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# Solve the equation $\frac{\sqrt {3}}{2}\sin(x) -\cos x=\cos^2x$

Solve the equation $\frac{\sqrt {3}}{2}\sin x-\cos x=\cos^2x$

My approach $\cos^2x=1-\sin^2x$

$\frac{\sqrt {3}}{2}\sin x-\cos^2x =\cos x$

$\frac{\sqrt {3}}{2}\sin x+\sin^2x -1=\cos x$

$\sqrt {3}\sin x+2\sin^2x-2=2\cos x$

Though the equation comes in form of $\sin x$ from here onward after squaring still not getting the answer.

Just to flesh out A. Pongrácz's answer, let $c=\cos x,\,s=\sin x$ so $\frac{s\sqrt{3}}{2}=c(1+c)$ and $3(1-c^2)=4c^2(1+c)^2$. After some rearrangment, $(c+1)(c-\frac{1}{2})(2c^2+3c+3)=0$, with the quadratic factor lacking real roots. We must be careful with the signs of $c,\,s$. One solution is $c=-1,\,s=0$; the other is $c=\frac{1}{2},\,s=\frac{\sqrt{3}}{2}$. In other words, the real $x$ allowed are $x=\pi(2k+1),\,x=\frac{\pi (1+6k)}{3}$ for $k\in\mathbb{Z}$.
• @SamarImamZaidi Solve on $[0,\,2\pi]$ first. – J.G. Aug 19 '18 at 14:34
A straightforward approach is indeed to square the equation: $$\frac{\sqrt{3}}{2}\sin x = \cos^2 x +\cos x$$
and replace $\sin^2 x$ by $1-\cos^2 x$. You obtain a quartic equation in $\cos x$. Maybe not the most elegant solution, but you cannot go wrong with it. Also, one root is clear without calculation: $\cos x = 1/2$ is fine. So you can surely reduce to a cubic equation. (And possibly even further to a quadratic one.)
Hint; Use the so-called Weierstrass Substitution: $$\sin(x)=\frac{2t}{1+t^2}$$ $$\cos(x)=\frac{1-t^2}{1+t^2}$$
$$\tan(\frac{x}{2})=t$$ and you don't need square the equation. You will get the equation $$-\sqrt {3} \left( \sqrt {3}t+{t}^{2}+2 \right) \left( -3\,t+\sqrt {3} \right) =0$$


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# Question 24a9d

Dec 21, 2017

Needless to say, they are different. The molecular weight is the sum of the atomic weight.

#### Explanation:

[What is the atomic mass?]
The atomic mass is the average mass of an element. It is the relative mass compared to ""^12C.

Example:
""^12C=12 amu (by definition)
""^13C=13.003355 amu
(http://www.sisweb.com/referenc/source/exactmas.htm)

The abundance(%) for ""^12C and ""^13C are $98.90$ and $1.10$, respectively.
Therefore, the atomic mass of carbon is
$12 \cdot 0.9890 + 13.00 \cdot 0.0110 = 12.011$.

[What is the molecular mass?]
The molecular mass is the average relative mass of an molecule, compared to ""^12C# atom. It is the sum of the atomic mass in the molecule.

For example, the atomic mass of hydrogen is $1.008$, and the molecular mass of hydrogen(${H}_{2}$) is $2 \cdot 1.008 = 2.016$.
The molecular mass of methane($C {H}_{4}$) will be $12.011 + 4 \cdot 1.008 = 16.043$.

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# Don't understand why my Taylor expansion results in a message and an unexpected result

Trying to do something simple: Taylor expand a generic function of t around a point t and substitute t0 + h for t. Here's my expression:

y1[t0_, h_] = Module[{t}, Series[y[t], {t, t0, 2}] /. t -> t0 + h]

This results the message

SeriesData::sdatv: First argument h + t0 is not a valid variable.

and it gives the result

y[t0] + y′[t0]((h + t0) - t0) + 1/2 y′′[t0] ((h + t0) - t0)^2 + O[(h + t0) - t0]^3

This is what I'm looking for, except all of the (h + t0) - t0 stuff. Why is this not simplifying to h? In fact, if I substitute 6 for t0, I get (h + 6) - 6. So there is enough information there to know that when it substitutes for t0, it substitutes for all t0, but it still treats one of the t0's as different from the other for purposes of simplification.

I sense that I'm missing something basic.

Change the argument of y to t0 + h
y1[t0_, h_] = Module[{t}, Series[y[h + t0], {h, 0, 2}]]

• Why not just y1[t0_, h_] = Series[y[h + t0], {h, 0, 2}] – m_goldberg Feb 14 '15 at 2:56
• @m_goldberg left the expression enclosed in the Module, in case the OP had some reason to have it there. You are correct that it does nothing in this toy problem. – bbgodfrey Feb 14 '15 at 2:59


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# Quantum Transmission Coefficient

1. ### PineApple2

34
Hello. I have a question, mathematical in nature.
Considering a potential step of height V0 and width a, the amplitude coefficient is
$$t=\frac{2k_1k_2e^{-ik_1a}}{2k_1k_2\cos{k_2a}-i(k_1^2+k_2^2)\sin(k_2a)}$$
Now the transmission coefficient is
$$T=|t|^2$$
So I need to find the absolute value of this expression. I thought about taking the complex conjugate of the denominator and multiply both the numerator and the denominator by this factor (in order to make the denominator real). but this is very messy.
Is there a simpler way to find the absolute value of this expression?
Thanks

2. ### Bill_K

4,160
Much simpler. Take the complex conjugate of both the numerator and the denominator separately. If t = N/D, then |t|2 = N*N/D*D

3. ### PineApple2

34
You are of course correct. Thanks for this obvious answer :-)


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# Random walks on graphs: Cover time and blanket time

Winkler and Zuckerman conjectured that the blanket time is within a constant factor of the cover time. The conjecture was recently proved. The cover time $C$ is the expectation of the first time $t$ that the walk has seen every vertex. The blanket time $B_\delta$, where $0<\delta<1$ is some constant, is the expectation of the first time $t$ such that each vertex $v$ has been visited at least $\delta \pi_v t$ times. That is, it is the expected time for all the vertices to have been seen roughly as expected by the stationary distribution.

So their now-proven conjecture was that $B_\delta \leq a C$ where $a$ is some constant.

One remark in their paper that I can't see the justification of is the claim that this implies that the expectation of the first time that each vertex $v$ has been visited $\pi_vC$ times is $O(C)$. I was wondering if anyone can offer some insight.

The remark is near the bottom of page 3 in their paper http://www.cs.utexas.edu/~diz/pubs/blanket.ps

For what it's worth, this question is related to another question I asked here A type of stochastic jump process

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That's interesting. Is $a$ independent of $\delta$? – Tom LaGatta Dec 22 '10 at 9:43
No, $a$ is dependent on $\delta$ – Probabilist Dec 22 '10 at 10:37

With probability at least $1/2$ the blanket time $B_{1/2}$ is at most $2aC$. It only takes a Geom$(1/2)$ such blocks to get $\pi_v C$ visits to each $v$.

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Omer, it takes $Geom(1/2)$ to blanket the graph, but that blanket could be less than $2aC$. – Probabilist Dec 22 '10 at 21:30

Suppose there exist $a$ and $\delta$ such that $B_\delta\le aC$ for all graphs.

Fix a graph and start a random walk at an arbitrary vertex. Repeatedly wait for a $\delta$-blanketting and reset the visit counts until the total time taken exceeds $C/\delta$. The expected time for this is at most $(1/\delta+a)C$ because the initial blanketting phases take in total at most $C/\delta$ steps (deterministically) and the expected time of the final blanketting is (using the strong Markov property) at most $B_\delta$. In particular $T=O(C)$.

By construction we have $T\ge C/\delta$. Also since $T$ is obtained by concatenating a sequence of $\delta$-blankettings, it is itself a $\delta$-blanketting. This means that each vertex $v$ is covered at least $\delta\pi_v T\ge \pi_v C$ times.

-
The problem is that it doesn't seem to be true that the final round of blanketing actually has an expectation $O(C)$. What we are talking about is a series of jumps on the number line until we cross $K=C/\delta$, and we want to know what in the final jump, the value $t-K=O(C)$. However, as can be seen from the second counter example in the link to my previous question - this is not necessarily the case. – Probabilist Dec 22 '10 at 21:37


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# All possible linear combinations of positive half-integers with coefficients +/- 1

This will be a simple problem on paper, but the brute force method is not really suitable for a computer, so I'm after a tricky algorithm that works in practice too: if $n$ positive half-integers $p_i$ are given, one can form $2^n$ linear combinations with coefficients $\pm 1$:

$$P = e_1 p_1 + \ldots + e_n p_n$$

Where $e_i = \pm 1$. With given $p_i$ one obtains $2^n$ values for $P$. Let's call $\mu(P)$ the multiplicity of $P$, the number of linear combinations that gives $P$. My goal is to know this $\mu(P)$. On a computer I could just loop over $2^n$ items given by n-length strings of +1 and -1's and record each $P$, but if $n$ is large ($n>64$) this becomes problematic if 64 bit integers are used (unsigned long long int in C). And also it becomes time consumong if $n$ is larger than 64 and some arbitrary precision library is used.

Is there a tricky way to obtain $\mu(P)$ that does not require a loop over $2^n$ elements?

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For arbitrary $p_i$, calculating the value of $\mu(P)$ for some $P$ is equivalent to the subset sum problem, an NP-Complete problem. –  Yoav Kallus Mar 6 '13 at 0:39
Indeed, and as is the case for subset sum, there is a pseudo-polynomial time solution using dynamic programming that might help answer the posted question. Gerhard "Ask Me About System Design" Paseman, 2013.03.05 –  Gerhard Paseman Mar 6 '13 at 1:43

Indeed it is the Subset Sum Problem (or closely related to it). This problem asks essentially whether $\mu(P)>0$. It is only weakly NP-complete, and there is a pseudopolynomial dynamic-programming algorithm for it. You can find it in any textbook on algorithm design. More precisely, if the sum of the $p_i$'s is $B$, you can solve it in $O(nB)$ time. Whether this is practical depends on the size of the numbers. (The other suggested solutions, which multiply polynomials, are another way to express the same algorithm. Your question suggests that $\mu(P)$ is not just 0 or 1, and apparently, the numbers are not so big.)

-

Hm, maybe rewriting this by adding $p_1+p_2+\dots$ on both sides, and dividing by two, then this is isomorphic to $P' = f_1 p_1 + \dotsc$ where $f_i$ is either 0 or 1. Thus, it is the number of ways to construct $P'$ as a sum using 0 or one $p_1.$ Thus, expand $\prod_i (1+x^{p_i})$ and look for the coefficient of $x^{P'}$ to find the number you are looking for.

I don't know if this helps you, but maybe this interpretation helps a bit. What do you know of the numbers $p_i$?

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This is very helpful, thanks. The $p_i$ numbers are not large always smaller than 10. And $n$ is always less than 100. –  Daniel Mar 6 '13 at 10:00

I don't know about tricky, but you can use Per's suggestion more directly. The problem is that is mu is uniform, you will have 2^n possiblities for P, and not enough memory to store them.

The idea is to compute product over i of (x^-p_i + x^p_i). If all the p_i are 1, this will have mu(P) resemble a row of Pascal's triangle, with each new row formed by adding translates of the previous row. This is the best case, since you will need to store O(n) coefficients. Hopefully, you will need as few as O(n^k) coefficients for small k, but there is the potential for exponentially many coefficients, one for each possible value of P. If the p_i occupy a small interval though, it should not be too bad.

Your problem is equivalent to a well-known problem in banking. Transform your problem as suggested by Per. Now imagine a bank holding a portfolio of bonds issed by obligors numbered $i=1,2,,\ldots,n$ and the bank owns bonds worth $p_1,p_2,\ldots,p_n$ issued respectively by these obligors. The $f_i$ variables correspond to indicator variables: $f_i=1$ means obligor $i$ defaulted (in a twelve month period, say), otherwise the value is 0. The value $\mu(P')$ is proportional the probability that the portfolio will sustain a loss of $P'$.
This discrete distribution approaches a continuous curve as $n\to\infty$. There are well-known theories to estimate this curve which come under the heading "Saddlepoint Approximations". For example, see "Saddlepoint Approcimations with Applications" by Ronald Butler, publ by Cambridge Unic Press. The classic is "Saddlepoint Approximations" by Jens Lensen publ by Oxford, but it's expensive.


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# R seasonal time series

I use the decompose function in R and come up with the 3 components of my monthly time series (trend, seasonal and random). If I plot the chart or look at the table, I can clearly see that the time series is affected by seasonality.

However, when I regress the time series onto the 11 seasonal dummy variables, all the coefficients are not statistically significant, suggesting there is no seasonality.

I don't understand why I come up with two very different results. Did this happen to anybody? Am I doing something wrong?

I add here some useful details.

This is my time series and the corresponding monthly change. In both charts, you can see there is seasonality (or this is what I would like to assess). Especially, in the second chart (which is the monthly change of the series) I can see a recurrent pattern (high points and low points in the same months of the year).

Below is the output of the decompose function. I appreciate that, as @RichardHardy said, the function does not test whether there is actual seasonality. But the decomposition seems to confirm what I think.

However, when I regress the time series on 11 seasonal dummy variables (January to November, excluding December) I find the following:

    Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5144454056  372840549  13.798   <2e-16 ***
Jan     -616669492  527276161  -1.170    0.248
Feb     -586884419  527276161  -1.113    0.271
Mar     -461990149  527276161  -0.876    0.385
Apr     -407860396  527276161  -0.774    0.443
May     -395942771  527276161  -0.751    0.456
Jun     -382312331  527276161  -0.725    0.472
Jul     -342137426  527276161  -0.649    0.520
Aug     -308931830  527276161  -0.586    0.561
Sep     -275129629  527276161  -0.522    0.604
Oct     -218035419  527276161  -0.414    0.681
Nov     -159814080  527276161  -0.303    0.763

Basically, all the seasonality coefficients are not statistically significant.

To run linear regression I use the following function:

lm.r = lm(Yvar~Var$Jan+Var$Feb+Var$Mar+Var$Apr+Var$May+Var$Jun+Var$Jul+Var$Aug+Var$Sep+Var$Oct+Var$Nov) where I set up Yvar as a time series variable with monthly frequency (frequency = 12). I also try to take into account the trending component of the time series including a trend variable to the regression. However, the result does not change.  Estimate Std. Error t value Pr(>|t|) (Intercept) 3600646404 96286811 37.395 <2e-16 *** Jan -144950487 117138294 -1.237 0.222 Feb -158048960 116963281 -1.351 0.183 Mar -76038236 116804709 -0.651 0.518 Apr -64792029 116662646 -0.555 0.581 May -95757949 116537153 -0.822 0.415 Jun -125011055 116428283 -1.074 0.288 Jul -127719697 116336082 -1.098 0.278 Aug -137397646 116260591 -1.182 0.243 Sep -146478991 116201842 -1.261 0.214 Oct -132268327 116159860 -1.139 0.261 Nov -116930534 116134664 -1.007 0.319 trend 42883546 1396782 30.702 <2e-16 ***  Hence my question is: am I doing something wrong in the regression analysis? • @forecaster, here is an answer to your question (OP says decompose function in R is used). – Richard Hardy Mar 12 '15 at 14:23 • Reading the help file of the decompose function, it seems that the function does not test whether there is seasonality. Instead, it just obtains averages for each season, subtracts the mean and calls this the seasonal component. So it would produce a seasonal component regardless of whether there is true underlying seasonal component or just noise. Nevertheless, this does not explain why your dummies are insignificant though you say the seasonality is visible from a plot of the data. Could it be that your sample is too small to get significant seasonal dummies? Are they jointly significant? – Richard Hardy Mar 12 '15 at 14:30 • You need to look at the scales, seasonal chart shows seasonal variation is between -0.02 and +0.04, while the actual values range from 4 billion to 6 billion. Decompose function forces your data to show some seasonality that is why it is showing a value that is negligible. There is no seasonality in your data. – forecaster Mar 12 '15 at 21:01 • I thought seasonality is more about movements with a certain "fixed" frequency rather than the magnitude of the movement. The scales are different as the first chart shows a balance (in terms of pounds) and the second is the change (expressed in percentage terms). I've just tried to re-run the regression: if I take into account a polynomial trend, some coefficient starts to be significant. I guess, as @danno suggested, the trend is very significant. – mattiace Mar 12 '15 at 21:11 • The Canova and Hansen test may give your some further information about the presence and stability of a seasonal pattern in your data. For some applications of this test see for example this post, which also gives the link to the original paper and to sample code. – javlacalle Mar 12 '15 at 21:40 ## 3 Answers Are you doing the regression on the data after you've removed the trend? You have a positive trend, and your seasonal signature is likely masked in your regression (variance due to trend, or error, is larger than due to month), unless you've accounted for the trend in Yvar... Also, I'm not terribly confident with time series, but shouldn't each observation be assigned a month, and your regression look something like this? lm(Yvar ~ Time + Month)  Apologies if that makes no sense... Does regression make the most sense here? • I read on an Econometrics textbook (Wooldridge) that if you introduce a "time" factor in the regression, it's like de-trending your original time series. For "time" factor I mean a vector such as [1, 2, ..., n] with n = to the number of observations. I don't really get your second point. My second regression looks like the following: lm.r = lm(Yvar~Var$Time+Var$Jan+Var$Feb+...+Var\$Nov). Is this what you meant? – mattiace Mar 12 '15 at 20:13
• It would help to see a few lines of your data. But, I imagine that you've got three columns, Time, Yvar, Month. The rows are observations. Thus, I would think the factors in your lm() would be Time and Month (not the factor levels of Month). Also, I think there are some issues with lm() on time series... the observation are not independent - temporally autocorrelated. – danno Mar 14 '15 at 20:18

In your graphical depiction of the time series, it is obvious that "trend"--a linear component in time--is the singlemost substantial contributor to the realization. We would comment that the most important aspect of this time series is the stable rise each month.

After that, I would comment that the seasonal variation is miniscule by comparison. It is not surprising, therefore, with monthly measures taken over 6 years (a total of only 72 observations) the linear regression model fails to have the precision to identify any of the 11 month-contrasts as statistically significant. It is furthermore not surprising that the time effect does achieve statistical significance, because it is the same approximately consistent linear increase occurring over all 72 observations, conditional upon their seasonal effect.

The lack of statistical significance for any of the 11 month contrasts does not mean that there are no seasonal effects. In fact, if you were to use a regression model to determine whether there is any seasonality, the appropriate test is the nested 11 degree of freedom test which simultaneously assesses the statistical significance of each month contrast. You would obtain such a test by conducting an ANOVA, likelihood ratio test, or robust Wald test. For instance:

 library(lmtest) model.mt <- lm(outcome ~ time + month) model.t <- lm(outcome ~ time) aov(model.mt, model.t) lrtest(model.mt, model.t) library(sandwich) ## autoregressive consistent robust standard errors waldtest(lrtest, lmtest, vcov.=function(x)vcovHAC(x))

I don't know if it's your case, but that happened to me when I started analyzing time series in R and the issue was that I hadn't correctly stated the time series period when creating the time series object to decompose it. There is a parameter in the time series function that lets you specify its frequency. Doing so, it correctly decomposes its seasonal trends.

• jmnavarro, I defined the frequency correctly in the decompose function (=12 as I have monthly data). In fact, I'm happy with the outcome of this function. My question is why I don't find the same result (seasonality is significant) when I do linear regression using dummy variables. I did it both with R and excel and the results is the same: dummy coefficients not statistically significant. This is against what I have previously found in the decompose function. I don't understand if I am missing something out here – mattiace Mar 12 '15 at 12:08
• True, sorry, I did not completely understand your question. It would help if you could post your code, so we could try and reproduce it. – jmnavarro Mar 12 '15 at 13:32


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# Non-hyperbolic fixed points in multidimensional systems

Consider first a one-dimensional dynamical system given by $dx/dt = f(x)$. Suppose that the origin is a fixed point, i.e. $f(0)=0$. Suppose that we're interested in whether trajectories that start from positive initial values of $x$ (hereafter "positive trajectories") will converge towards the origin or diverge away from it.

If $df/dx$ is negative at 0 then the fixed point is locally asymptotically stable; if it's positive then it's unstable. If $df/dx=0$ at the origin then the fixed point is called "non-hyperbolic". If $d^2 f/dx^2 <0$ then it may or may not be Lyapunov stable (because negative trajectories might diverge), but positive trajectories will converge toward the origin, albeit more slowly than a linearly stable fixed point. In general positive trajectories will converge towards the origin iff the lowest non-zero derivative is negative.

My question is about how to generalise this to the $n$-dimensional case. I'm dealing with systems where no trajectory starting in the non-negative orthant can leave the non-negative othant, and I'm interested only in trajectories that are confined to it. ("Nonnegative trajectories".)

In the case where the origin is a hyperbolic fixed point it's clear how to proceed. One creates the linear approximation $\mathbf{\dot{x}} = A\mathbf{x}$. The Jacobian matrix $A$ is the analog of $dx/dt$ - if all its eigenvalue have real part strictly less than 0 then the fixed point is stable, and hence all trajectories (positive or otherwise) will converge toward the origin. If one of the eigenvalues has real part greater than $0$ then the fixed point is unstable. (The condition that trajectories are confined to the non-negative orthant implies that $A$ can have negative entries only on its diagonal, which in turn means that its leading eigenvalue will always have a corresponding eigenvector in the nonnegative orthant, by the Perron-Frobenius theorem applied to $e^{\varepsilon A}$ for sufficiently small $\varepsilon$; hence we don't need to worry about the possibility that it might be unstable only for non-nonnegative trajectories.)

However, I'm currently dealing with systems where the leading eigenvalue of $J$ is exactly 0, i.e. the fixed point is non-hyperbolic. In this case I guess I have to move up to a quadratic approximation: $$\frac{dx_i}{dt} = \sum_j A_{ij}x_j + \sum_{jk}B_{ijk}x_jx_k.$$ I can calculate the values of $A_{ij}$ and $B_{ijk}$ for my systems. (They are generated computationally, and in my systems $n$ is of the order 1000 or so.) In addition to $A$ having negative entries only on its diagonal, the condition of being confined to the positive orthant implies that if $B_{ijk}$ is non-zero then it is negative if and only if $i=j$ or $i=k$ (or both).

The problem is that I don't know how to tell from these values whether the system is stable or not. I'm looking both for an algorithm to determine this, and for necessary and sufficient conditions that are easy to compute. It strikes me that there's probably a nicely worked out theory that deals with this, but I don't know its name, so I can't find any information about it.

• Do you mean real part positive/negative instead of $<1, >1$? Apr 14, 2014 at 3:25
• @ChristianRemling d'oh, fixed. Apr 14, 2014 at 3:27
• I don't think you're criterion in 1D is right. $\dot x=-x^2$ satisfies the criterion, but 0 is not Lyapunov stable. In fact, in one dimension, I think it's a necessary condition that the second derivative is 0. The third derivative then can give a sufficient condition for Lyapunov stability. Apr 14, 2014 at 5:44
• @AnthonyQuas you're right of course. I'll fix it in a bit. (I was only thinking about trajectories where $x$ is positive, since that's what I'm most interested in the multidimensional case.) Apr 14, 2014 at 5:59
• The first thing you need to search for is center manifold theory. Apr 14, 2014 at 17:05

Unfortunately you'll have to face a «no» answer, as there is no general/generic way to handle your problem…

First of all, your definition of «non-hyperbolic» is not clear to me when $n>1$. As far as I know a singular point like yours is deemed hyperbolic when $0$ does not belong to the real convex hull of the spectrum (in $\mathbb C$) of the linear part $A$. In case the singularity is hyperbolic, Poincaré's linearization theorem ensures that the system is dynamically equivalent to its linear part, as you refer to in the question. Certainly when you have null eigenvalues the system is not hyperbolic in that sense.

Now if the singular point is not hyperbolic, then a lot of bad behaviors can crop up, even in the case $n=2$. The presence of resonances ($\mathbb Q$-linear relations between the eigenvalues of $A$) or quasi-resonances (irrational relations which are «well approximated» by rational ones) can mess things up. Even from a formal viewpoint the answer is not clear, and certainly not computable without restrictive conditions on $f$ (e.g. polynomial, and even then…). In particular it is not true that the system $\dot x=f(x)$ is conjugate (dynamically equivalent) to a system where $f$ is algebraic using changes of coordinates which are merely $C^1$ at the singular point. Hence it is not sufficient to take a finite jet (let alone the second jet) to answer your question when there are resonances (as is the case here when at least one eigenvalue vanishes).

Dynamical systems with at least one zero eigenvalue are called «saddle-nodes», so you might wish to search for this keyword. Notice, though, that the theory of saddle-nodes is complete only in the smooth case (that is, $f$ smooth) for $n=2$, and partially covered when $n>2$. The more resonances (e.g. zero eigenvalues) the more difficult the problem is, even from a purely theoretical viewpoint…

So, without more information on your specific system, nothing can be said.

EDIT: You still can say something regarding stability. For instance look at the system $$\dot x(t) = x(t)^{k+1} \\ \dot y(t)=-y(t)$$ with $x, y\in\mathbb R$. Depending on the parity of $k$, either all trajectories are stable or only a single trafectory is (that's $y=0$). In these problems you need to identify directions in the $x$-variable where you have contributions of exponential terms $\exp(x^{-k}/k)$. This can be done by inspecting the linear part, and in general governs «how stable» the solutions will be. I don't have a recipe when $n>1$, though.

• Ok, that's useful to know. It's actually quite encouraging if systems like this are deeply non-trivial. Note that my systems do have quite substantial restrictions compared to the general case, though, due to trajectories being restricted to the non-negative orthant. Whether that's enough to make a difference I have no idea. Even if nothing can be said in the general case, I'm still looking for necessary or sufficient conditions for (in)stability. (Besides the obvious existence of a Lyapunov function, which I don't have a recipe to compute.) Apr 14, 2014 at 11:40
• @Nathaniel: see my edit. Apr 14, 2014 at 11:54


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# Commenting code - completely useless?

This topic is 4144 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

I'd like to think not... but I've been learning a lot about agile dev lately, and Refactoring mentions that comments are a good indicator that a refactoring would be useful, even if it's just to extract a single line of code into a function with a clearer name. This makes sense, and all is well and good from here. Furthermore the source code of an entire project (or subsystem or once in a while even class) might not always be self-explanatory, but for something like that I would think you'd want to collect it in some other documentation (whether external or interwoven a la Literate Programming, something a bit bigger than //). I haven't yet started applying refactoring in my own code, so I've got the theory but not the practice -- can anyone offer an example or two of when comments are warranted vs. the alternatives? Thanks

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It is an interesting point as a perfect bit of code would be written in such a way that it reads like English and hence needs no comments e.g. instead of writing:

// Fire the bullet
DoStuff()

you write

FireTheBullet()

it now needs no comment. Of course a very obvious example but it bears up in most cases. Whenever you write a comment in a function you are explaining something or maybe reminding yourself to do something. In the first case better naming of functions and variables will often do away with the need for a comment and in the second a proper to do tracking system is preferred.

I put comments above functions (using Doxygen notation) that can then help create automatic documentation but I avoid all comfents in functions as, like you say, they tend to indicate something that could be refactored.

A lot comes down to naming things correctly. Call your functions exactly what they do and do not worry about long names - nowadays with all the intellisense features in IDEs you rarely have to type out a whole name anyway. e.g. instead of

CalcTraj()

write

CalculateTrajectory()

Proper naming also allows you to spot refactorings, e.g. if you find you are putting an 'and' in the name it usually means you need two functions eg.

FireBulletAndCheckForCollision()

Should be split into two functions (refactors) and then called like:

FireBullet();
CheckForCollision();

The same goes for variable names. Always name a variable exactly what it is! If you are unsure then you are not clear on your own design and need to sort that out first. This also prevents the horrendous activity of reusing variables e.g. temp

Another use of comments may be to say 'only ever call this function with a valid pointer'. Again rather than put the comment up enforce it. e.g. in this case I would put in an assert:

assert(pointer!=0);

You need to be clear what you want to code, clear on the restrictions of each function and the one taks the function does and then name the function correctly, enforce the restrictions via error tests and verbosely name your variables etc.

Anyway an interesting topic... :)

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Well, I think that if some capability has been included in a language, then its for a good reason.

I believe that commenting every single line of code is, by and large, 'useless'. If you need to do that in order to understand your code, then its a good indication that it should be broken up into smaller, more manageable pieces.

However, a single comment for a block of code, a function or a module, is completely warranted. You can only gain so much by simply reading the name of a function or module. The comments should give an extra level of detail concerning the code's function and its use. Rather than trying to interpret several dozen (or more) lines of a code, reading a simple, clear and concise comment can save you loads of time when working on a larger project.

If the documentation is external, than I've got to always have that documentation open, switch to it when I need the information, and then search for the appropriate location in the document. Its much easier and faster when its right there. I'm not saying external documentation isn't helpful; it can be very helpful. But the two should probably be working in conjunction to provide the programmer what they need.

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You may want to take a look at Javadoc or Doxygen. Those tools can generate documentation based on your comments.

looking something like this:

/** * a normal member taking two arguments and returning an integer value. * @param a an integer argument. * @param s a constant character pointer. * @see Test() * @see ~Test() * @see testMeToo() * @see publicVar() * @return The test results */int testMe(int a,const char *s);

even if the comments might seem slightly redundant it does result in pretty impressive documentation (Fully browsable html covering all classes and methods in your project).

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Simian: Doxygen can generate most of that information without special comments. I can build references of where your function is called as well.

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• // StartPage should be called before DrawPageassert (\$skeleton_status === SKELETON_ACTIVE);

My failed assertion handler in PHP automatically displays the line of code with the error, as well as the preceding comment (and a backtrace), making error messages simpler to understand than only file-line information.
• // TODO: replace with actual function once it's doneMockDisplay();

During development, in-code notes from programmer to programmer (and information about how to resume work) are very useful. Arguably, quite less so during maintenance.
• (* Little optimization trick: keep the pair *)| (x,0) as p when x > 0 -> right(p)

Small code-level information might not deserve moving to its own function (especially since it's usually a small modification), but it's interesting to leave behind for the less clever programmers.
• // Foo's algorithm: after the loop, store is never nullassert (store);return *store;

Although the overall algorithm may be described in detail in paper X, properties such as the above are best repeated in a comment, so that someone who does not wish to wade through an entire page of properties still understands why the function "magically" expects the store to be non-null.

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Yeah sure ideally we'd never need comments, but ideally we'd only to write simple code like FireZeMeeesiles(). You're going to have to use comments a lot of times when there's no easier way to quickly tell what an algorithm does, or when it's not efficient to wrap a piece of code into a easy-to-read-function.

I only use comments though because it makes my code more colorful, green and friendly. Makes it feel alive.

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I like to think that the majority of my code is easily understandable, so I use comments quite sparingly. I do however think they are extremely useful for separating out blocks of code, just so at a glance I can tell which block does what if I'm scrolling through a page. I prefer that to having a large chunk of code in a single code block, or just separating with whitespace.

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Ideals and reality rarely see eye-to-eye.

Architectures with complex template schemes, hierarchies, and special cases usually benenfit from a little comment blurb explaining how and why they work the way they do.

Even for a well-named and coded function, comments can help to clarify what is going on, especially for mathmatical or physics algorithms that other coders may not be familiar with.

Comments are usually a good idea when answers to the questions "why" or "how" about a particular piece of code are not immediately obvious.

Think about something like bump mapping and all the steps involved with it. It might make perfect sense to you, and the technique might even be well known to someone else, but how you implement it in actual code might be confusing to someone that didn't write it. It might even be confusing to you if you come back to it 4 or 5 months later.

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The one thing that I think that should be commented but almost never is; the legal values for all paramters. Can you have a null for that? are negative numbers ok? Will an empty string break things.

Doing this helps the testers and also helps people that use your functions.

theTroll

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# zbMATH — the first resource for mathematics

On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics. (English) Zbl 1295.35122
Summary: In this paper, we establish an optimal blow-up criterion for classical solutions to the incompressible resistive Hall-magnetohydrodynamic equations. We also prove two global-in-time existence results of the classical solutions for small initial data, the smallness conditions of which are given by the suitable Sobolev and the Besov norms respectively. Although the Sobolev space version is already an improvement of the corresponding result in [D. Chae et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31, No. 3, 555–565 (2014; Zbl 1297.35064)], the optimality in terms of the scaling property is achieved via the Besov space estimate. The special property of the energy estimate in terms of $$\dot B_{2,1}^s$$ norm is essential for this result. Contrary to the usual MHD the global well-posedness in the $$2\frac{1}{2}$$ dimensional Hall-MHD is wide open.

##### MSC:
 35B44 Blow-up in context of PDEs 35L60 First-order nonlinear hyperbolic equations 35K55 Nonlinear parabolic equations 76W05 Magnetohydrodynamics and electrohydrodynamics 35Q35 PDEs in connection with fluid mechanics 35Q60 PDEs in connection with optics and electromagnetic theory
well-posedness
Full Text:
##### References:
 [1] Acheritogaray, M.; Degond, P.; Frouvelle, A.; Liu, J.-G., Kinetic formulation and global existence for the Hall-magneto-hydrodynamics system, Kinet. Relat. Models, 4, 901-918, (2011) · Zbl 1251.35076 [2] Balbus, S. A.; Terquem, C., Linear analysis of the Hall effect in protostellar disks, Astrophys. J., 552, 235-247, (2001) [3] Berselli, L. C., On a regularity criterion for the 3D Navier-Stokes equations, Differential Integral Equations, 15, 1129-1137, (2002) · Zbl 1034.35087 [4] Chae, D.; Degond, P.; Liu, J.-G., Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincare Anal. Non Lineaire, (2013), in press [5] Chae, D.; Lee, J., Global well-posedness in the super-critical dissipative quasi-geostrophic equations, Comm. Math. Phys., 233, 297-311, (2003) · Zbl 1019.86002 [6] Chae, D.; Schonbek, M., On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255, 3971-3982, (2013) · Zbl 1291.35212 [7] Forbes, T. G., Magnetic reconnection in solar flares, Geophys. Astrophys. Fluid Dyn., 62, 15-36, (1991) [8] Homann, H.; Grauer, R., Bifurcation analysis of magnetic reconnection in Hall-MHD systems, Phys. D, 208, 59-72, (2005) · Zbl 1154.76392 [9] Kenig, C.; Ponce, G.; Vega, L., Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4, 323-347, (1991) · Zbl 0737.35102 [10] Kozono, H.; Taniuchi, Y., Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235, 173-194, (2000) · Zbl 0970.35099 [11] Lighthill, M. J., Studies on magneto-hydrodynamic waves and other anisotropic wave motions, Philos. Trans. R. Soc. Lond. Ser. A, 252, 397-430, (1960) · Zbl 0097.20806 [12] Majda, A. J.; Bertozzi, A. L., Vorticity and incompressible flow, Cambridge Texts Appl. Math., (2002), Cambridge · Zbl 0983.76001 [13] Mininni, P. D.; Gómez, D. O.; Mahajan, S. M., Dynamo action in magnetohydrodynamics and Hall magnetohydrodynamics, Astrophys. J., 587, 472-481, (2003) [14] Ogawa, T., Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J. Math. Anal., 34, 1318-1330, (2003) · Zbl 1036.35082 [15] Serrin, J., The initial value problem for the Navier-Stokes equations, (Langer, R. E, Nonlinear Probl., Proc. Sympos., Madison 1962, (1963), Univ. Wisconsin Press Madison), 69-98 [16] Shalybkov, D. A.; Urpin, V. A., The Hall effect and the decay of magnetic fields, Astron. Astrophys., 321, 685-690, (1997) [17] Triebel, H., Theory of function spaces I, (1983), Birkhäuser Basel [18] Wardle, M., Star formation and the Hall effect, Astrophys. Space Sci., 292, 317-323, (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
how to solve it by residue theorem

i have to solve the folowing integral by using contour integration & residue theorem $$\int_{-\infty}^{\infty}\frac{\sin x}{x^2+2x+2}\ dx$$ i factorised $x^2+2x+2=(x+1+i)(x+1-i)$ where $x=-1+i$ & $x=-1-i$ are single poles $$\int_{-\infty}^{\infty}\frac{\sin x}{(x+1+i)(x+1-i)}\ dx$$ i got stuck here, don't know which contour is to to be used? please help me solve it

$$I=\int \limits^{\infty }_{-\infty }\frac{\sin \left( x\right) }{x^{2}+2x+2} dx=\Im \left[ \int \limits^{\infty }_{-\infty }\frac{e^{iz}}{z^{2}+2z+2} dz\right] =\Im \left[ 2\pi i \sum Res\left( f\left( z\right) , z_{0}\right) \right]$$ you have one pole inside your contour $z=i-1$ so your Residue will be $$2\pi i\lim \limits_{z\to i-1}\left( z+1-i\right) \frac{e^{iz}}{\left( z+1-i\right) \left( z+i+1\right) } =\Im \left( 2\pi i \left( \frac{e^{i\left( i-1\right) }}{2i} \right) \right) =\Im\left( \frac{\pi \left( \cos \left( 1\right) -i \sin \left( 1\right) \right) }{e}\right)$$ So the integral equal to $$\int \limits^{\infty }_{-\infty }\frac{\sin \left( x\right) }{x^{2}+2x+2} dx=\boxed{\frac{-\pi \sin \left( 1\right) }{e}}$$
• please tell me how you knew the pole $i-1$ is inside contour? – Bhaskara-III Sep 11 '16 at 12:04
• in the function $R(z) e^{imz}$ . Let $\sum Res(f(z), z_{0})$ be the sum of the residues of $R(z)e^{imz}$ in the upper half plane. So that $z=-i-1$ not in your contour. – Refaat M. Sayed Sep 11 '16 at 12:14
• You can use the lower half plane. but in this case your function will be $$\Im \left[ \int \limits^{\infty }_{-\infty }\frac{e^{-iz}}{z^{2}+2z+2} dz\right]$$ and you will take the second pole $z=-1-i$ read this (math.stackexchange.com/questions/530204/…) – Refaat M. Sayed Sep 11 '16 at 13:34
There are different ways to proceed. I think the easiest is to note that the ${\rm Re }\; e^{ix} = \sin (x)$ so that you may write the integral as the imaginary part of another one for which the contour becomes more 'obvious'.
• Refaat has already spelled out parts of the calculation, but you should add the (closed) contour-integral which consists of a real segment $[-R,R]$ and a half-circle (of radius $R$) in the upper half-plane. Note that in the upper half-plan $e^{iz}$ tends to zero fast so you proceed to show that the half-circle contribution goes to zero as $R\rightarrow \infty$ – H. H. Rugh Sep 11 '16 at 12:04


text:
Fundamentals of Fluid Mechanics B Questions and Answers
No, there are no other assumptions. These strain rates can be used for any fluid. Also, the same applies with the strain rates in cylindrical and spherical coordinates in the tables. The strain rates can be used in the general case.
 03.24.20
 Question by AME536A Student Could you also give us the final answers for Assignment 5 Question 3 (d) and (e)?
Let me see if I can find them.. If I do, I'll post them soon.
 Question by AME536A Student I'm still having problems to solve Assignment#1 Question#2(b), the prove of Crocco's theorem. In class you gave us this hint: say $a=b-c$ $$da = d(b-c)$$ If $da=0$ and if s-s and uniform properties at some point upstream: $$\nabla a = 0$$ Then $$db-dc =0$$ and $$\nabla (b-c)=0$$ $$\nabla b = \nabla c$$ I don't understand how we can use this without ending up with a temperature gradient term in the equation.
 03.25.20
Another hint. For two streamlines near each other and at a certain location, one has a temperature $T$, and the other the temperature $T+dT$. Expand terms and get rid of the terms that are necessarily much smaller than the others.
 Question by Student AME536B Do we need to memorize proofs for this midterm?
The midterm is open book so you can consult your assignments, class notes, books if you want. However, if I were you, I would stick to consulting the tables: you won't waste time and score higher this way.
 Question by Student AME536B I was attempting to figure out the mean free path derivation question you gave us previously on how to get the $\sqrt2$ term. What I came up with using your tip of $$q_{rms}=c*q_{avg}$$ is, $$\overline{V_{rel}^{2}}=\overline{V_1^{2}}+\overline{V_2^{2}}$$ Is this process correct?
I don't understand where your equation (2) is coming from. You need to clarify this.
 Question by Student AME536B On the tables, the equation given for the strain rate $S_{\theta r}$ in spherical coordinates has a term of $\frac{v_{\theta}}{r}$; is that a typo? Shouldn't the shear strain equation for spherical coordinates be given by: $$S_{r\theta} = \frac{1}{2} \left( \frac{1}{r}\frac{\partial v_r}{\partial \theta} + \frac{\partial v_{\theta}}{\partial r} \right)$$
 04.06.20
The equation in the tables is correct: there is no typo.
 Question by AME536A Student For question 1. b. on Assignment 6, do you want us to derive the expression you gave for the streamfunction? Or just start from the given equation?
If it's given in the question don't derive it.
 Previous   1 ,  2   •  PDF 1✕1 2✕1 2✕2  •  New Question
 $\pi$


text:
# How Strong is Gravity?

1. Jun 19, 2015

### robbertypob

Something is confusing me...

1. If I drop a cannonball and a feather in a vacuum they will fall to Earth at the same rate. The mass and size of the objects do not appear to be affecting the 'strength' of the gravitational force acting upon it.

2. The Earth's gravity is stronger than the Moon's, because the Earth is more massive.

How can both these statements be true? I must be missing something. It looks like mass doesn't matter in the first statement but in the second statement it does?!

2. Jun 19, 2015

### phinds

What you are missing is that the feather moves the Earth less than the cannonball. We don't normally think of the Earth as moving in that transaction, but it does (albeit in a a trivial and in all likelihood unmeasurable amount). THAT's where the mass matters. It would be more obvious with the moon. If the moon were not orbiting the Earth, but were to magically appear (defying all laws of physics) in its current position but not orbiting, it and the Earth would fall towards each other, but the Earth would move more slowly than the moon and they would meet up closer to the starting point of the Earth than the starting point of the moon (LOTS closer, actually).

3. Jun 19, 2015

### Staff: Mentor

Mass does matter, but it takes more force to accelerate the cannonball than the feather. If they accelerate at the same rate, the forces must be unequal. The relationship if F=ma, force equals mass times acceleration. Therefore F/m for the cannonball is the same as F/m for the feather.

Does that make sense to,you?

4. Jun 19, 2015

### robbertypob

OK, let me see...

So, gravity is calculated as Force = Mass x Acceleration. The cannonball has a greater mass but that means it has a lower acceleration. The feather has a lesser mass but that means it has a higher acceleration.
Any differences in mass are offset by the difference in acceleration, meaning they appear to fall at exactly the same rate.

Is that right? Have I understood it?

5. Jun 19, 2015

### Staff: Mentor

Nope. F=ma is Newton's second law. It works for any force, not just gravity

"difference in acceleration, meaning they appear to fall at exactly the same rate.". That sentence is self contradictory. If there are differences in acceleration, then they don't fall at the same rate. The feather and the cannonball have the same acceleration.

6. Jun 19, 2015

### robbertypob

Thanks for explaining but I just don't understand.

I don't understand how greater mass means greater gravity and yet the cannonball falls at the same rate as the feather.
If the cannonball is more massive than the feather then it must have more gravity and therefore the gravitational attraction between the Earth and the cannonball is stronger than between the Earth and the feather. It just seems to me that the cannonball should be falling more quickly than the feather, even if that difference is marginal.

7. Jun 19, 2015

### phinds

Think of inertia. It's harder to get the cannonball moving, which offsets its "greater gravity" as you put it, so it falls at the same rate. It's all there in the math but perhaps the inertia consideration will make it more understandable to you.

8. Jun 19, 2015

### A.T.

Two cannonballs have greater mass than one cannonball. Should two cannonballs fall faster if glued to one object then when they are separate?

9. Jun 19, 2015

### robbertypob

Thanks for helping me to understand this.

I think I get it now - the cannonball has a greater mass but a slower acceleration and therefore falls at the same rate.

10. Jun 19, 2015

### phinds

No, rate and acceleration are the same thing. It does NOT have a slower acceleration (or a faster one) or else it would not fall at the same rate as the feather.

11. Jun 19, 2015

### robbertypob

My brain hurts! I need to read up on some more basic concepts before trying to comfortably understand this.

12. Jun 19, 2015

### Bandersnatch

Let's get back to the basics here.

Acceleration (the $a$ in $F=ma$) IS the rate of falling. The $m$ in $F=ma$ tells you that the higher the mass of the object whose motion you're considering, the higher force is required to move it. You'd normally call this $m$ inertia.

So for two objects of different masses, say $m_1$ and $m_2$, to fall at the same rate $a$, the force $F$ must be higher or lower proportional to the difference in masses.
Say, $m_1=1kg$ and $m_2=2kg$. You need twice the force for $m_2$ to accelerate it at the same rate as you would $m_1$.

This works for all forces. The $F$ in $F=ma$ doesn't tell you anything about which kind of force is it.

Now, look at the force in question. The force of gravity (law of gravity) is $F_G=GMm/R^2$. This force is a force that always acts between two bodies. These two bodies are represented by their masses $M$ and $m$. $G$ is a constant that is not very important here, $R$ is the distance between the bodies.

Notice, that the force of gravity increases if you increase either of the masses.

So, while $F=ma$ means that mass increases inertia, making it harder to accelerate, the law of gravity tells you that the force on an object being accelerated increases by the same proportion as inertia.

You can combine the two equations to get:
$ma=GMm/R^2$
You divide both sides by $m$ to get the equation of acceleration:
$a=GM/R^2$
which is independent of the mass $m$! It still is dependent on the mass $M$. So all objects, regardless of what exactly their mass $m$ is, will fall at the rate $a$ determined solely by the mass of the body $M$.

In all of this, an important assumption is made, that $M$ is much, much greater than any mass $m$ whose motion you're looking at.
This is to ensure that you don't have to worry about the mass $M$ being accelerated towards the smaller mass $m$. This is a good approximation for dropping spoons, feathers and cannon balls onto planets and moons, but in principle the Earth does get accelerated to a cannon ball faster than to a feather, and the two don't fall at exactly the same time as a result.
However, this kind of analysis is more involved, so until you understand the basic one described above, leave it be. All you need to remember is that there is a deeper level to falling at the same rate, that is negligible for small object falling onto a huge one.

13. Jun 19, 2015

### robbertypob

Thank you for such a detailed and understandable explanation! So the actual force of gravity is greater on the cannonball than the feather - that is fascinating to me.

14. Jun 19, 2015

### rumborak

I get the impression that a good amount of confusion comes from using the very vague term "rate". In reading replies I've seen people mean both the acceleration and the speed by it.
We should stick to the much less ambiguous terms "speed" and "acceleration".

15. Jun 19, 2015

### rumborak

BTW, this might make it a bit more intuitive why it is that way:
Imagine two feathers in a vacuum, a meter apart from each other. You drop them simultaneously. Obviously, they will come to the ground at the same speed. Now, bring the two feathers to each other, i.e. glue them together. Drop them now. What would happen? The same thing really. It's not as if one feather is trying to fall faster than the other, dragging the other with it. So, the two-feather blob takes the same time to drop as the two separate feathers. And of course twice the force was applied to the two-feather object, after all it's two of them now.
Well, now take 1,000 feathers and bunch them all together, and you got yourself a cannonball.

16. Jun 19, 2015

### robbertypob

Yes I see what you mean!

Gravity is really interesting to me. It just seems to behave in a bizarre way (i.e. the force is actually stronger on the cannonball than the feather), although of course it makes sense when you look at the equations. Do we understand how gravity works at a molecular level? What specifically is happening to attract the two masses towards each other?

17. Jun 19, 2015

### rumborak

Let me ask you, why is it bizarre? After all, both the feather and the cannonball are made of the same particles, just a different amount. Each particle on its own experiences the gravitational pull, and so each of them falls with the same acceleration. It doesn't matter whether those particles stick together or not.

18. Jun 19, 2015

### Bandersnatch

Pfft. Such lack of accuracy. On PF, of all places!
A broiler chicken has some 9000 feathers on average, which altogether weigh about 74 grams.
The lightest cannon ball according to Borgard's ordnance standards in the 18th century British Navy weighs 4 pounds.
That means you need on the order of 200 000 feathers for a cannonball.

19. Jun 19, 2015

### robbertypob

Ah yes it's helpful for me to think of it in that way - gravity is actually acting on each particle individually. It doesn't make any difference at all (at the level of cannonballs and feathers) what the overall mass of the object is. If the mass is greater then the force of gravity will be greater.

I'm sure it's not bizarre but it seems so to me because I can't think of any other force that behaves in that manner. If I were to have a marble in one hand and a baseball in the other and threw them both at a wall I would have to think very carefully about how much force to apply to both if I wanted them to reach the wall at the same time due to the differences in mass, but this doesn't matter with gravity. They will arrive at their destination at the same time. Wow!

20. Jun 19, 2015

### phinds

Nitpicker ... them's a lot of chickens


text:
## Statement

difficulty

A competitor shoots a plate by using a rifle of 2,5 Kg on a skeet shooting competition. Knowing that the ball is 23 g and it is shot horizontally at 350 m/s, what is the recoil of the rifle?

## Solution

Data

 mass of the rifle velocity before the shot velocity of the rifle after the shot mass of the bullet velocity of the bullet before the shot velocity of the bullet after the shot

Resolution

If we consider the bullet and the rifle as a system, the outer forces to it are cancel since, initially, both are held by the handle, therefore the conservation linear momentum principle, momentums before the shot (${\stackrel{\to }{p}}_{i}$) and after the shot (${\stackrel{\to }{p}}_{f}$) are the same:

For this reason  we are going to calculate the linear momentum of the system at that time, having into account the momentum of a system is equivalent to the sum of momentums of the bodies separatly:

 Momentum before the shot Momentum after the shot

Matching both linear momentums and solving for ${\stackrel{\to }{v}}_{fr}$, we can calculate the rifle´s  speed of recoil after the shot.

## Formulas worksheet

These are the main formulas that you must know to solve this exercise. If you are not clear about their meaning, we recommend you to check the theory in the corresponding sections. Furthermore, you will find in them, under the Formulas tab, the codes that will allow you to integrate these equations in external programs like Word or Mathematica.

Formulas
Related sections
$\stackrel{\to }{p}=m·\stackrel{\to }{v}$


text:
# 前川 泰則

プロフィール
(日本語)

(英語)
My research area is mathematical analysis of partial differential equations related to fluid mechanics. As for education, I make much account for basic things for undergraduate course. For graduate course, I make an effort so that students can achieve the ability to analyze objects mathematically by themselves. I will contribute to society by revealing mathematical structures underlying various natural phenomena.
researchmap URL
https://researchmap.jp/7000006958

(日本語)

(英語)
My research field is mathematical analysis of partial differential equations related to fluid mechanics, such as the Navier-Stokes equations for viscous incompressible flows, the vorticity equations, etc. I am also interested in real analysis, Fourier analysis, and functional analysis, which are fundamental tools in studying partial differential equations.

(日本語)

(英語)
My current research topics are: (1) Inviscid limit problem for viscous incompressible flows (2) Stability and instability of boundary layers in fluid dynamics, such as Prandtl boundary layer and Ekman boundary layer (3) Stability of viscous flows in two-dimensional exterior domains, such as Oseen flows and Lamb-Oseen flows (4) Mathematical analysis of flows around rotating obstacle (5) Study of nonlinear partial differential equations under scaling invariance

キーワード(日本語) キーワード(英語)

Yasunori Maekawa, Hideyuki Miura, and Christophe Prange Yasunori Maekawa, Hideyuki Miura, and Christophe Prange Yasunori Maekawa, Hideyuki Miura, and Christophe Prange Local energy weak solutions for the Navier-Stokes equations in the half-spacenergy weak solutions for the Navier-Stokes equations in the half-space Local energy weak solutions for the Navier-Stokes equations in the half-spacenergy weak solutions for the Navier-Stokes equations in the half-space Local energy weak solutions for the Navier-Stokes equations in the half-space Commun. Math. Phys., 367, 517-580 Commun. Math. Phys., 367, 517-580 Commun. Math. Phys., 367, 517-580 2019 英語 研究論文(学術雑誌)
Isabelle Gallagher, Mitsuo Higaki, and Yasunori Maekawa Isabelle Gallagher, Mitsuo Higaki, and Yasunori Maekawa Isabelle Gallagher, Mitsuo Higaki, and Yasunori Maekawa On stationary two-dimensional flows around a fast rotating disk On stationary two-dimensional flows around a fast rotating disk On stationary two-dimensional flows around a fast rotating disk Mathematische Nachrichten, 292, 273-308 Mathematische Nachrichten, 292, 273-308 Mathematische Nachrichten, 292, 273-308 2019 英語 研究論文(学術雑誌)
Yasunori Maekawa, Hideyuki Miura Yasunori Maekawa, Hideyuki Miura Yasunori Maekawa, Hideyuki Miura On isomorphism for the space of solenoidal vector fields and its application to the Stokes problem On isomorphism for the space of solenoidal vector fields and its application to the Stokes problem On isomorphism for the space of solenoidal vector fields and its application to the Stokes problem SIAM Journal on Mathematical Analysis, 50, 339-353 SIAM Journal on Mathematical Analysis, 50, 339-353 SIAM Journal on Mathematical Analysis, 50, 339-353 2018/01 英語 研究論文(学術雑誌)
Mitsuo Higaki, Yasunori Maekawa, Yuu Nakahara Mitsuo Higaki, Yasunori Maekawa, Yuu Nakahara Mitsuo Higaki, Yasunori Maekawa, Yuu Nakahara On stationary Navier-Stokes flows around a rotating obstacle in two-dimensions On stationary Navier-Stokes flows around a rotating obstacle in two-dimensions On stationary Navier-Stokes flows around a rotating obstacle in two-dimensions Archive for Rational Mechanics and Analysis, 228, 603-651 Archive for Rational Mechanics and Analysis, 228, 603-651 Archive for Rational Mechanics and Analysis, 228, 603-651 2018 英語 研究論文(学術雑誌)
David Gerard-Varet, Yasunori Maekawa, and Nader Masmoudi David Gerard-Varet, Yasunori Maekawa, and Nader Masmoudi David Gerard-Varet, Yasunori Maekawa, and Nader Masmoudi Gevrey stability of Prandtl expansions for 2D Navier-Stokes flows Gevrey stability of Prandtl expansions for 2D Navier-Stokes flows Gevrey stability of Prandtl expansions for 2D Navier-Stokes flows Duke Math. J., 167, 13, 2531-2631 Duke Math. J., 167, 13, 2531-2631 Duke Math. J., 167, 13, 2531-2631 2018 英語 研究論文(学術雑誌)
Yasunori Maekawa, Hideyuki Miura Yasunori Maekawa, Hideyuki Miura Yasunori Maekawa, Hideyuki Miura On domain of Poisson operators and factorization for divergence form elliptic operators On domain of Poisson operators and factorization for divergence form elliptic operators On domain of Poisson operators and factorization for divergence form elliptic operators Manuscripta Mathematica, 152, 459-512 Manuscripta Mathematica, 152, 459-512 Manuscripta Mathematica, 152, 459-512 2017/03 英語 研究論文(学術雑誌)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa On stability of steady circular flows in a two-dimensional exterior disk On stability of steady circular flows in a two-dimensional exterior disk On stability of steady circular flows in a two-dimensional exterior disk Archive for Rational Mechanics and Analysis, 225, 287-374 Archive for Rational Mechanics and Analysis, 225, 287-374 Archive for Rational Mechanics and Analysis, 225, 287-374 2017 英語 研究論文(学術雑誌)
Mitsuo Higaki, Yasunori Maekawa, Yuu Nakahara Mitsuo Higaki, Yasunori Maekawa, Yuu Nakahara Mitsuo Higaki, Yasunori Maekawa, Yuu Nakahara On the two-dimensional steady Navier- Stokes equations related to flows around a rotating obstacle On the two-dimensional steady Navier- Stokes equations related to flows around a rotating obstacle On the two-dimensional steady Navier- Stokes equations related to flows around a rotating obstacle “Workshop on the Boltzmann Equation, Microlocal Analysis and Related Topics”. RIMS Kokyuroku Bessatsu, B67, 107-122 “Workshop on the Boltzmann Equation, Microlocal Analysis and Related Topics”. RIMS Kokyuroku Bessatsu, B67, 107-122 “Workshop on the Boltzmann Equation, Microlocal Analysis and Related Topics”. RIMS Kokyuroku Bessatsu, B67, 107-122 2017 英語 研究論文(学術雑誌)
Yasunori Maekawa, Jonas Sauer Yasunori Maekawa, Jonas Sauer Yasunori Maekawa, Jonas Sauer Maximal regularity of the time-periodic Stokes operator on unbounded and bounded domains Maximal regularity of the time-periodic Stokes operator on unbounded and bounded domains Maximal regularity of the time-periodic Stokes operator on unbounded and bounded domains Journal of the Mathematical Society of Japan, 69, 1403-1429 Journal of the Mathematical Society of Japan, 69, 1403-1429 Journal of the Mathematical Society of Japan, 69, 1403-1429 2017 英語 研究論文(学術雑誌)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa Remark on stability of scale-critical stationary flows in a two-dimensional exterior disk Remark on stability of scale-critical stationary flows in a two-dimensional exterior disk Remark on stability of scale-critical stationary flows in a two-dimensional exterior disk Mathematics for Nonlinear Phenomena: Analysis and Computation - In Honor of Yoshikazu Giga's 60th Birthday, Springer Proceedings in Mathematics & Statistics, 215, 105-130 Mathematics for Nonlinear Phenomena: Analysis and Computation - In Honor of Yoshikazu Giga's 60th Birthday, Springer Proceedings in Mathematics & Statistics, 215, 105-130 Mathematics for Nonlinear Phenomena: Analysis and Computation - In Honor of Yoshikazu Giga's 60th Birthday, Springer Proceedings in Mathematics & Statistics, 215, 105-130 2017 英語 研究論文(国際会議プロシーディングス)
Humiya Kosaka, Yasunori Maekawa Humiya Kosaka, Yasunori Maekawa Humiya Kosaka, Yasunori Maekawa On vorticity formulation for viscous incompressible flows in R^3_+ On vorticity formulation for viscous incompressible flows in R^3_+ On vorticity formulation for viscous incompressible flows in R3<inf>+</inf> Advances in Mathematical Fluid Mechanics, Recent Development of Mathematical Fluid Mechanics, 351-374 Advances in Mathematical Fluid Mechanics, Recent Development of Mathematical Fluid Mechanics, 351-374 Advances in Mathematical Fluid Mechanics, Recent Development of Mathematical Fluid Mechanics, 351-374 2016/01 英語 研究論文(国際会議プロシーディングス)
Yasunori Maekawa, Hideyuki Miura Yasunori Maekawa, Hideyuki Miura Yasunori Maekawa, Hideyuki Miura On Poisson operators and Dirichlet-Neumann maps in Hs for divergence form elliptic operators with Lipschitz coefficients On Poisson operators and Dirichlet-Neumann maps in Hs for divergence form elliptic operators with Lipschitz coefficients On Poisson operators and Dirichlet-Neumann maps in Hs for divergence form elliptic operators with Lipschitz coefficients Transactions of the American Mathematical Society, 368, 9, 6227-6252 Transactions of the American Mathematical Society, 368, 9, 6227-6252 Transactions of the American Mathematical Society, 368, 9, 6227-6252 2016/01 英語 研究論文(学術雑誌)
Yasunori Maekawa, Anna Mazzucato Yasunori Maekawa, Anna Mazzucato Yasunori Maekawa, Anna Mazzucato The Inviscid Limit and Boundary Layers for Navier-Stokes Flows The Inviscid Limit and Boundary Layers for Navier-Stokes Flows The Inviscid Limit and Boundary Layers for Navier-Stokes Flows Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer International Publishing Switzerland 2016 Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer International Publishing Switzerland 2016 Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer International Publishing Switzerland 2016 2016 英語 研究論文(学術雑誌)
Thierry Gallay, Yasunori Maekawa Thierry Gallay, Yasunori Maekawa Thierry Gallay, Yasunori Maekawa Existence and Stability of Viscous Vortices Existence and Stability of Viscous Vortices Existence and Stability of Viscous Vortices Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer International Publishing Switzerland 2016 Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer International Publishing Switzerland 2016 Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer International Publishing Switzerland 2016 2016 英語 研究論文(学術雑誌)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa On asymptotic stability of global solutions in weak L^2 space for the two-dimensional Navier-Stokes equations On asymptotic stability of global solutions in weak L^2 space for the two-dimensional Navier-Stokes equations On asymptotic stability of global solutions in the weak L2 space for the two-dimensional Navier-Stokes equations Analysis: International mathematical journal of analysis and its applications, 35, 4, 245-257 Analysis: International mathematical journal of analysis and its applications, 35, 4, 245-257 Analysis, 35, 4, 245-257 2015/01 英語 研究論文(国際会議プロシーディングス)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa Remark on $C_0$ -semigroups with scaling invariance Remark on $C_0$ -semigroups with scaling invariance Remark on $C_0$ -semigroups with scaling invariance Nonlinear Dynamics in Partial Differential Equations, ASPM 64, 337-345 Nonlinear Dynamics in Partial Differential Equations, ASPM 64, 337-345 Nonlinear Dynamics in Partial Differential Equations, ASPM 64, 337-345 2015 英語 研究論文(国際会議プロシーディングス)
Hiroshi Takeda, Yasunori Maekawa, Shuichi Kawashima Hiroshi Takeda, Yasunori Maekawa, Shuichi Kawashima Hiroshi Takeda, Yasunori Maekawa, Shuichi Kawashima Asymptotic profile of solutions to a hyperbolic Cahn-Hilliard equation Asymptotic profile of solutions to a hyperbolic Cahn-Hilliard equation Asymptotic profile of solutions to a hyperbolic Cahn-Hilliard equation Bulletin of the Institute of Mathematics, Academia Sinica, 10, 479-539 Bulletin of the Institute of Mathematics, Academia Sinica, 10, 479-539 Bulletin of the Institute of Mathematics, Academia Sinica, 10, 479-539 2015 英語 研究論文(学術雑誌)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half plane On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half plane On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane Comm. Pure and Applied Math., 67, 7, 1045-1128 Comm. Pure and Applied Math., 67, 7, 1045-1128 Communications on Pure and Applied Mathematics, 67, 7, 1045-1128 2014/01 英語 研究論文(学術雑誌)
Yasunori Maekawa, Hideyuki Miura Yasunori Maekawa, Hideyuki Miura Yasunori Maekawa, Hideyuki Miura Remark on the Helmholtz decomposition in domains with noncompact boundary Remark on the Helmholtz decomposition in domains with noncompact boundary Remark on the Helmholtz decomposition in domains with noncompact boundary Math. Ann., 359, 3-4, 1077-1095 Math. Ann., 359, 3-4, 1077-1095 Mathematische Annalen, 359, 3-4, 1077-1095 2014/01 英語 研究論文(学術雑誌)
Yoshikazu Giga, Pen-Yuan Hsu, Yasunori Maekawa Yoshikazu Giga, Pen-Yuan Hsu, Yasunori Maekawa Yoshikazu Giga, Pen Yuan Hsu, Yasunori Maekawa A Liouville theorem for the planer Navier-Stokes equations with the no-slip boundary condition and its application to a geometric regularity criterion A Liouville theorem for the planer Navier-Stokes equations with the no-slip boundary condition and its application to a geometric regularity criterion A Liouville Theorem for the Planer Navier-Stokes Equations with the No-Slip Boundary Condition and Its Application to a Geometric Regularity Criterion Comm. Partial Differential Equations, 39, 10, 1906-1935 Comm. Partial Differential Equations, 39, 10, 1906-1935 Communications in Partial Differential Equations, 39, 10, 1906-1935 2014/01 英語 研究論文(学術雑誌)
Thierry Gallay, Yasunori Maekawa Thierry Gallay, Yasunori Maekawa Thierry Gallay, Yasunori Maekawa Long-time asymptotics for two-dimensional exterior flows with small circulation at infinity Long-time asymptotics for two-dimensional exterior flows with small circulation at infinity Long-Time asymptotics for two-dimensional exterior flows with small circulation at infinity Analysis and PDE, 6, 4, 973-991 Analysis and PDE, 6, 4, 973-991 Analysis and PDE, 6, 4, 973-991 2013/12 英語 研究論文(学術雑誌)
Yasunori Maekawa, Hideyuki Miura Yasunori Maekawa, Hideyuki Miura Yasunori Maekawa, Hideyuki Miura On fundamental solutions for non-local parabolic equations with divergence free drift On fundamental solutions for non-local parabolic equations with divergence free drift On fundamental solutions for non-local parabolic equations with divergence free drift Advances in Mathematics, 247, 123-191 Advances in Mathematics, 247, 123-191 Advances in Mathematics, 247, 123-191 2013/11 英語 研究論文(学術雑誌)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa Solution formula for the vorticity equations in the half plane with application to high vorticity creation at zero viscosity limit Solution formula for the vorticity equations in the half plane with application to high vorticity creation at zero viscosity limit Solution formula for the vorticity equations in the half plane with application to high vorticity creation at zero viscosity limit Advances in Differential Equations, 18, 101-146 Advances in Differential Equations, 18, 101-146 Advances in Differential Equations, 18, 101-146 2013/09 英語 研究論文(学術雑誌)
Yasunori Maekawa, Hideyuki Miura Yasunori Maekawa, Hideyuki Miura Yasunori Maekawa, Hideyuki Miura Upper bounds for fundamental solutions to non-local diffusion equations with divergence free drift Upper bounds for fundamental solutions to non-local diffusion equations with divergence free drift Upper bounds for fundamental solutions to non-local diffusion equations with divergence free drift Journal of Functional Analysis, 264, 10, 2245-2268 Journal of Functional Analysis, 264, 10, 2245-2268 Journal of Functional Analysis, 264, 10, 2245-2268 2013 英語 研究論文(学術雑誌)
Pen-Yuan Hsu, Yasunori Maekawa Pen-Yuan Hsu, Yasunori Maekawa Pen-Yuan Hsu, Yasunori Maekawa On nonexistence for stationary solutions to the Navier-Stokes equations with a linear strain On nonexistence for stationary solutions to the Navier-Stokes equations with a linear strain On nonexistence for stationary solutions to the Navier-Stokes equations with a linear strain Journal of Mathematical Fluid Mechanics, 15, 2, 317-333 Journal of Mathematical Fluid Mechanics, 15, 2, 317-333 Journal of Mathematical Fluid Mechanics, 15, 2, 317-333 2013 英語 研究論文(学術雑誌)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa Recent developments in the mathematical analysis of the Burgers vortex Recent developments in the mathematical analysis of the Burgers vortex Recent developments in the mathematical analysis of the Burgers vortex Gakuto International Series, Mathematical Sciences and Applications, 35, 103-115 Gakuto International Series, Mathematical Sciences and Applications, 35, 103-115 Gakuto International Series, Mathematical Sciences and Applications, 35, 103-115 2012 英語 研究論文(国際会議プロシーディングス)
Yoshiyuki Kagei, Yasunori Maekawa Yoshiyuki Kagei, Yasunori Maekawa Yoshiyuki Kagei, Yasunori Maekawa On asymptotic behaviors of solutions to parabolic systems modelling chemotaxis On asymptotic behaviors of solutions to parabolic systems modelling chemotaxis On asymptotic behaviors of solutions to parabolic systems modelling chemotaxis Journal of Differential Equations, 253, 11, 2951-2992 Journal of Differential Equations, 253, 11, 2951-2992 Journal of Differential Equations, 253, 11, 2951-2992 2012 英語 研究論文(学術雑誌)
Thierry Gallay, Yasunori Maekawa Thierry Gallay, Yasunori Maekawa Thierry Gallay, Yasunori Maekawa Three-dimensional stability of Burgers vortices Three-dimensional stability of Burgers vortices Three-dimensional stability of Burgers vortices Communications in Mathematical Physics, 302, 2, 477-511 Communications in Mathematical Physics, 302, 2, 477-511 Communications in Mathematical Physics, 302, 2, 477-511 2011 英語 研究論文(学術雑誌)
Yoshiyuki Kagei, Yasunori Maekawa Yoshiyuki Kagei, Yasunori Maekawa Yoshiyuki Kagei, Yasunori Maekawa Asymptotic behaviors of solutions to evolution equations in the presenceof translation and scaling invariance Asymptotic behaviors of solutions to evolution equations in the presenceof translation and scaling invariance Asymptotic behaviors of solutions to evolution equations in the presenceof translation and scaling invariance Journal of Functional Analysis, 260, 10, 3036-3096 Journal of Functional Analysis, 260, 10, 3036-3096 Journal of Functional Analysis, 260, 10, 3036-3096 2011 英語 研究論文(学術雑誌)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa Spectral properties of the linearization at the Burgers vortex in the high rotation limit Spectral properties of the linearization at the Burgers vortex in the high rotation limit Spectral properties of the linearization at the Burgers vortex in the high rotation limit Journal of Mathematical Fluid Mechanics, 13, 4, 515-532 Journal of Mathematical Fluid Mechanics, 13, 4, 515-532 Journal of Mathematical Fluid Mechanics, 13, 4, 515-532 2011 英語 研究論文(学術雑誌)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa On Gaussian decay estimates of solutions to some linear elliptic equations and their applications On Gaussian decay estimates of solutions to some linear elliptic equations and their applications On Gaussian decay estimates of solutions to some linear elliptic equations and their applications Z. Angew. Math. Phys., 62, 1, 1-30 Z. Angew. Math. Phys., 62, 1, 1-30 Z. Angew. Math. Phys., 62, 1, 1-30 2011 英語 研究論文(学術雑誌)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa Existence of asymmetric Burgers vortices and their asymptotic behavior at large circulations Existence of asymmetric Burgers vortices and their asymptotic behavior at large circulations Existence of asymmetric Burgers vortices and their asymptotic behavior at large circulations Mathematical Models and Methods in Applied Sciences, 19, 5, 669-705 Mathematical Models and Methods in Applied Sciences, 19, 5, 669-705 Mathematical Models and Methods in Applied Sciences, 19, 5, 669-705 2009 英語 研究論文(学術雑誌)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa On the existence of Burgers vortices for high Reynolds numbers On the existence of Burgers vortices for high Reynolds numbers On the existence of Burgers vortices for high Reynolds numbers Journal of Mathematical Analysis and Applications, 349, 1, 181-200 Journal of Mathematical Analysis and Applications, 349, 1, 181-200 Journal of Mathematical Analysis and Applications, 349, 1, 181-200 2009 英語 研究論文(学術雑誌)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa On spatial decay estimates for derivatives of vorticities with application to large time behavior of the two dimensional Navier-Stokes flow On spatial decay estimates for derivatives of vorticities with application to large time behavior of the two dimensional Navier-Stokes flow On spatial decay estimates for derivatives of vorticities with application to large time behavior of the two dimensional Navier-Stokes flow Journal of Mathematical Fluid Mechanics, 10, 1, 89-105 Journal of Mathematical Fluid Mechanics, 10, 1, 89-105 Journal of Mathematical Fluid Mechanics, 10, 1, 89-105 2008 英語 研究論文(学術雑誌)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa A lower bound for fundamental solutions of the heat convection equations A lower bound for fundamental solutions of the heat convection equations A lower bound for fundamental solutions of the heat convection equations Archive for Rational Mechanics and Analysis, 189, 1, 45-58 Archive for Rational Mechanics and Analysis, 189, 1, 45-58 Archive for Rational Mechanics and Analysis, 189, 1, 45-58 2008 英語 研究論文(学術雑誌)
Yasunori Maekawa Yasunori Maekawa Yasunori Maekawa On a free boundary problem of viscous incompressible flows On a free boundary problem of viscous incompressible flows On a free boundary problem of viscous incompressible flows Interfaces and Free Boundaries, 9, 549-589 Interfaces and Free Boundaries, 9, 549-589 Interfaces and Free Boundaries, 9, 549-589 2007 英語 研究論文(学術雑誌)
Yasunori Maekawa, Yutaka Terasawa Yasunori Maekawa, Yutaka Terasawa Yasunori Maekawa, Yutaka Terasawa The Navier-Stokes equations with initial data in uniformly local spaces The Navier-Stokes equations with initial data in uniformly local spaces The Navier-Stokes equations with initial data in uniformly local spaces Differential Integral Equations, 19, 369-400 Differential Integral Equations, 19, 369-400 Differential Integral Equations, 19, 369-400 2006 英語 研究論文(学術雑誌)

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タイトル タイトル(日本語) タイトル(英語) 会議名 会議名(日本語) 会議名(英語) 主催者 主催者(日本語) 主催者(英語) 開催年月日 記述言語 会議種別
Navier-Stokes方程式における非粘性極限問題とPrandtl 境界層展開[招待あり] Navier-Stokes方程式における非粘性極限問題とPrandtl 境界層展開 [招待あり] Navier-Stokes方程式における非粘性極限問題とPrandtl 境界層展開 [招待あり] 2019年度日本数学会年会 2019年度日本数学会年会 2019年度日本数学会年会 2019/03 日本語 口頭発表(招待・特別)
On Finn’s starting problem for the two-dimensional Navier-Stokes flow[招待あり] On Finn’s starting problem for the two-dimensional Navier-Stokes flow [招待あり] On Finn’s starting problem for the two-dimensional Navier-Stokes flow [招待あり] Maximal regularity and nonlinear PDE Maximal regularity and nonlinear PDE Maximal regularity and nonlinear PDE 2019/03 英語 口頭発表(招待・特別)
Stability of the shear boundary layer in the inviscid limit for the Navier-Stokes flow[招待あり] Stability of the shear boundary layer in the inviscid limit for the Navier-Stokes flow [招待あり] Stability of the shear boundary layer in the inviscid limit for the Navier-Stokes flow [招待あり] Workshop on Theory of Boundary Layers and Related Topics Workshop on Theory of Boundary Layers and Related Topics Workshop on Theory of Boundary Layers and Related Topics 2018/12 英語 口頭発表(招待・特別)
On inviscid limit around shear boundary layers for the steady Navier-Stokes equations[招待あり] On inviscid limit around shear boundary layers for the steady Navier-Stokes equations [招待あり] On inviscid limit around shear boundary layers for the steady Navier-Stokes equations [招待あり] RIMS共同研究：流体と気体の数学解析 RIMS共同研究：流体と気体の数学解析 RIMS workshop: Mathematical Analysis in Fluid and Gas Dynamics 2018/07 英語 口頭発表(招待・特別)
Axisymmetrization and boundary layer of the two-dimensional exterior flow around a fast rotating disk[招待あり] Axisymmetrization and boundary layer of the two-dimensional exterior flow around a fast rotating disk [招待あり] Axisymmetrization and boundary layer of the two-dimensional exterior flow around a fast rotating disk [招待あり] Workshop on Hyperbolic and Dispersive PDEs in Fukuoka, Kyushu University, Fukuoka Workshop on Hyperbolic and Dispersive PDEs in Fukuoka, Kyushu University, Fukuoka Workshop on Hyperbolic and Dispersive PDEs in Fukuoka, Kyushu University, Fukuoka 2018/02/10 口頭発表(招待・特別)
On stability of shear flows with high Reynolds number[招待あり] On stability of shear flows with high Reynolds number [招待あり] On stability of shear flows with high Reynolds number [招待あり] 奈良女子大学PDEセミナー，奈良女子大学，奈良 奈良女子大学PDEセミナー，奈良女子大学，奈良 PDE Seminar at Nara Wemen University 2018/02/07 公開講演、セミナー、チュートリアル、講習、講義
On stability of some nearly inviscid flows[招待あり] On stability of some nearly inviscid flows [招待あり] On stability of some nearly inviscid flows [招待あり] The 35th Kyushu Symposium on PDEs The 35th Kyushu Symposium on PDEs The 35th Kyushu Symposium on PDEs 2018/01/29 口頭発表(招待・特別)

On axisymmetrization and boundary layer in the two-dimensional exterior flow around a fast rotating obstacle[招待あり] On axisymmetrization and boundary layer in the two-dimensional exterior flow around a fast rotating obstacle [招待あり] On axisymmetrization and boundary layer in the two-dimensional exterior flow around a fast rotating obstacle, HMAセミナー・冬の研究会2018，広島大学，広島，2018.1.19. [招待あり] HMAセミナー・冬の研究会2018，広島大学，広島 HMAセミナー・冬の研究会2018，広島大学，広島 HMAセミナー・冬の研究会2018，広島大学，広島 2018/01/19 口頭発表(招待・特別)
On stationary two-dimensional flows around a fast rotating disk[招待あり] On stationary two-dimensional flows around a fast rotating disk [招待あり] On stationary two-dimensional flows around a fast rotating disk [招待あり] International Workshop on the Multi-phase Flow; Analysis, Modeling and Numerics. Waseda University, 東京 International Workshop on the Multi-phase Flow; Analysis, Modeling and Numerics. Waseda University, 東京 International Workshop on the Multi-phase Flow; Analysis, Modeling and Numerics. Waseda University, 東京 2017/11/28 口頭発表(招待・特別)
On the Prandtl boundary layer expansion in a Gevrey class[招待あり] On the Prandtl boundary layer expansion in a Gevrey class [招待あり] On the Prandtl boundary layer expansion in a Gevrey class [招待あり] Fluids,Dispersion and Blow-up. IHP, Paris, France Fluids,Dispersion and Blow-up. IHP, Paris, France Fluids,Dispersion and Blow-up. IHP, Paris, France 2017/10/13 口頭発表(招待・特別)
On the stability of the physically reasonable solution to the two-dimensional Navier-Stokes equations[招待あり] On the stability of the physically reasonable solution to the two-dimensional Navier-Stokes equations [招待あり] On the stability of the physically reasonable solution to the two-dimensional Navier-Stokes equations [招待あり] Seminaire du Laboratoire Jacques-Louis Lions. Paris 6, France Seminaire du Laboratoire Jacques-Louis Lions. Paris 6, France Seminaire du Laboratoire Jacques-Louis Lions. Paris 6, France 2017/06/16 公開講演、セミナー、チュートリアル、講習、講義
On stability of physically reasonable solutions to the Navier-Stokes equations in a two-dimensional exterior domain[招待あり] On stability of physically reasonable solutions to the Navier-Stokes equations in a two-dimensional exterior domain [招待あり] On stability of physically reasonable solutions to the Navier-Stokes equations in a two-dimensional exterior domain [招待あり] Analysis Seminar. Max Planck Institute, Leipzig, Germany Analysis Seminar. Max Planck Institute, Leipzig, Germany Analysis Seminar. Max Planck Institute, Leipzig, Germany 2017/06/13 公開講演、セミナー、チュートリアル、講習、講義
On Prandtl expansion for the Navier-Stokes flows in the half plane[招待あり] On Prandtl expansion for the Navier-Stokes flows in the half plane [招待あり] On Prandtl expansion for the Navier-Stokes flows in the half plane [招待あり] MFO Workshop: Geophysical Fluid Dynamics. Oberwolfach, Germany MFO Workshop: Geophysical Fluid Dynamics. Oberwolfach, Germany MFO Workshop: Geophysical Fluid Dynamics. Oberwolfach, Germany 2017/05/08 口頭発表(招待・特別)
On stability of physically reasonable solutions to the Navier-Stokes equations in a two-dimensional exterior domain[招待あり] On stability of physically reasonable solutions to the Navier-Stokes equations in a two-dimensional exterior domain [招待あり] On stability of physically reasonable solutions to the Navier-Stokes equations in a two-dimensional exterior domain [招待あり] Analysis Seminar. Cergy University, France Analysis Seminar. Cergy University, France Analysis Seminar. Cergy University, France 2017/04/24 公開講演、セミナー、チュートリアル、講習、講義
On some stability results for two-dimensional exterior flows[招待あり] On some stability results for two-dimensional exterior flows [招待あり] On some stability results for two-dimensional exterior flows [招待あり] International Workshop on the Multi-Phase Flow; Analysis, Modeling and Numerics, 早稲田大学 International Workshop on the Multi-Phase Flow; Analysis, Modeling and Numerics, 早稲田大学 International Workshop on the Multi-Phase Flow; Analysis, Modeling and Numerics, Waseda University 2016/11/07 英語 口頭発表(招待・特別)
On Prandtl expansion in the inviscid limit for the Navier-Stokes flows in Gevrey class[招待あり] On Prandtl expansion in the inviscid limit for the Navier-Stokes flows in Gevrey class [招待あり] On Prandtl expansion in the inviscid limit for the Navier-Stokes flows in Gevrey class [招待あり] The 14th Linear and Nonlinear Waves, プラザ淡海, 滋賀 The 14th Linear and Nonlinear Waves, プラザ淡海, 滋賀 The 14th Linear and Nonlinear Waves 2016/11/02 日本語 口頭発表(招待・特別)
Gevrey stability of Prandtl expansion for the Navier-Stokes flows in the half plane[招待あり] Gevrey stability of Prandtl expansion for the Navier-Stokes flows in the half plane [招待あり] Gevrey stability of Prandtl expansion for the Navier-Stokes flows in the half plane [招待あり] 7th Euro-Japanese Workshop on Blow Up, Banach center (Bedlewo, Poland) 7th Euro-Japanese Workshop on Blow Up, Banach center (Bedlewo, Poland) 7th Euro-Japanese Workshop on Blow Up, Banach center (Bedlewo, Poland) 2016/09/05 英語 口頭発表(招待・特別)
On Prandtl expansion for the Navier-Stokes flows in the half plane On Prandtl expansion for the Navier-Stokes flows in the half plane On Prandtl expansion for the Navier-Stokes flows in the half plane RIMS研究集会「流体と気体の数学解析」, 京都大学数理解析研究所 RIMS研究集会「流体と気体の数学解析」, 京都大学数理解析研究所 RIMS workshop: Mathematical Analysis in Fluid and Gas Dynamics, RIMS 2016/07/06 英語 口頭発表(招待・特別)
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On viscous incompressible flows around a rotating obstacle in two-dimensions[招待あり] On viscous incompressible flows around a rotating obstacle in two-dimensions [招待あり] On viscous incompressible flows around a rotating obstacle in two-dimensions [招待あり] Workshop on the Boltzmann Equation, Microlocal Analysis and Related Topics, 京都大学 Workshop on the Boltzmann Equation, Microlocal Analysis and Related Topics, 京都大学 Workshop on the Boltzmann Equation, Microlocal Analysis and Related Topics, Kyoto University 2016/05/27 英語 口頭発表(招待・特別)

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2018年JMSJ論文賞 The 2018 JMSJ Outstanding Paper Prize 日本数学会 Mathematical Society of Japan 2018/03/19
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2018年JMSJ論文賞 日本数学会 2018/03/19
2019年度日本数学会春季賞 日本数学会 2019/03/18


text:
# Math Help - Algebraic Simplification

1. ## Algebraic Simplification

Show that $\frac{(-1+(a)^2)}{(2-y)}=\frac{17}{(2+y)^3}$ where

$a=\frac{(1-x)}{(2-y)}$ and also $x^2 - y^2 - 2x +4y -20 =0$

Can anyone do? lol
p.s. This is the end simplification of another problem which I just don't know how to do.

Looking in particular to learn useful methods to do this efficiently for future reference.

Edit: Still need help...

3. ## Re: Algebraic Simplification

Sorry, it's

Show that (-1+(a)^2)/(2-y)=17/(x+3)^3 where a=(1-x)/(2-y)

4. ## Re: Algebraic Simplification

Originally Posted by Umair
Sorry, it's

Show that (-1+(a)^2)/(2-y)=17/(x+3)^3 where a=(1-x)/(2-y)

This is manifestly untrue. Simply pick x = 1, y = 1. Then a = 0. The LHS does not equal the RHS.

5. ## Re: Algebraic Simplification

Simplify $\displaystyle -1+\left(\frac{1-x}{2-y}\right)^2$

Do you get the RHS?

6. ## Re: Algebraic Simplification

Oh man, I'm so sorry, let me re-type it LOL. I got the RHS denominator bit wrong...

It's supposed to be:

Show that (-1+(a)^2)/(2-y)=17/(2+y)^3 where a=(1-x)/(2-y)

7. ## Re: Algebraic Simplification

Originally Posted by Umair
Oh man, I'm so sorry, let me re-type it LOL. I got the RHS denominator bit wrong...

It's supposed to be:

Show that (-1+(a)^2)/(2-y)=17/(2+y)^3 where a=(1-x)/(2-y)
Still not true. I'm always suspicious when I see x's on one side of an equation that don't seem to have any way of canceling, and then I don't see them on the RHS of the equation. I do get the following:

$\frac{a^{2}-1}{2-y}=-\frac{(1+x-y)(x+y-3)}{(2-y)^{3}}.$

8. ## Re: Algebraic Simplification

Nup, this should definitely be true... unless my University maths professor made the mistake lol

Urm, its actually the end bit of another problem. Maybe you could do the full problem?:

Equation of a circle centre (1, -2) and radius 5 is given by
x^2-y^2-2x+4y-20=0

Show that (d^2.y)/(d.x^2 )=17/(2+y)^3

At the original post, the LHS is d^2y/dx^2, and a = dy/dx.

9. ## Re: Algebraic Simplification

Originally Posted by Ackbeet
Still not true. I'm always suspicious when I see x's on one side of an equation that don't seem to have any way of canceling, and then I don't see them on the RHS of the equation. I do get the following:

$\frac{a^{2}-1}{2-y}=-\frac{(1+x-y)(x+y-3)}{(2-y)^{3}}.$
The Bottom part of the RHS is $(2+y)^3$, not $(2-y)^3$ :S

10. ## Re: Algebraic Simplification

Originally Posted by Umair
Nup, this should definitely be true... unless my University maths professor made the mistake lol

Urm, its actually the end bit of another problem. Maybe you could do the full problem?:

Equation of a circle centre (1, -2) and radius 5 is given by
x^2-y^2-2x+4y-20=0

Show that (d^2.y)/(d.x^2 )=17/(2+y)^3

At the original post, the LHS is d^2y/dx^2, and a = dy/dx.
not a circle ...

Equation of a circle centre (1, -2) and radius 5 is given by
$(x-1)^2 + (y+2)^2 = 25$

start from here ...

11. ## Re: Algebraic Simplification

Both of those equations are the same circle, lol

12. ## Re: Algebraic Simplification

Hmm. I've moved this thread to Calculus, now that we're talking about second derivatives obtained implicitly. I get the following:

$x^{2}-y^{2}-2x+4y-20=0$

$y'=\frac{x-1}{y-2},$

which appears to be your 'a'.

Using the quotient rule now, we get

$y''=\frac{1-(y')^{2}}{y-2}=\frac{(1+x-y)(x+y-3)}{(2-y)^{3}}.$

As skeeter pointed out, the equation

$x^{2}-y^{2}-2x+4y-20=0$

is not a circle. It is an hyperbola, since the signs of the $x^{2}$ and $y^{2}$ terms are opposite.

13. ## Re: Algebraic Simplification

Ohh quite a few things make sense now! Thank you very much! One question though, is your value of y' supposed to be that? I've re-done my calculation for dy/dx many times and checked that you didn't get my answer for a?

14. ## Re: Algebraic Simplification

I did my calculations in Post # 12 based on the (incorrect) hyperbola equation, not the (correct) circle equation exhibited by skeeter in post # 10. For the correct circle equation, I get the following:

$y'=\frac{1-x}{2+y},$

and

$y''=\frac{(2+y)(-1)-(1-x)(y')}{(2+y)^{2}}=-\frac{y+2-(x-1)y'}{(2+y)^{2}}=-\frac{5-2x+x^{2}+4y+y^{2}}{(y+2)^{3}}$

$=-\frac{x^{2}-2x+1+y^{2}+4y+4}{(y+2)^{3}}=-\frac{(x-1)^{2}+(y+2)^{2}}{(y+2)^{3}}.$

The x's don't cancel, and I didn't think they would.

15. ## Re: Algebraic Simplification

Originally Posted by Ackbeet
I did my calculations in Post # 12 based on the (incorrect) hyperbola equation, not the (correct) circle equation exhibited by skeeter in post # 10. For the correct circle equation, I get the following:

$y'=\frac{1-x}{2+y},$

and

$y''=\frac{(2+y)(-1)-(1-x)(y')}{(2+y)^{2}}=-\frac{y+2-(x-1)y'}{(2+y)^{2}}=-\frac{5-2x+x^{2}+4y+y^{2}}{(y+2)^{3}}$

$=-\frac{x^{2}-2x+1+y^{2}+4y+4}{(y+2)^{3}}=-\frac{(x-1)^{2}+(y+2)^{2}}{(y+2)^{3}}.$

The x's don't cancel, and I didn't think they would.

The fact it may be a circle or not is trivial in the question I think, but it's the "incorrect" equation we need to use.

For the equation of $x^2-y^2-2x+4y-20=0$,

I got:

$2x -2y\frac{dy}{dx} - 2 +4\frac{dy}{dx} = 0$

which simplifies to: $(2-y)\frac{dy}{dx}=1-x$

and finally $y'=\frac{dy}{dx}=\frac{1-x}{2-y}$

I'm not sure if this is correct or not ,but it's how I did it... please elaborate

Edit:

For $d^2y/dx^2$, I got the following:

$(1) \frac{d}{dx}((2-y)\frac{dy}{dx}+x-1)=0$

$(2) -(\frac{dy}{dx})^2+(2-y)\frac{d^2y}{dx^2}+1=0$

$(3) \frac{d^2y}{dx^2}=\frac{-1+\(dy/dx)^2}{2-y}$

SOMEHOW simplifies to:

$(4) \frac{d^2y}{dx^2}=\frac{x^2-y^2-2x+4y-3}{(2+y)^3} = \frac{17}{(2+y)^3}$

Any idea how (3) simplifies to Middle section of (4)?

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# MESON2016

2-7 June 2016
Auditorium Maximum
Europe/Warsaw timezone
14th International Workshop on Meson Production, Properties and Interaction
Home > Timetable > Session details > Contribution details

# Contribution plenary talk

Auditorium Maximum - Medium lecture hall (A,B)

# The investigation of $K^+\pi^-$, $\pi^+K^-$ and $\pi^+\pi^-$ atoms

## Speakers

• Valeriy YAZKOV

## Primary authors

• Valeriy YAZKOV (Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University)

DIRAC

## Abstract content

Theory, using Low Energy QCD, predicts with high precision the pion-pion and pion-kaon scattering lengths. There is accurate relation between $\pi^+\pi^-$ atom lifetime and a difference of S-wave pion-pion scattering lengths with isospin 0 ($a_0$) and 2 ($a_2$). Similar relation exists for $K^+\pi^-$ and $\pi^+K^-$ atoms lifetime and pion-kaon scattering lengths with isospin 1/2 ($a_{1/2}$) and 3/2 ($a_{3/2}$). Experiment DIRAC at CERN PS detects $345 \pm 61$ pairs from $K^+\pi^-$ and $\pi^+K^-$ atoms breakup. It allows to achieve the first observation of exotic atoms consisted of pion and kaon. Measured values of $\pi K$ atom lifetime and corresponding pion-kaon scattering length difference are presented. It is shown, that experimental accuracy for pion-kaon scattering length difference could be significantly improved with an experiment at SPS energy.

The experimental observation of the long-lived $\pi^+\pi^-$ atoms is shown. The scheme to use the obtained data for the long-lived atom lifetime measurement will be described. Possibility of the long-lived $\pi^+\pi^-$ atom Lamb shift measurement and evaluation of pion-pion scattering length new combination ($|2 \cdot a_0 + a_2|$) is discussed.


text:
# Synopsis: How Much Does a Cell Weigh?

Optical microscopes can be adapted to measure the mass of individual cells.

If you had to estimate the mass of the average human cell, you could cheat and look it up on Wikipedia, or do a back-of-the-envelope calculation with a few assumptions about the mass and volume of a person (a cylinder) and the size of cells (spheres). But researchers need more precise measures of a cell’s mass so they can detect small changes as it grows or responds to its environment. In Physical Review Letters, Kevin Phillips and colleagues at the Oregon Health & Science University in Portland report a simple way to measure the mass of individual red blood cells with a commercial microscope.

To do this, Phillips et al. have developed a technique called tomographic bright-field imaging, which reconstructs the 3D optical index of refraction of a microscopic object, like a cell or plastic bead, and then converts this information into the object’s mass. The team places a narrow aperture in front of a microscope light source to make a unidirectional plane wave and passes this light through a glass slide containing red blood cells. A CCD camera on the opposite side of the slide maps out the light intensity. Phillips et al. feed a series of these intensity maps, each corresponding to the slide at different distances from the focal point of the light source, into a numerical model that reconstructs the 3D index of refraction of the cells. They then use tabulated values to relate the index to the mass density of the red blood cells (and some other tricks to find their volumes), arriving at a mass of roughly 27 picograms (1 picogram $={10}^{-12}$ gram) per cell.

Measuring cell masses optically is less destructive than other methods and could, according to the authors, be incorporated into lab-on-a-chip studies of bacteria, human cell lines, and patient samples. – Jessica Thomas

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