exam
Part 2: Observations on the PARCC sample Algebra II exam
Part 3: Observations on the PARCC sample Geometry exam

Calculator part: 18 of 25

Use the information provided in the animation to answer the questions about the geometric construction.

To pause the animation, select the animation window.

The students are supposed to watch a video of a construction and then say things about the proof enacted through the constructions. This is a very specific skill that needs to be practiced. Daniel Schneider kindly sent me a link to a website with a large number of construction animations (along with proofs) in case you need more to use in class.

However, there’s a serious interface problem. Here’s what the video looks like when paused, as well as a question to go with it:

Point “C” is completely covered. Whoops.

Non-calculator part: 6 of 7

This is one of those simple-looking questions which has enough of a trick to it I’m not sure how many students will get it right.

Part A requires students to work a double-completing-the-square manipulation, hopefully not getting sidetracked by the presence of b on the right hand side:

$x^2 + y^2 - 4x + 2y = b$
$x^2 - 4x + y^2 + 2y = b$
$x^2 - 4x + 4 + y^2 + 2y + 1 = b + 4 + 1$
$(x-2)^2 + (y+1)^2 = b + 5$

Part B requires noticing that a radius of 7 means the right hand side will be 49, so $b + 5 = 49$ and thus $b = 44$.

In principle this problem is solvable, but the lack of partial credit on a problem with a “trick” that I worry a student who can normally complete the square would still get no points due to the indirection.

Calculator part: 13 of 25

This problem’s rough for three reasons:

a.) Even with the phrase “the pipe is open at both ends” placed in there, this is something of a background knowledge problem; the students need to know the “outer surface” excludes the circles on the top and bottom.

b.) There are volume formulas on the formula sheet but not surface area formulas. Thus the students need to have memorized $\pi d h$ or be able to extrapolate it, and know enough to exclude the circles.

c.) If $\pi$ is set to be 3.14, there answer comes out to be 1356.48. If $\pi$ is set to be 3.14159 or something with more digits (not unusual since graphing calculators have a “pi” button) the answer comes out to be roughly 1357.168. Rounding to the nearest integer thus can give either 1356 or 1357 as an answer.

Non-calculator part: 7 of 7

Out of all the problems on the PARCC final exam for geometry, 28% are related to transformations.

I can understand a transformational emphasis in general: it leads to a function transformation understanding of graphs (which is far more powerful and useful than looking at each kind of graph individually). However, why do so many of the dilation and rotation problem — 4 out of the 9 — involve centers not at the origin? This is not rhetorical; I really want to know where the utility is.

Non-calculator part: 20 of 25

This is one of the easier problems on the test, but assumes background the students don’t necessarily have. I can assume what a “collar” means here (even though I’ve never heard the word used in this context) but my ELL students are more likely interpret it as gibberish.

Calculator part: 10 of 25

This is very similar to the other problem in relying somewhat on background knowledge. Technically speaking one can ignore all the external stuff about merchant vessels and probes and focus on the math, but the brain of the ELL student doesn’t have an easy time removing the context.

Also note the weirdness of the rounding; in problem 20 the rounding needed to be done to the nearest tenth, while in this problem the rounding needs to be done to the nearest integer off the list.

Calculator part: 3 of 25

I’m noting this one because nothing in my current textbook (Carnegie Learning, written for Common Core) has anything resembling this kind of problem. Anyone have a source with problems that are similar?

Calculator part: 23 of 25

I don’t think I’ve ever give this much emphasis to the vocabulary of proof. Getting my students to keep the reflexive, symmetric, and transitive properties of congruence straight is going to be a nightmare and a half.

Ok, one last problem, from HS Sample Math Items, 7 of 10 (so not the final exam, but the open response part):

Here the angle bisector video returns (complete with unhelpful play button covering the diagram when paused) but the student is supposed to free-write a proof.

Here is how you type the first line as given:

1.) Pick “geometry” on the side and pick the short line in the upper right; that’s a “line segment” and will give you a blank box under a line segment so you can type letters.

2.) Type the letters you want under the line segment. If you accidentally type more than two letters any extra keypresses will be ignored.

3.) Go to “relations” above “geometry” and find the congruence symbol. Pick that. This will give the congruence symbol and a blank box.

4.) Pick the “line segment” and it will take the blank box that just appeared and put a line segment over it.

5.) Type the letters you need for the other line segment.

Now you have one step of the proof, now you just need to give a reason and then do four more steps.

(What would constitute a valid reason here, by the way? The mathopenref site I linked to early in this post just states “They were both drawn with the same compass width” — would this be considered valid by graders?)

## Robot maze puzzle

The puzzle above has a robot (marked with an arrow pointing up, or “north”) that you can control with a set of command cards that either move the robot forwards a set number of spaces (the + numbers) or backwards (the – numbers). After using a command card, the card is used up and can’t be used again.

If the robot hits either the border of the grid or one of the black spaces, the robot stops moving and any remaining steps on the command card being used are ignored.

Landing on one of the spaces marked with circles causes your robot to turn 90 degrees. (That is, if the robot faces north it turns east, if east it turns south, if south it turns west, and if west it turns north.) The robot starts facing north. Can you get the robot to the star?

## Observations on the PARCC sample Algebra I exam

Similar to my breakdown of the PARCC Algebra II, here’s some question-by-question comment on the sample PARCC Algebra I Final Exam. I’m not too discouraged by the actual items on this one, but the interface needs work.

This is the second problem of the test. You click points to set a line, then click “solution set” to shade in one side. Note (just like the Algebra II test) the points snap to half-grid points, not to grid points, a circumstance I find hazardous. Also, if you click on solution set to color a side, then realize your line was wrong, you have to click back on “line” again but the color goes away, so you have to add the color back again after the line is fixed. This is true even if switching the line from solid to dotted.

I should add this is one case where I see the superiority of open response to multiple choice. Here’s an inequality graphing problem from our old state test:

The lack of choices makes the problem a dotted-or-solid / above-or-below question where the actual shape of the graph is given away.

The question here is fine, but what if a student drags in the wrong number and wants to fix their answer? Removing a number only works if you drag the number back to the original number boxes, just “tossing” the number to a random position outside the answer box doesn’t work.

One common technique in the PARCC interface is for students to fill in sentences with a drag-down menu. By my eye, though, the interface doesn’t look much like a sentence, and I could imagine a student not understanding they are placing words between f(2) and g(2) and so forth to produce something that is meant to be read from left to right.

I guess 4 answer boxes — clear overkill — is better than the situation with 2 answer boxes where the suggestion seems strong to fill both of them even if one of the answers turns out to be extraneous.

I’m pretty sure logarithms aren’t supposed to be on the Algebra I test? Also, the graph is drawn automatically through the points, unless it can’t like in the example above. It took me a bit of deciphering to realize there’s an asymptote on there (right on the y-axis) and the asymptote can be slid around, so the reason the graph wasn’t showing up is the points were on opposite sides of the asymptote.

Do the blanks really have to be so large? I admit to getting confused because the symbols spread out in a single function looked to me like function-break-really small expression-break-random parenthesis and I had to do a double-take before I realized what was going on.

I hope students have their window large enough to realize (or least deduce from there being a “Part A”) that there is a “Part B” to the question.

There’s a truly weird option to change colors of things. Sometimes I can get it to trigger but I’m not sure how. The upper right inequality in pink shows what things look like after you’ve messed with the color.

There’s even an interface for systems of linear inequalities. Notice how there’s still a snap-to-half-grid feature even when the y-axis goes up by 5.

Why does one “find the zeros” question have a drag-and-drop interface, while this one gives a list?

## Mathematics classes do not teach mathematics literacy

Both the phrases “mathematics classes” and “mathematics literacy” can be ambiguous, so I am meaning–

“Mathematics classes”: The high school curriculum delivered to a typical student in the United States. This usually excludes classes like “Statistics” or “Financial Math”.

“Mathematics literacy”: The kind of “good citizen” math that people refer to in articles like Headlines from a Mathematically Literate World. The word can also mean “ability to problem solve”, but that’s not what I mean here.

Taking the Headlines article and the classes of a typical high school mathematics student, how many of the headlines would a a student understand?

At the very least, understanding the entire list requires knowing about: correlation vs. causation, inflation, experimental replication, estimation of large numbers, incompatibility of comparisons with different conditions, understanding how tax brackets work, meaninglessness of predictions within a margin of error, statistically unlikely events, and reversion to the mean.

None of these will ever occur in an traditional math class. In other words, in the list of supposed math literacies, the typical math student in the US receives zero of them. (Some might possibly show up in a class labelled “Economics” or “Free Enterprise”, but those don’t get called Math Classes).

It’d be fair to argue I’m being highly specific in my starting definitions, but I often see the “good citizen” argument used during a general “why are we teaching math” type discussion which assumes a traditional math class track. That sort of argument only works if people are prepared to also overhaul the curriculum (by putting, for example, statistics before calculus as Arthur Benjamin discusses at TED).

## Why is a negative divided by a negative a positive?

So there’s a whole lot of posts, including one from this very blog, which give intuitive explanations of why a negative times a negative is a positive.

I haven’t seen nearly as much material for a negative divided by a negative. One can certainly appeal to the inverse — since $1 \times -1 = -1$, $\frac{-1}{-1} = 1$. Google searching leads to answers like that, but I’ve found nothing like the multiplication picture above.

Can anyone explain directly, at an intuitive level, why a negative divided by a negative is a positive? Or is the only way to do it to refer to multiplication?

## The evolution of mathematical exposition

More rambly and unsubstantiated than usual, apologies –

[Source.]

Theory: Mathematical exposition has evolved just like fiction writing has. However, tradition has held stronger in mathematics (likely due to a need for precision) and it means that clarity in writing is if not actively discouraged at least passively devalued.

Theory: We are not anywhere near the threshold of simplest and clearest explanations in the exposition of mathematical subjects.

Still, what used to be difficult is now considered easier. Various subjects have shifted their supposed level. For instance, not long ago College Algebra was the prestige class at the top of the high school level.

Furthermore, our raw definitions of what each class is has shuffled the actual content of subjects; Algebra I from the 1940s is not the Algebra I of today.

Theory: It would be possible to take a “hard” subject like group theory or transcendental number theory and make it comprehensible at a lower level. However, as there is no requirement to do so there is little motivation to make the subject easier. When a curriculum shift happens to move topics to a lower level, mathematical exposition evolves to catch up.

## Book update (and call for testers)

Don’t get too excited yet — the big one (Why Algebra Works, you’re best off reading this post for an idea of how it is being written) isn’t done.

However, in the midst of work I realized I was assuming the readers would remember how integer operations went, and it was quite possible they forgot, so I worked on a appendix. The appendix ballooned into a full fledged … short story? novella? … and got to the point that I even separated addition/subtraction from multiplication/division.

So the first part (addition/subtraction of integers) is close to ready, and it does follow my percepts, namely–

1.) that it should have a smooth writing style modeled after popular math articles (like Steven Strogatz or Martin Gardner) rather than textbooks

2.) that there are no “problems” but rather “puzzles”, roughly defined as anything that wouldn’t be out of place alongside a Sudoku book or in the middle of a Professor Layton game

3.) that there is a strong emphasis on meta-thinking; that is, having readers examine whatever mental model they are using in a particular part of mathematics and diagnosing where misconceptions may come about at the internal level.

As an example of #3, I start by asking the reader to add 2 + 2 (really), examining the possible ways of visualizing it and which ways might be more or less helpful.

In any case, everything is so unlike the textbook approach that I need some beta testers. In particular, while I would like some people who are adept in mathematics, I would also like some people who think that are not good at math or even actively dislike it. I’m guessing the latter don’t read this blog I’m going to need some help — if you know someone who might be a good candidate, could you send the word along? I’ll get back to everyone in a few weeks.

You can either post here or toss a line to my email over at my About Page.

## Impossible Learning: watch someone trying to learn calculus from scratch

David Wees retweeted this from Jared Cosulich:

It turns out Impossible Learning is a just-started-last-month blog where Jared is trying to learn Calculus and post about his struggles. It’s terrific and you should read it. See him ask the perennial question When Will I Use This?

And I immediately found myself saying “come on, when am I ever going to have to find the limit of this random equation”.

I felt like I was back in High School again.

But seriously, why is this one of the first things I’m directed to learn when I want to know more about Calculus? Why is it so hard for me to find some practical applications of this material? I know there is value in understanding the abstract math, but I’d like to balance that with at least some understanding of how this works practically…

I don’t think this actually counts as a proof, but it definitely made the “Power Rule” click for me a bit more. Basically it’s saying that the derivative of a square (x²) is two lines (2x) and the derivative of a cube (x³) is three squares (3x²).

So for a square to get a tiny bit bigger you need to add on two lines (one to the top and one to a side). Similarly for a cube to get a tiny bit bigger you need to add a square to three sides (e.g. top, right, and front).

## Counting puzzle from the first US puzzle championship

My students had fun with this one today. Part d is what showed up in the actual championship.

One night, in an attempt to combat insomnia, you begin counting (1, 2, 3, …), but you decide to do it digit by digit. As you go along, for example, the 15th digit you count is the 2 of the number 12.
1 2 3 4 5 6 7 8 9 10 11 12

a.) What’s the 50th digit you count?
b.) What’s the 100th digit you count?
c.) What’s the 1000th digit you count?
d.) What’s the 1,000,000th digit you count?

## Commentary on “Wrong Answer: The Case Against Algebra II”

There’s a new article by the author Nicholson Baker that is not raising as much a fuss as “Is Algebra Necessary?” from The New York Times last year, probably because it’s at Harper’s behind a paywall. Also, as I write this children are fleeing from algebra all over magazine stands:

The title strikes me as odd, given that the general argument is aimed at all algebra, and the text constantly references algebra as a whole rather than the quite US-specific version of Algebra II. (As far as I’m aware no other country has such a thing; most of them integrate all forms of mathematics together.) What is Algebra II anyway, and where is the cutoff point where Nicholson Baker considers algebra to be too hard to handle? He hints at what he’d like as a replacement, which strikes me as a math appreciation analogous to music appreciation courses:

We should, I think, create a new, one-year teaser course for ninth graders, which would briefly cover a few techniques of algebraic manipulation, some mindstretching geometric proofs, some nifty things about parabolas and conic sections, and even perhaps a soft-core hint of the innitesimal, change-explaining powers of calculus. Throw in some scatter plots and data analysis, a touch of mathematical logic, and several representative topics in math history and math appreciation.

This seems like a truly random collection of topics, but it sounds like Mr. Baker is basing his list off the notion of using The Joy of x by Steven Strogatz as a possible text.

I have had some personal experience with using popular texts; when I team-taught statistics at the University of Arizona we used The Drunkard’s Walk, Fooled by Randomness, and Struck by Lightning: The Curious World of Probabilities. While it led to interesting discussions, there just wasn’t enough meat to have students do problem solving based solely on the text. The math from the books was purely a passive experience. (We still did just fine by augmenting with our own material.)

I still hold forward the absurd idea that students still solve math problems in a math class. If you’re designing a freshman mathematics-teaser course, I might humbly suggest Problem Solving Strategies: Crossing the River with Dogs, which has the virtue of steering away from algebra as the sole touchstone for problem solving.

Back to Mr. Baker’s attempt to define Algebra I:

Six weeks of factoring and solving simple equations is enough to give any student a rough idea of what the algebraic ars magna is really like, and whether he or she has any head for it.

Mr. Baker himself seems to have a confused idea of what algebra is like, but he’s not alone. (See: my prior post on what is algebra and why you might have the wrong idea.) Also of note: some countries don’t bother to factor quadratics. (I haven’t made a map comparing countries, but it seems to be continental Europe that ignores it and just says “use the quadratic formula”.)

I could see solving equations managed in six weeks, but in a turgid, just-the-rules style that would be the opposite of what we’d want in this kind of appreciation class in the first place. Just the concept of a variable can take some students a month to wrap their head around, making me disturbed by the notion that six weeks would be enough for a student to find “whether he or she has any head for it”. (This isn’t even touching the issue of just how much is internal to the student. I’ve heard dyscalculia estimates of up to 7% of all students, but excluding that group a great deal of the attitude seems culturally specific. Allegedly in Hungary [I don't have a hard research paper or anything, this is just from personal anecdotes] it is quite common for folks to say mathematics was their favorite subject in school and the level of disdain isn’t remotely comparable to the US.)

They are forced, repeatedly, to stare at hairy, squarerooted, polynomialed horseradish clumps of mute symbology that irritate them

The article’s invective along these lines makes me wonder again how much the visual aspect to mathematics is the source of hatred, as opposed to the mathematics itself.

This picture is from the Adventure Time episode “Slumber Party Panic” and is supposed to represent the ultimate in math difficulty. However, the math symbolism is strewn truly at random and there is no real problem here. I am guessing a good chunk of the audience couldn’t even tell the difference and for them, any difficult math problem looks like random symbolic gibberish.

This is related to another issue, that of bad writing. Here’s Mr. Baker quoting a textbook:

A rational function is a function that you can write in the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions. The domain of f(x) is all real numbers except those for which Q(x) = 0.

I claim the above definition is simply bad writing, and a cursory check of the Internet reveals several definitions I’d peg as clearer in a students-who-don’t-like-math-are-trying-to-read sense:

any function which can be written as the ratio of two polynomial functions.

a function that is the quotient of two polynomials

“Rational function” is the name given to a function which can be represented as the quotient of polynomials, just as a rational number is a number which can be expressed as a quotient of whole numbers.

Hardy’s A Course of Pure Mathematics, page 38:

the quotient of one polynomial by another

Godfrey Harold Hardy admittedly goes on from there, but his text is written for mathematicians. High school texts have no such excuse.

In a similar vein, the article later quotes a 7th grade Common Core standard:

solve word problems leading to equations of the form px + q = r and p(x+q)=r, where p, q, and r are specific rational numbers

which I suppose is meant to be horrifying, but in this case the standards are written for the teachers and aim to remove any ambiguity. Here’s problems that matches the standard:

1. You bought 3 sodas for 99 cents each and paid 10 cents in tax. How much money did you pay?
2. You bought 4 candies for 1.50 each and paid \$6.20. How much was tax?

The standards can’t simply produce many examples and gesture vaguely. They must be exact. Just because 4th grade specifies that students “Explain why a fraction a/b is equivalent to a fraction (n×a)/(n×b)” does not mean students are using variables to do so. (In case you’re curious what it does look like to the 4th graders, Illustrative Mathematics has tasks here and here matching the standard.)

There’s lots more to comment on, but let me leave off for the moment on this quote, because I’m curious…

Math-intensive education hasn’t done much for Russia, as it turns out.

…is this statement (in the last paragraph of the article) accurate?


text:
## Lipid compositions for all the bilayersa used in the present work.

2014-08-26T07:22:20Z (GMT) by
a
The subscripts “MD” refer to the systems studied only by regular (unbiased) molecular dynamics simulations, thus they were not used in free energy calculations. The subscripts “FE” refer to the systems used for free energy calculations. Lo and Ld stand for liquid-ordered and liquid-disordered phases, respectively.
b
Area per lipid calculated by dividing the time averaged total area of each simulation system by the number of lipids per leaflet.

The numbers of each lipid in a given system, the phase, and the area per lipid are indicated.

Lipid compositions for all the bilayersa used in the present work.


text:
# Interpreting the expectation values of $\langle p \rangle$ and $\langle p^2 \rangle$

Why the expectation value of momentum $$\langle p \rangle$$ is zero for the one dimensional ground-state wave function of an infinite square well? And why $$\langle p^2 \rangle = \frac{\hbar^2 \pi^2}{L^2}$$? I am not asking for a proof. I am trying to interpret physical their meanings. I understand $$\langle p \rangle = 0$$ does not imply that momentum itself is zero, though at first I thought that way. I would really appreciate if you can help me build a good intuition. I have seen other questions related to this matter here, but they have not given the intuition I am looking for.

• An infinite square well is symmetrical. Why would the particle have an average momentum to the right or the left? – G. Smith Apr 5 at 3:30
• I don't know that is why I am asking this, and if the average momentum is zero why $\langle p^2 \rangle \neq 0$? – Rob Apr 5 at 3:32
• What is the average value of the numbers -1 and +1? What is the average value of their squares? – Prahar Apr 5 at 3:33
• Think about the bell curve of a normal distribution around zero. Its average is zero. Its mean square average is not. – G. Smith Apr 5 at 3:35
• @Prahar I was hoping to see a physical explanation. I knew that. – Rob Apr 5 at 3:35

## 3 Answers

To be $$0$$ on average, a quantity must either be always $$0$$ as you suggest or else it must have positive or negative outcomes, for instance $$\{-2,-1,3\}$$. Of course it is possible for a particle in $$1d$$ to have positive or negative values of momentum, with negative values corresponding to motion in the negative direction.

In the case of steady state solution to the Schrödinger equation (as the ground state of the infinite well), the distribution of positions does not change with time, so $$\langle x\rangle$$ does not depend on $$t$$; as a result, the average momentum $$\langle p\rangle= m\frac{d}{dt}\langle x\rangle=0$$. If you solution is not a steady-state solution then $$\langle p\rangle\ne 0$$ at all times.

On the other hand, $$\langle p^2\rangle$$ is the average of non-negative quantities since $$p^2\ge 0$$; this average cannot be negative, v.g. the average of $$\{4,1,9\}$$ is certainly not $$0$$. The precise actual value of $$\langle p^2\rangle$$ just happens to work out to $$\hbar^2\pi^2/L^2$$ and is basically related to the energy of your system: the ground state energy is $$E_1= \pi^2\hbar^2/(2mL^2)$$; in an infinite well the energy is completely kinetic. Since $$E_1=\langle p^2\rangle/(2m)$$, it's easy matter to find $$\langle p^2\rangle= \pi^2\hbar^2/L^2$$.

So when trying to build intuition about a subject, ideally you want something experiential to help you understand it. By the statement of your question you seem to get why the math woks out the way it does (please correct me if this is not the case) but don't get what it's telling you about the system. Let's see if we can construct some intuition from experiences that a lot of have, driving. Let's say you live 5 miles away from work, which for the sake of this example takes you 20 minutes to drive to work and 20 minutes to drive back (obviously this is hardly ever the case, but we're idealizing here). On average you're driving at 15 mph to and from work. Though because you spend equal amounts of time driving to work as you do driving home. I can't say I expect you to be do one more than the other, so my expectation value would have to be equally weighted between both possibilities. However, that doesn't mean I can't have an idea on how fast you're going (15 mph) which is definite and positive.

In terms of the particle in the box, classically the particle can be viewed as traveling from one side of the box just as much as it does to the other. Therefore, if we had any nonzero value we would be claiming that it would be spending more time traveling to one side than the other (though in reality "time" isn't the right way to think about it, but it's a good starting point). However, it does have a nonzero speed so if we average that and square it, it will still be nonzero (multiple by mass squared to get $$\langle p ^{2} \rangle$$. The statements of the first quantity $$\langle p \rangle$$ are more a value about where the particle is on average, while the latter $$\langle p^{2} \rangle$$ are more about how fast the particle is going.

This all of course is simplifying quite a bite, but hopefully it's enough to get the gears moving.

• That was more or less what I needed. As you pointed out, I understand why the math works. This gave me intuition. The only part I was bothered with was using miles in your answer (Haha I've used metric system my entire life). Jokes aside, it certainly got the gears moving. Thanks! – Rob Apr 5 at 4:26

Expectation value is nothing but an average value ,So

is the average of many p values in that state , there can even be positive momentum's or negative momentum's as reference to the direction (in mathematical perspective),So

=0 but for p square it's no 0


text:
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Some come in the form of a simple percentage, while others are much more complicated. However, the typical commission rate for sales starts at about 5%, which usually applies to sales teams that have a generous base pay. The average in sales, though, is usually between 20-30%. What is a good commission rate for sales?

We have our commission rate or earned per sale, which is $100 per sale for this employee. Understand the commission definition with examples of how to calculate commission using a commission rate formula and examples. This approach is only used under the accrual basis of accounting, and ensures that the expense is recorded in the same period as the sales transaction that triggered the commission. If more than one salesperson is involved in a sale, then the commission is split between them. It is also possible that the manager of a sales region will earn a portion of the commissions of the salespeople working in that region. Meanwhile, others base the commission calculation on base salaries. So, if you want to know the ideal commission structure for your sales agents, keep reading this article. We’ll share the pros and cons of each type of commission while discussing applicable team sizes and industries. In this step the system identifies if more than one broker agent overlaps with the time validity of the policy enrollment product and the calculation period segment. If that is the case, each overlap results into a unique commission period. The system identifies the applicable broker agent for each commission period and determines if it has to account for the switch from one broker agent to another. You work as a real estate agent and rely solely upon the commission to generate an income. Robert C. Young began writing professionally in 1989 as a copywriter for an advertising specialty company. From 2000 to 2007 he operated a real-estate development and construction company. His work has been published online at SFGate and various other websites. He graduated with a Bachelor of Business Administration in economics from Georgia State University. In some cases there may be fees charged to the agent and paid to other entities, such as professional photography, independent transaction coordinators, or the client. For example, you may have to pay franchise fees or royalty fees if your brokerage is part of a larger entity, or if referral fees to another brokerage are necessary. There may be other situations where a seller’s agent negotiates a lower fee. So you take the “Allocation” amount and multiply it by the “Rate” column to get the commission earned for a bucket and sum up those to get the total commission. Changing the Sales Commission % will apply the change to all sales commission calculations. It’s important to remember that commission is included in the cost of sale, it’s not an extra fee. Now that you have a default commission setup, the invoices and estimates in your account will display the commission amount on the Cost Basis page of the Invoice Record. Navigate to an invoice https://www.bookstime.com/ and click on ‘Cost Basis’ in the header. Moreover, they can have some control over how much they want to earn. Along that line, they will be able to clearly assess their job performance. ## All You Need to Know About Sales Commissions A commission is the amount of money paid to an employee for selling something. Payment for some jobs include an amount per hour as well as a commission on total sales. The commission is a motivation or reward for the employee to sell products so the company has more sales and can make more money. The commission payment rate could be, for example, 5% of the selling price of all goods sold. Alternatively, your company might offer a flat commission rate (for example,$25 for every sale you make).

For example, she must buy lemons, add sugar, and pay for the glasses. These costs are expenses, and they’re the cost of earning revenue. Each glass of lemonade that Janelle’s sister sells costs her 20 cents. Let’s calculate the cost of the sales made by Janelle’s sister. She sold 30 glasses of lemonade and each glass cost her 20 cents. The total cost for Sunday’s sales were \$6.00 (or 30 x 0.20).

## What Are the Disadvantages of a Gross Margin Commission?

You want to be fair with your sales team when determining their rates. It is important to note that where a company offers a large salary, commissions are low and vice versa. The starting point has to be estimating the expected sales amount.


text:
# Characterization of Pre-Measures

## Theorem

Let $X$ be a set, and let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

Let $\varnothing \in \mathcal S$.

Denote $\overline \R_{\ge 0}$ for the set of positive extended real numbers.

A mapping $\mu: \mathcal S \to \overline \R_{\ge 0}$ is a pre-measure iff:

$(1):\quad \mu \left({\varnothing}\right) = 0$
$(2):\quad \mu$ is finitely additive
$(3):\quad$ For every increasing sequence $\left({E_n}\right)_{n \in \N}$ in $\mathcal S$, if $E_n \uparrow E$ for some $E \in \mathcal S$, then:
$\mu \left({E}\right) = \displaystyle \lim_{n \to \infty} \mu \left({E_n}\right)$

where $E_n \uparrow E$ denotes limit of increasing sequence of sets.

Alternatively, and equivalently, $(3)$ may be replaced by either of:

$(3'):\quad$ For every decreasing sequence $\left({E_n}\right)_{n \in \N}$ in $\mathcal S$ for which $\mu \left({E_1}\right)$ is finite, if $E_n \downarrow E$ for some $E \in \mathcal S$, then:
$\mu \left({E}\right) = \displaystyle \lim_{n \to \infty} \mu \left({E_n}\right)$
$(3''):\quad$ For every decreasing sequence $\left({E_n}\right)_{n \in \N}$ in $\mathcal S$ for which $\mu \left({E_1}\right)$ is finite, if $E_n \downarrow \varnothing$, then:
$\displaystyle \lim_{n \to \infty} \mu \left({E_n}\right) = 0$

where $E_n \downarrow E$ denotes limit of decreasing sequence of sets.