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# Soft Maximum

Published:

In applications you often want to take the maximum of two numbers. But the simple function

$f(x, y) = \max(x, y)$

can be difficult to work with because it has sharp corners. Sometimes you want an alternative that sands down the sharp edges of the maximum function. One such alternative is the soft maximum. Here are some questions this post copied from [1] we will answer.

• What exactly is a soft maximum and why is it called that?
• How is it related to the maximum function?
• In what sense does the maximum have “sharp edges”?
• How does the soft maximum round these edges?
• What are the advantages to the soft maximum?
• Can you control the “softness”?

I’ll call the original maximum the “hard” maximum to make it easier to compare to the soft maximum.

The soft maximum of two variables is the function

$g(x, y) = \log(\exp(x) + \exp(y) )\;.$

This can be extended to more than two variables by taking $g(x_1, \ldots, x_n) = \log( \exp(x_1) + \cdots + \exp(x_n)\;.$

The soft maximum approximates the hard maximum. Why? If $x$ is a little bigger than $y$, $\exp(x)$ will be a lot bigger than $\exp(y)$. That is, exponentiation exaggerates the differences between $x$ and $y$. If $x$ is significantly bigger than $y$, $\exp(x)$ will be so much bigger than $\exp(y)$ that $\exp(x) + exp(y)$ will essentially equal $\exp(x)$ and the soft maximum will be approximately $\log(\exp(x) ) = x$, the hard maximum. Here’s an example. Suppose you have three numbers: -2, 3, 8. Obviously the maximum is 8. The soft maximum is 8.007.

The soft maximum approximates the hard maximum but it also rounds off the corners. Let’s look at some graphs that show what these corners are and how the soft maximum softens them.

Here are 3-D plots of the hard maximum f(x, y) and the soft maximum g(x, y). First the hard maximum:

Now the soft maximum:

Next we look at a particular slice through the graph. Here’s the view as we walk along the line $y=0$. Here we see how the hard and soft maxima compare. Also, the difference between $g$ and $f$.

The soft maximum approximates the hard maximum and is a convex function just like the hard maximum. But the soft maximum is smooth. It has no sudden changes in direction and can be differentiated as many times as you like. These properties make it easy for convex optimization algorithms to work with the soft maximum. In fact, the function may have been invented for optimization; that’s where I first heard of it.

Notice that the accuracy of the soft maximum approximation depends on scale. If you multiply $x$ and $y$ by a large constant, the soft maximum will be closer to the hard maximum. For example, $g(1, 2) = 2.31$, but $g(10, 20) = 20.00004$. This suggests you could control the “hardness” of the soft maximum by generalizing the soft maximum to depend on a parameter $k$.

$g(x, y, k) = \log( \exp(kx) + \exp(ky) ) / k$

You can make the soft maximum as close to the hard maximum as you like by making $k$ large enough. For every value of $k$ the soft maximum is differentiable, but the size of the derivatives increase as $k$ increases. In the limit the derivative becomes infinite as the soft maximum converges to the hard maximum.

# References

[1] https://www.johndcook.com/blog/2010/01/13/soft-maximum/

format: plain


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Definition:Orientation (Graph Theory)

Definition

Let $G = \struct {V, E}$ be a simple graph.

Let $H = \struct {V, A}$ be a directed graph.

Then $H$ is an orientation of $G$ if and only if both of the following hold:

$G$ is a simple digraph. That is, $A$ is antisymmetric.
$\forall x, y \in V: \paren {\set {x, y} \in E \iff \tuple {x, y} \in A \lor \tuple {y, x} \in A}$

Also see

Note that every simple digraph is an orientation of exactly one simple graph, but a simple graph may have more than one orientation.


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# Tag Info

As mentioned in the comments, the translation is shown in: Volker Diekert and Paul Gastin. "First-order definable languages." (2008) http://www.lsv.fr/Publis/PAPERS/PDF/DG-WT08.pdf And it goes via a characterization of $LTL$ as $FO[<]$.


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# Complex Analysis: Identity Theorem

1. Mar 4, 2015

### nateHI

1. The problem statement, all variables and given/known data
Let f be a function with a power series representation on a disk, say $D(0,1)$. In each case, use the given information to identify the function. Is it unique?
(a) $f(1/n)=4$ for n=1,2,$\dots$
(b) $f(i/n)=-\frac{1}{n^2}$ for n=1,2,$\dots$

A side question:
Is corollary 1 from my textbook given below just the Uniqueness Theorem?
2. Relevant equations
Corollary 1
Let $g_1$ and $g_2$ be holomorphic on the open connected set $O$, such that $g_1=g_2$ on a set that has an accumulation point within $O$. Then $g_1=g_2$ throughout $O$.

3. The attempt at a solution
(a)
Consider $g(z)=f(z)-4$. $g$ has a zero set $\{1/n\}$ that converges to $0$. Therefore, by, the identity theorem
$$g(z)=0\implies f(z)=4$$
Finally, we note that $f(z)=4$ satisfies the conditions given in the problem statement for part (a). Also, in solving the first part of this problem we've demonstrated the hypothesis for Corollary 1, thereby showing that $f(z)$ is unique.
(b)
Consider $g(z)=f(z)-z^2$. $g$ has a zero set $\{i/n\}$ that converges to $0$. Therefore, by the Identity Theorem,
$$g(z)=0\implies f(z)=z^2$$
Finally, we note that $z^2$ satisfies the conditions given in the problem statement for part (b). Also, in solving the first part of this problem we've demonstrated the hypothesis for Corollary 1, thereby showing that $f(z)$ is unique.

2. Mar 4, 2015

### Staff: Mentor

I'm not sure what you mean with "zero set [...] that converges to 0" - a set does not converge, and what is a zero set?
You can probably skip some steps you make twice, but the proof is fine.

3. Mar 4, 2015

### nateHI

A zero sequence would be a more apt descriptor then. So, a zero sequence is a sequence $\{z_k\}$ such that $f(z_k)=0$ for all k and where $\lim_{k\to\infty} z_k\to a$. In my case a=0.

4. Mar 4, 2015

Ah, okay.


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ODESolStruc - Maple Help

ODESolStruc

Reductions of order returned by dsolve

 Calling Sequence ODESolStruc(ans, [{reduced_ODE}, {TR}, {ITR}])

Parameters

 ans - answer expressed in terms of (r, s(r)), for an ODE originally in y(x). {reduced_ODE} - set containing the reduced_ODEs for s(r). {TR} - set containing the transformation equations from (r,s(r)) to (x,y(x)), where (r, s(r)) and (x, y(x)) represent any two pairs of independent/dependent variables. {ITR} - set containing the transformation equations from (x,y(x)) to (r,s(r)).

Description

 • For high order ODEs it may happen that dsolve succeeds in reducing the order of the ODE but not in solving the problem to the end. In those cases, the answer is expressed using ODESolStruc. The motivation for this routine is that the user may be able to obtain a solution for the reduced ODE, by manipulating it using the tools available in DEtools, or as a series expansion, or by other means. If a solution to the reduced ODE is obtained, a solution to the original problem can be built using DEtools[buildsol].
 • ODESolStruc is a function of two arguments. The first argument is the true mathematical value of the structure, written in terms of new variables (herein called r and s(r)) introduced by dsolve during the solving process. The second argument is a list with three sets, respectively containing: the reduced ODE for s(r); the transformation of variables (TR) used in the reduction process, from {r, s(r)} to {x, y(x)} (herein representing the variables of the original problem); and the inverse transformation (ITR).
 • The transformation sets TR and ITR are defined as follows: changing variables in the reduced ODE using TR must yield the original ODE; changing variables in the original ODE using ITR must yield the reduced ODE (see the example below). To perform the change of variables see PDEtools[dchange].
 • The differentiation rule of ODESolStruc is defined as follows. When calculating the derivative, only the first argument (true mathematical value of the structure) is differentiated, and it is considered a function of x, the old independent variable. The derivative is then calculated using the chain rule, taking into account the transformation equations TR. Note that the new variables {r, s(r)} are considered "dummies" (like the integration variable in a definite integral), and then the derivative of the ODESolStruc with respect to the new independent variable r is 0.

Examples

A second order nonlinear ODE

 > $\mathrm{ode}\left[1\right]≔\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right),x\right),x\right)=-\frac{{\mathrm{diff}\left(y\left(x\right),x\right)}^{2}{y\left(x\right)}^{2}+x}{{y\left(x\right)}^{3}}$
 ${{\mathrm{ode}}}_{{1}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}{}{{y}{}\left({x}\right)}^{{2}}{+}{x}}{{{y}{}\left({x}\right)}^{{3}}}$ (1)

This ODE is reduced by dsolve to a first order ODE of Abel type

 > $\mathrm{ans}\left[1\right]≔\mathrm{dsolve}\left(\mathrm{ode}\left[1\right]\right)$
 ${{\mathrm{ans}}}_{{1}}{≔}{y}{}\left({x}\right){=}\left({\mathrm{_a}}{}{\left({{ⅇ}}^{\frac{{4}{}\left({\int }{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}\right)}{{3}}{+}\frac{{4}{}{\mathrm{_C1}}}{{3}}}\right)}^{{3}}{{4}}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left[\left\{\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{_b}{}\left({\mathrm{_a}}\right){=}\frac{{2}{}\left({3}{}{{\mathrm{_a}}}^{{4}}{+}{8}\right){}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{3}}}{{9}{}{{\mathrm{_a}}}^{{3}}}{+}\frac{{8}{}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{2}}}{{3}}{+}\frac{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}{{\mathrm{_a}}}\right\}{,}\left\{{\mathrm{_a}}{=}\frac{{y}{}\left({x}\right)}{{{x}}^{{3}}{{4}}}}{,}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}\frac{{3}{}{{x}}^{{3}}{{4}}}}{{4}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{x}{-}{3}{}{y}{}\left({x}\right)}\right\}{,}\left\{{x}{=}{{ⅇ}}^{\frac{{4}{}\left({\int }{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}\right)}{{3}}{+}\frac{{4}{}{\mathrm{_C1}}}{{3}}}{,}{y}{}\left({x}\right){=}{\mathrm{_a}}{}{\left({{ⅇ}}^{\frac{{4}{}\left({\int }{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}\right)}{{3}}{+}\frac{{4}{}{\mathrm{_C1}}}{{3}}}\right)}^{{3}}{{4}}}\right\}\right]$ (2)

The display of ODESolStruc uses where to display the first operand (the answer) on the left hand side and the list with the reduced ODE and the transformation equations on the right hand side.

Answers containing ODESolStruc can also be tested using odetest

 > $\mathrm{odetest}\left(\mathrm{ans}\left[1\right],\mathrm{ode}\left[1\right]\right)$
 ${0}$ (3)

Each piece of the ODESolStruc above can be selected using the mouse, or through the following command:

 > $\mathrm{reduced_ODE}≔\mathrm{op}\left(\left[2,2,1,1\right],\mathrm{ans}\left[1\right]\right)$
 ${\mathrm{reduced_ODE}}{≔}\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{_b}{}\left({\mathrm{_a}}\right){=}\frac{{2}{}\left({3}{}{{\mathrm{_a}}}^{{4}}{+}{8}\right){}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{3}}}{{9}{}{{\mathrm{_a}}}^{{3}}}{+}\frac{{8}{}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{2}}}{{3}}{+}\frac{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}{{\mathrm{_a}}}$ (4)
 > $\mathrm{TR}≔\mathrm{op}\left(\left[2,2,2\right],\mathrm{ans}\left[1\right]\right)$
 ${\mathrm{TR}}{≔}\left\{{\mathrm{_a}}{=}\frac{{y}{}\left({x}\right)}{{{x}}^{{3}}{{4}}}}{,}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}\frac{{3}{}{{x}}^{{3}}{{4}}}}{{4}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{x}{-}{3}{}{y}{}\left({x}\right)}\right\}$ (5)
 > $\mathrm{ITR}≔\mathrm{op}\left(\left[2,2,3\right],\mathrm{ans}\left[1\right]\right)$
 ${\mathrm{ITR}}{≔}\left\{{x}{=}{{ⅇ}}^{\frac{{4}{}\left({\int }{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}\right)}{{3}}{+}\frac{{4}{}{\mathrm{_C1}}}{{3}}}{,}{y}{}\left({x}\right){=}{\mathrm{_a}}{}{\left({{ⅇ}}^{\frac{{4}{}\left({\int }{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}\right)}{{3}}{+}\frac{{4}{}{\mathrm{_C1}}}{{3}}}\right)}^{{3}}{{4}}}\right\}$ (6)
 > $\mathrm{new_vars}≔\mathrm{map}\left(\mathrm{lhs},\mathrm{TR}\right)$
 ${\mathrm{new_vars}}{≔}\left\{{\mathrm{_a}}{,}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\right\}$ (7)
 > $\mathrm{old_vars}≔\mathrm{map}\left(\mathrm{lhs},\mathrm{ITR}\right)$
 ${\mathrm{old_vars}}{≔}\left\{{x}{,}{y}{}\left({x}\right)\right\}$ (8)

The original ODE (ode[1] above) can be reobtained by changing variables in the reduced_ODE, using TR as follows:

 > $\mathrm{ODE}≔\mathrm{PDEtools}\left[\mathrm{dchange}\right]\left(\mathrm{TR},\mathrm{reduced_ODE}\right):$
 > $\mathrm{original_ODE}≔\mathrm{normal}\left(\mathrm{isolate}\left(\mathrm{ODE},\mathrm{diff}\left(y\left(x\right),x,x\right)\right)\right)$
 ${\mathrm{original_ODE}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}{}{{y}{}\left({x}\right)}^{{2}}{+}{x}}{{{y}{}\left({x}\right)}^{{3}}}$ (9)

The reduced_ODE, in turn, can be obtained from the original_ODE by changing variables using the inverse transformation ITR:

 > $\mathrm{RDE}≔\mathrm{PDEtools}\left[\mathrm{dchange}\right]\left(\mathrm{ITR},\mathrm{original_ODE},\left[\mathrm{_a},\mathrm{_b}\left(\mathrm{_a}\right)\right]\right):$
 > $\mathrm{collect}\left(\mathrm{isolate}\left(\mathrm{RDE},\mathrm{diff}\left(\mathrm{_b}\left(\mathrm{_a}\right),\mathrm{_a}\right)\right),\mathrm{_b}\left(\mathrm{_a}\right),\mathrm{normal}\right)$
 $\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{_b}{}\left({\mathrm{_a}}\right){=}\frac{{2}{}\left({3}{}{{\mathrm{_a}}}^{{4}}{+}{8}\right){}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{3}}}{{9}{}{{\mathrm{_a}}}^{{3}}}{+}\frac{{8}{}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{2}}}{{3}}{+}\frac{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}{{\mathrm{_a}}}$ (10)


text:
mersenneforum.org Factoring Problem
 Register FAQ Search Today's Posts Mark Forums Read

 2003-08-27, 06:31 #1 asdf     Sep 2002 6010 Posts Factoring Problem I'm having trouble with this problem: Factor: (4)(a^2)(c^2) - (a^2 - b^2 + c^2)^2 Thanks
 2003-08-27, 07:59 #2 andi314     Nov 2002 2·37 Posts ==> (4)(a^2)(c^2) - (a^2 - b^2 + c^2)^2 = - (a + b + c)·(a + b - c)·(a - b - c)·(a - b + c) hope i could help you andi314
 2003-08-29, 06:10 #3 asdf     Sep 2002 22·3·5 Posts I have checked and the answer is correct, and I have accidentally found a way to find the factors, but it was through major guessing. Could anyone show how to factor this out fully?
2003-08-29, 06:34   #4

"Richard B. Woods"
Aug 2002
Wisconsin USA

170148 Posts

There are online factorers. The one I frequently use is "Factoris" at http://wims.unice.fr/wims/wims.cgi It can factor an integer, a rational number, a polynomial or a rational function.

When given:

4(a^2)*(c^2)-(a^2-b^2+c^2)^2

Factoris returned:

Quote:
 Warning. You have entered an ambiguous formula, whose interpretation by wims may differ from what you meant. Please check the formula. Please always use *'' for multiplication, and always enclose function arguments in parentheses. (For experts. If you don't want wims to interprete your expression, start it with the string 1-1+''.) Factorization of the polynomial 4*(a^2)*(c^2)-(a^2-b^2+c^2)^2 into irreducible components over Z : -a4 + (2c2 + 2b2) a2 + (-c4 + 2b2 c2 - b4)= -(c-b-a)(c-b+a)(c+b-a)(c+b+a) Remark. Factorization of multivariate polynomials is only implemented over Z for the time being.
In the next-to-last line, Factoris superscripts the exponents, but that formatting is lost in my cut-and-pasting the text here. That is, what is meant is -a^4 + (2c^2 + 2b^2) a^2 + (-c^4 + 2b^2 c^2 - b^4) = -(c-b-a)(c-b+a)(c+b-a)(c+b+a)

Factoris doesn't use "*" for multiplication in its output even though it wants you to use it in input. :)

Of course, Factoris doesn't tell you how to factor an expression. But one might get clues from its answers.

2003-08-30, 17:56   #5
axn

Jun 2003

3×17×101 Posts
Re: Factoring Problem

Quote:
 Originally Posted by asdf I'm having trouble with this problem: Factor: (4)(a^2)(c^2) - (a^2 - b^2 + c^2)^2 Thanks
Soln: (I am omitting the * for multiplication)

(4)(a^2)(c^2) - (a^2 - b^2 + c^2)^2

==> X^2 - Y^2, where X = 2ac and Y = (a^2 - b^2 + c^2)

==> (X+Y)(X-Y)

==> (2ac + a^2 - b^2 + c^2)(2ac - a^2 + b^2 - c^2)

==> [(a+c)^2 - b^2] [b^2 - (a-c)^2]

==>[(a+c+b)(a+c-b)] [(b+a-c)(b-a+c)]

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The application of Appropriate Technology

# Part 7: Pumps and Turbines – The Bernoulli Equation

Essentially a pump adds energy to a system and a turbine takes it away. Therefore typically in the Bernoulli Equation the pump pressure (Pp) is added to the left-hand side of the equation and the turbine pressure (Pt) is added to the right. So for a system containing a pump and a turbine the Bernoulli equation would look something like this:

$P_1 + \frac{1}{2}.\rho.v_1^2 + \rho.g.h_1 + P_p = P_2 + \frac{1}{2}.\rho.v_2^2 + \rho.g.h_2 + f_h.\rho.g + P_t$

(16)

If you know the power (Wout) that is delivered by a pump, then it is possible to calculate the pressure it represents (Pp).

From Equations (6) and (7):

$W_{out} = F.S/t$

From Equation (1):

$v = S/t$

So we can write:
look something like this:

$W_{out} = F.v$

(17)

Now from Equation (4):

$P = F/A$

so:

$F = P.A$

Combining this with Equation (17):

$W_{out} = P.A.v$

From Equation (8):

$Q = A.v$

So we can write:

$W_{out} = P.Q$

Thus the energy the pump provides as a pressure (Pp) is given by:

$P_p = W_{out}/Q$

(18)

Typically pumps have an efficiency (γ)which is the ratio of the power out (Wout ) to the power in (Win ). It represents the losses in the pump due to friction and electrical efficiency. It is usually expressed as a percentage and will always be less than 100%. It should be applied in the equation as a fraction. We can write therefore:
look something like this:

$W_{out} = W_{in}.\gamma$

(19)

Combining Equations (18) and (19) gives:
look something like this:

$P_p = W_{in}.\gamma /Q$

(20)

This can be substituted into the Bernoulli Equation (16) and allows the determination of the pump power requirement or alternatively the flow rate in a system for a given pump power.

Analysis of turbines will not be dealt with in detail here but is very similar to that of pumps (but in reverse).


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## Introduction

Weather impacts both crop yield quantity and quality and is thus a driving force of farm income volatility1. While yield quantity risks and their determinants are usually well documented, yield quality risks are widely neglected. Possible effects of weather on parameters that impact the quality of a crop are, for example, the outer appearance2,3,4, specific valuable constituents5,6 or the textural structure7 of the harvest. So far, the market valuation of these effects remains unquantified. What ultimately matters for farmers’ incomes is how changes in the quality of products translate into price changes. Ignoring the important channel on how weather extremes translate into income losses due to a change in quality (and associated prices) potentially biases the impact of extreme weather events and climate change on agricultural production.

We use orchard and year-specific price shocks to assess how weather impacts apple quality and ultimately farm revenue. Our unique data set gives prices for orchards that each grow one particular apple variety over several years. This allows us to construct a causal link between weather and the quality-induced income effect. Intuitively, say both orchards A and orchards B grow Golden Delicious apples for several years. The average prices the owners of each orchard receive might differ due to differences in average quality or simply because one has better business acumen, e.g., superior supply chain management arrangements. We therefore look at the evolution of price differentials over time. If orchard A gets on average 5 cents more per kg of apple, but this difference narrows to 2 cents in a given year, we attribute the 3 cents price decline to the observed weather the orchard faces. In technical terms, the orchard fixed effects account for time-invariant differences of each orchard and by differencing with other orchards through the inclusion of year-fixed effects we account for market wide price movements. Since weather is random and exogenous, it should not be correlated with other time varying factors and hence the resulting change in price differentials can be causally attributed to the observed weather fluctuation. We estimate how much deviations from optimum quality traits translate into drops in realized producer prices. By doing so we are able to disentangle the overall effect of weather on revenues into its yield and price components. While the quality channel has been mentioned before8, we offer a novel approach to estimate it empirically.

Amid lacking site-specific realized price information, past literature has been unable to quantify the economic effect of weather extremes on harvest quality, while the agronomic mechanisms how weather can negatively influence crop quality are only known from field experiments [e.g.9,10,11].

Our data consists of a unique orchard-level panel dataset on apple yield and price records consisting of 2′462 observations from 1997 to 2014 in Switzerland. This allows us to investigate how late spring frost events impact apple quality. Spring frost events are known to cause morphologically damaged apples (see Fig. 1 and supplementary section S1 for an overview on plant physiological mechanisms). We match orchard-level price and yield observations with regional tree phenology and historical daily temperature records. We use a piecewise linear regression approach proposed by Schlenker & Roberts12 that puts a special focus on extreme temperature events, which in our case turn out to be low temperature events. Furthermore, our model controls for other temperature impacts besides spring frost on apple trees, such as beneficial winter chilling and summer heat, as these might be correlated with spring frost. We use plot level fixed effects to control for time invariant single orchards characteristics, such as variety, production system, soil conditions. To control for overall market movements in prices we use year dummies.

Our results show that idiosyncratic spring frost events induce only minor drops in yields while they cause farm gate price and thus revenue reductions of up to 2.05% per hour of exposure. Our findings highlight that frost effects on crop yield quality clearly outweigh effects on quantity. Never being quantified in any region for any crop before, these findings on weather induced quality reductions are of particular relevance for agricultural risk and risk management literature but also add to the climate change impacts literature, as late frosts are more likely to occur in the future and effects of climate change on crop quality have received limited attention so far9,13,14.

## Results

We empirically estimate the influence of temperatures on both apple quantity and quality. The results are shown in Fig. 2. The piecewise linear function is akin to the concept of degree days, which measure how much and how long temperatures exceed or fall below a certain threshold. For example, degree days below 0 °C measure how long and by how much temperatures fall below the threshold. Being 4 hours at −1 °C or two hours at −2 °C would both result in 4 degree days below 0 °C. Figure 2 shows temperature impacts across different temperature intervals. We find the impact of temperature exposure during flowering on prices (blue line) being highly nonlinear with large drops under freezing conditions. Here, idiosyncratic farm gate prices start dropping at slightly below 0 °C. This effect increases up to a loss of 1.76% when the apple orchard is exposed for one hour to −4 °C (See appendix Figure S1 for revenue drops of −2.05% for this temperature). As market (supply and demand) induced price shifts are controlled for by year dummies, this effect is idiosyncratic, only affecting single orchards. We thus conclude that this is determined by a frost induced downgrading of a certain share of the apple harvest. This share is increasing with decreasing temperatures. On the other hand, the red line in Fig. 2 shows that there is no similar relationship for the apple quantity produced (See Table S2 for full regression results and Figure S9 summaries on the temperature exposure across different temperatures). In summary, quality losses induced by weather extremes substantially outweigh quantity losses in our case study example of spring frost in apple production. In fact, for moderate spring frosts during apple flowering we find a negative but not statistically significant effect on yields while realized selling prices substantially drop.

Supplementary Figures S2 and S3 show that our findings are robust to changes in the temperature interval choice. We also experimented with flexible cubic splines that do not impose a linear relationship, but the estimated relationship at the extremes closely mirrors a linear relationship.

## Discussion

Many studies estimated the impact of temperature extremes on crop yields and the resulting monetary implications for farmers12,15,16. While this helped to better understand how weather extremes impact volumetric crop yield losses, farmers’ revenues are also substantially driven by the realized selling price of the harvest, which itself depends on the quality grade of the final product8,17. Multiple examples on weather induced quality losses exist, however, proxies to quantify the impact of weather on quality economically are missing5,9,10,11. Our understanding of how weather extremes impact agricultural production is incomplete.

Our findings fill this gap by proposing that the realized site-specific selling price deviations can serve as proxy for the quality of the harvest, and the price effect of the quality changes is what farmers are ultimately interested in. We see it as a strength that we are relying on price data instead of measurements of an apple’s quality without knowing how the market ultimately prices such quality differences. The main empirical weather condition in our setting of apple production that impacts an apples quality is spring frost. In this instance, economic consequences of quality losses can outweigh those on quantity substantially. Our empirical procedure allows to identify this effect for an orchard growing a specific variety across years by linking it to random weather variation, which are idiosyncratic and random. The mechanism are morphological damages such as frost rings (Fig. 1) but can also be caused by pests that are supported by the frost events that might manifest in rotten fruits18.

Supplementary Figures S4S8 estimate the relationship for particular apple varieties. In a placebo test we find that the price of processing varieties for which the outer appearance is of minor important, such as Boskoop, does not respond to frost exposure as expected, highlighting the validity of our assumption that frost is uncorrelated with other factors. We observe the market’s response to a reduction in apple price only for varieties that have shown to be impacted by the apple frost in this way. The novelty of our study is to monetize this effect, and measure the income implications for farmers. Given the limited number of observations for some apple varieties, the error bands tend to widen if we limit the data to a particular variety and the coefficient estimates are mostly not significantly different from the average effect.

Leaving this important mechanism unconsidered could dramatically underestimate the impact of extreme events on agricultural production and could bias farmers’ expected response to climate change. Simulation studies that predict farmers’ response to climate change should consider all climate change induced alterations in the risk exposure and its monetary consequences to avoid biased predictions19,20. Indeed, focusing on yield losses only, ignoring realized prices as important part of farmers’ revenues, which constitutes farmers’ most important goal variable, makes us unable to simulate farmers’ behavior under climate change conditions.

We estimate a reduced form model that does make assumptions on the relationship between weather and outcome variables21. Thus, the apple producers might apply agro-managerial practices and our estimated relationship allows for these short-term adaptations22. The resulting effect can be interpreted as the spring frost response of yields and prices after short-term adaptation. Moreover, production costs might be affected by spring frost events, which itself impacts overall profits, but our dataset does not include costs at the plot level. Our estimates are hence a lower bound on the detrimental effects of spring frost, as they include the beneficial effects of short-term adaptation, but exclude the cost of such measures. We see this as an interesting topic for future research. Note that weather induced costs as well as the weather induced yield quantity and quality losses can, for example, be reduced by designing appropriate weather index insurance solutions. Such insurance would pay out to farmers in a frost event, independently of the physical damage. Thus, producers can receive an insurance payout even without facing yield losses due to the use of preventive, but costly, measures23,24,25.

We here use heat stress (number of day >30 °C) as a control variable. When being particularly interested in the impact of extreme heat, it can be modelled more flexibly, e.g. with a non-linear specification such as regressions splines, the piecewise linear approach used here for the frost impact or polynomial specifications (see Blanc & Schlenker22 for an overview). As we do neither expect nor find any correlation between spring frost and summer heat events we decided for this more straightforward summer heat control variable. It should be noted that simple statistical models outperformed biophysical models in an out-of sample prediction exercise where various modeling groups predicted yields for real world test plots before knowing their yields26.

Important for apple trees, climate change is expected to cause increases in winter temperatures leading to changes in the fulfillment of cooling (chilling) and heat requirements (forcing) that induce the end of the apple trees’ winter dormancy13,27,28,29,30. Thus, apple blooming is more likely to be affected by spring frost events across temperature regimes analyzed here, constituting an increase in the downside risk exposure of apple producers31,32,33,34,35.

A complete picture of extreme weather impacts on agricultural production is a crucial prerequisite for measuring the full effects of climate change. So far, weather induced crop quality losses are largely ignored although agronomic knowledge suggests strong relationships. Neglecting these quality losses that likely occur also in other crops, downward biases our understanding of the impact of weather extremes on agricultural incomes and therefore threatens food security. Future research should estimate the impact of weather events on quality also beyond our apple example, when site specific price data becomes available.

Many policy instruments (extension, education, financial support, subsidies to insurance premiums) are targeted on the reduction of farmers’ financial risk exposure, but often ignore the role of crop quality losses. Thus, our results deliver important insights for policy makers to design better polices that aim at supporting farmers in low income situations.

## Empirical Methods

We use regression analysis to estimate the impact of temperature exposure during apple flowering on apple price $${p}_{it}$$ and yield $${y}_{it}$$ at orchard i in year t. Therefore, we estimate the following models:

$$\log \,({p}_{it})=f({T}_{ti},\beta )+\delta {z}_{it}\,{+}_{i}{+}_{t}+{\varepsilon }_{it}$$
(1)
$$\log \,({y}_{it})=g({T}_{ti},b)+d{z}_{it}\,+{a}_{i}+{c}_{t}+{e}_{it}$$
(2)

where log (pit) and log (Yit) are the natural logarithms of the apple price [CHF/kg] and yield [kg/ha] of orchard i in year t that grows a specific apple variety36. The function $$f({T}_{ti},\beta )$$ and $$g({T}_{it},b)$$ allow for the potentially non-linear effects of temperature exposure during various time periods of the growing season, e.g., flowering. The controls $${z}_{it}$$ denote an orchard and year specific matrix of temperature variables that have previously been shown to impact apple output, i.e., chilling hours in the winter and heat spells in the summer. All time invariant unobserved heterogeneity is controlled for in the orchard-level fixed effects i and $${a}_{i}$$. Hence, our estimates are not affected by orchard characteristics that lead to shifts in average orchard price or yield levels (such as variety, production method (i.e. organic), surface texture or microclimate). In our sample, observations are structured in a way that only one variety is grown on a single orchard. We are thus able to explicitly control for variety effects. Moreover, see appendix Figures S4S8 for subsample results for the five most common varieties. We include y and $${c}_{t}\,$$to capture all systemic price and yield movements that occur across all orchards jointly in each year. Consequently, we model the impact of temperature exposure on yields and prices when these deviate from their long-term orchard average and from the other observations in a given year (i.e. systemic price movements e.g. due to policy changes or market effects and systemic yield movements through pestoutbreaks in the entire country). The estimated impact is thus idiosyncratic. The error terms $${\varepsilon }_{it}$$ and $${e}_{it}$$ are likely heteroskedastic, spatially correlated within each year, and temporally correlated within one orchard over the years. We therefore use heteroscedasticity robust standard errors that are two-way clustered by year and orchard37. We follow Schlenker and Roberts12 and use a piecewise linear approach to model $$f({T}_{ti},\beta )$$ and $$g({T}_{it},b)$$ Therefore the temperature impacts on prices and yields during the flowering phase are modelled as:

$$f({T}_{ti},\beta )={\beta }_{1}{T}_{(-\propto ,-2]{}^{^\circ }C}+{\beta }_{2}{T}_{[-2,0]{}^{\circ }C}+{\beta }_{3}{T}_{[0,15]{}^{\circ }C}+{\beta }_{4}{T}_{[15,){}^{\circ }C}$$
(3)
$$g({T}_{ti},b)={b}_{1}{T}_{(-\propto ,-2]{}^{\circ }C}+{b}_{2}{T}_{[-2,0]{}^{\circ }C}+{b}_{3}{T}_{[0,15]{}^{\circ }C}+{b}_{4}{T}_{[15,){}^{\circ }C}$$
(4)

In (3), $${T}_{(-\propto ,-2]{}^{\circ }C}$$ measures by how much and for how long temperatures fall below −2 °C, the same concept underlying degree days that are often used as factors driving agricultural output. For example, being 4 hours at $$-3^\circ C$$ would result in an exposure of 4, as would being one hour at $$-6\,^\circ C$$. Temperature intervals with two bounds, e.g., $${T}_{[-2,0]^\circ C}$$,measure by how much and for how long temperatures exceed the lower bound (−2 °C), while temperatures are truncated at the upper bound (0 °C), i.e., any temperature above 0 °C will be recorded as two (difference between upper and lower bound for the interval in question). The analogous definition applies for the temperature interval $${T}_{0^\circ C-15^\circ C}$$. Finally, $${T}_{[15,\propto )^\circ C}$$ measures for how long and by how much temperatures exceed 15 °C. The motivation of these temperature intervals come from a more flexible regression using cubic splines (third-order approximations) that exhibited clear strong nonlinearities at both ends of the temperature distribution. (See Supplementary Figures S2 and S3 for results using different interval specifications and section S1 for background information on the plant physiological importance to distinguish between different freezing temperatures). For impacts of temperature exposure on overall revenues (price multiplied by yield) see appendix Figure S1.

## Data

### Orchard data

Economic orchard-level panel data on apple production were provided by the Swiss Federal research station Agroscope38. This dataset is unique in capturing site specific realized price information. We are not aware of a comparable data set that gives the same fine-scaled (orchard-level) market assessment of apple quality. The unbalanced dataset includes 2′462 observations containing information on 55 apple varieties planted on 505 orchards across ten cantons (Swiss Federal States) during the years 1997–2014. Observed variables include yields (2′444 observations) [kilogram (kg)/ hectare (ha)], revenues (2′389 observations) [CHF (Swiss Francs)/ha] (1 CHF = 0.93 EUR), farm-gate prices (2′389 observations) [CHF/kg], variety, municipality and a dummy on whether the orchard produces organic (See Table S1 for descriptive statistics). For 18 observations we have no information on either yields, prices or revenues. Thus, these 18 observations are dropped from the further analysis. The farm-gate prices are orchard specific average realized prices. Thus, a larger share of downgraded apples leads to lower prices. Most common varieties in our sample were Golden Delicious (16% of all orchards), Gala (10%) and Jonagold (8%). We apply a multivariate outlier detection procedure to identify and remove outliers in our data using the bacon algorithm based on Mahalanobis distances39. We thus removed four outlying price and three yield observations. For the respective variables our final dataset thus results in 2′441 observations for yields and 2′385 for prices respectively, which we include in our analysis. For further information on the orchard data see the supplementary section S2 Economic orchard data.

### Apple tree phenology data

Start and end dates of the apple blooming period (i.e. from growth stages BBCH 61 “first flowers open” to BBCH 69 “End of flowering: all petals fallen”40) in each orchard, year and variety were found using phenological records of experimental sites across our study region. More specifically, we obtain our tree phenology data from 33 different experimental stations across Switzerland. The dataset includes location information of the experimental station and occurrence dates of growth stages along the BBCH scale. The data can be accessed via an online form at http://www.agrometeo.ch. We use phenology information to find start and end dates of flowering (BBCH 61 to BBCH 69) and chilling (BBCH 97 and BBCH 00; see section on weather data for further information on temperature control variables)41. Dependent on data availability we consider the following four steps of assigning phenology data to single orchards36.

• First we match the single orchard information with variety specific phenology data within the same year and canton (here we find 1,176 matches). We then use the earliest/latest date within BBCH 61 and BBCH 69 to find the start/end date of flowering for the single orchard.

• Second, in case of insufficient data for step one, we match orchard information with all available variety phenology (except the early flowering variety Boskoop) within the same year and canton (here we find 1′020 further matches). We then use the earliest/latest date within BBCH 61 and BBCH 69 to find the start/end date of flowering for the single orchard.

• Third, in case of insufficient data for step two, we match orchard information with variety specific phenology within the same year in all cantons (here we find 141 further matches). We then use the earliest/latest date within BBCH 61 and BBCH 69 to find the start/end date of flowering for the single orchard.

• Fourth, in case of insufficient data for step three, we match orchard information with phenology information within the same year across all varieties from all cantons (here we find 107 further matches). We then use the earliest/latest date within BBCH 61 and BBCH 69 to find the start/end date of flowering for the single orchard.

See Figure S11 for a histogram of start and end dates of flowering. Our results are robust against dropping the 248 observations from the last two nationwide matching steps (See Supplementary Figure S9).

### Weather data

Daily minimum and maximum temperatures within tree phenology stages were obtained from a gridded (raster) dataset with 2.5 ×2.5 km resolution provided by the Swiss Meteorological Office42. The interpolation method to produce the temperature raster data specifically considers Swiss specific texture characteristics and thus nonlinear temperature changes across elevation levels. Furthermore, “valley-scale cold-air pools” can be realistically displayed, taking into account site specific micro climates36.

To obtain time spend in the different temperature intervals described earlier, we use the procedure proposed by43. We fit a sine curve between daily maximum and minimum temperature and derive the daily exposure spent in each temperature interval. The sum of all daily exposures during the flowering period across the intervals defined in the methods section is then used in the regression analysis. See Figure S10 for boxplots of the temperature exposure [in days per degree].

We further use a set of variables to control for other than frost damage effects of temperature. More specifically, we control for damaging summer heat and beneficial winter chill. Regarding summer heat, we use the number of days with a maximum air temperature above 30 °C to control for summer heat spells that potentially damage apple fruits (see Racsko & Schrader44 for an overview). Regarding winter chill, we control for beneficial cooling effects by including a winter chill control variable. Here we use the available chilling hours (hours with air temperature between 0 °C and 7.2 °C) between leaf fall and bud development (BBCH 97 and BBCH 00) obtained from the phenology data, according to above matching procedure. For the chilling variable we use a similar procedure as for the temperature during flowering variable. We estimate a sine curve between the daily maximum and minimum temperature variables to derive hourly temperatures43. From that we derive the exposure time between 0 °C and 7.2 °C.


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# What is 51/365 as a decimal?

## Solution and how to convert 51 / 365 into a decimal

51 / 365 = 0.14

The basis of converting 51/365 to a decimal begins understanding why the fraction should be handled as a decimal. Decimals and Fractions represent parts of numbers, giving us the ability to represent smaller numbers than the whole. Choosing which to use starts with the real life scenario. Fractions are clearer representation of objects (half of a cake, 1/3 of our time) while decimals represent comparison numbers a better (.333 batting average, pricing: \$1.50 USD). Now, let's solve for how we convert 51/365 into a decimal.

## 51/365 is 51 divided by 365

The first step of teaching our students how to convert to and from decimals and fractions is understanding what the fraction is telling is. 51 is being divided into 365. Think of this as our directions and now we just need to be able to assemble the project! Fractions have two parts: Numerators on the top and Denominators on the bottom with a division symbol between or 51 divided by 365. We use this as our equation: numerator(51) / denominator (365) to determine how many whole numbers we have. Then we will continue this process until the number is fully represented as a decimal. This is how we look at our fraction as an equation:

### Numerator: 51

• Numerators represent the number of parts being taken from a denominator. Any value greater than fifty will be more difficult to covert to a decimal. 51 is an odd number so it might be harder to convert without a calculator. Large two-digit conversions are tough. Especially without a calculator. So how does our denominator stack up?

### Denominator: 365

• Denominators represent the total number of parts, located below the vinculum or fraction bar. Larger values over fifty like 365 makes conversion to decimals tougher. But 365 is an odd number. Having an odd denominator like 365 could sometimes be more difficult. Ultimately, don't be afraid of double-digit denominators. Now let's dive into how we convert into decimal format.

## Converting 51/365 to 0.14

### Step 1: Set your long division bracket: denominator / numerator

$$\require{enclose} 365 \enclose{longdiv}{ 51 }$$

We will be using the left-to-right method of calculation. This method allows us to solve for pieces of the equation rather than trying to do it all at once.

### Step 2: Extend your division problem

$$\require{enclose} 00. \\ 365 \enclose{longdiv}{ 51.0 }$$

Because 365 into 51 will equal less than one, we can’t divide less than a whole number. Place a decimal point in your answer and add a zero. Now 365 will be able to divide into 510.

### Step 3: Solve for how many whole groups you can divide 365 into 510

$$\require{enclose} 00.1 \\ 365 \enclose{longdiv}{ 51.0 }$$

How many whole groups of 365 can you pull from 510? 365 Multiple this number by our furthest left number, 365, (remember, left-to-right long division) to get our first number to our conversion.

### Step 4: Subtract the remainder

$$\require{enclose} 00.1 \\ 365 \enclose{longdiv}{ 51.0 } \\ \underline{ 365 \phantom{00} } \\ 145 \phantom{0}$$

If there is no remainder, you’re done! If you have a remainder over 365, go back. Your solution will need a bit of adjustment. If you have a number less than 365, continue!

### Step 5: Repeat step 4 until you have no remainder or reach a decimal point you feel comfortable stopping. Then round to the nearest digit.

Remember, sometimes you won't get a remainder of zero and that's okay. Round to the nearest digit and complete the conversion. There you have it! Converting 51/365 fraction into a decimal is long division just as you learned in school.

### Why should you convert between fractions, decimals, and percentages?

Converting between fractions and decimals depend on the life situation you need to represent numbers. They each bring clarity to numbers and values of every day life. This is also true for percentages. It’s common for students to hate learning about decimals and fractions because it is tedious. But each represent values in everyday life! Without them, we’re stuck rounding and guessing. Here are real life examples:

### When you should convert 51/365 into a decimal

Investments - Comparing currency, especially on the stock market are great examples of using decimals over fractions.

### When to convert 0.14 to 51/365 as a fraction

Carpentry - To build a table, you must have the right measurements. When you stretch the tape measure across the piece of wood, you won't see 10.6 inches. You'll see a tick mark at 10 and 3/5 inches.

### Practice Decimal Conversion with your Classroom

• If 51/365 = 0.14 what would it be as a percentage?
• What is 1 + 51/365 in decimal form?
• What is 1 - 51/365 in decimal form?
• If we switched the numerator and denominator, what would be our new fraction?
• What is 0.14 + 1/2?


text:
# Cos 2 Pi 7 Cos 4 Pi 7 Cos 8 Pi 7 1 8

Cos 2 Pi 7 Cos 4 Pi 7 Cos 8 Pi 7 1 8. If the hypotenuse of a right – angled triangle is four times the length of the perpendicular drawn from the opposite vertex to it, then the difference of the two acute angles will be. How do you find the domain and range of sine, cosine, and tangent?

How do you find the domain and range of sine, cosine, and tangent? If the hypotenuse of a right – angled triangle is four times the length of the perpendicular drawn from the opposite vertex to it, then the difference of the two acute angles will be.

### If the hypotenuse of a right – angled triangle is four times the length of the perpendicular drawn from the opposite vertex to it, then the difference of the two acute angles will be.

How do you find the domain and range of sine, cosine, and tangent?

somme de cos

cos.(2pi)/(7)+cos.(4pi)/(7)+cos.(6pi)/(7) – YouTube

How to calculate the product of: [math]\cos\left(\frac{2 …

If A=sin((2pi)/7)+sin((4pi)/7)+sin((8pi)/7) and `B=cos …


text:
# American Institute of Mathematical Sciences

May  2014, 34(5): 1951-1959. doi: 10.3934/dcds.2014.34.1951

## An extended discrete Hardy-Littlewood-Sobolev inequality

 1 Department of Applied Mathematics, University of Colorado at Boulder, Colorado, United States, United States

Received  May 2013 Revised  July 2013 Published  October 2013

Hardy-Littlewood-Sobolev (HLS) Inequality fails in the critical'' case: $μ=n$. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: $μ=n$ and $p=q$, by limiting the inequality on a finite domain. The best constant in the inequality and its corresponding solution, the optimizer, are studied. First, we obtain a sharp estimate for the best constant. Then for the optimizer, we prove the uniqueness and a symmetry property. This is achieved by proving that the corresponding Euler-Lagrange equation has a unique nontrivial nonnegative critical point. Also, by using a discrete version of maximum principle, we prove certain monotonicity of this optimizer.
Citation: Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951
##### References:
 [1] W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev Inequalities and Systems of Integral Equations,, Discrete and Continuous Dynamical Systems, (2005), 164.   Google Scholar [2] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar [3] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar [4] W. Chen and C. Li, Indefinite elliptic problems in a domain,, Discrete Contin. Dynam. Systems, 3 (1997), 333.  doi: 10.3934/dcds.1997.3.333.  Google Scholar [5] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Discrete Contin. Dyn. Syst., 30 (2011), 1083.  doi: 10.3934/dcds.2011.30.1083.  Google Scholar [6] W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Comm Pure Appl Anal, 4 (2005), 1.   Google Scholar [7] W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality,, Proc. AMS, 136 (2008), 955.  doi: 10.1090/S0002-9939-07-09232-5.  Google Scholar [8] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Commun. in Partial Differential Equations, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar [9] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Disc. & Cont. Dynamics Sys., 12 (2005), 347.   Google Scholar [10] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure and Appl. Math., 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar [11] L. E. Fraenkel, An introduction to maximum principles and symmetry in elliptic problems,, Cambridge Tracts in Mathematics, (2000).  doi: 10.1017/CBO9780511569203.  Google Scholar [12] B. Gidas, W. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $R^n$,, Mathematical Analysis and Applications, (1981), 369.   Google Scholar [13] F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math Res Lett, 14 (2007), 373.  doi: 10.4310/MRL.2007.v14.n3.a2.  Google Scholar [14] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661.  doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar [15] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities,, Cambridge at the University Press, (1952).   Google Scholar [16] J. Kigami, Analysis on Fractals,, Cambridge Tracts in Mathematics, (2001).  doi: 10.1017/CBO9780511470943.  Google Scholar [17] Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations,, Calc. Var. Partial Differential Equations, 45 (2012), 43.  doi: 10.1007/s00526-011-0450-7.  Google Scholar [18] Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation,, Discrete Contin. Dyn. Syst., 30 (2011), 547.  doi: 10.3934/dcds.2011.30.547.  Google Scholar [19] C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, Commun. Pure Appl. Anal, 6 (2007), 453.  doi: 10.3934/cpaa.2007.6.453.  Google Scholar [20] C. Li and J. Villavert, An extension of the Hardy-Littlewood-Pólya inequality,, Acta Mathematica Scientia, 31 (2011), 2285.  doi: 10.1016/S0252-9602(11)60400-1.  Google Scholar [21] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar [22] O. Perron, Zur Theorie der Matrices,, Mathematische Annalen, 64 (1907), 248.  doi: 10.1007/BF01449896.  Google Scholar [23] E. B. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503.  doi: 10.1512/iumj.1958.7.57030.  Google Scholar

show all references

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# Scattering in Schrödinger picture [closed]

If we look at a scattering process in the Schrödinger picture for a Hamiltonian $H = H_0(t) + V(t)$ where $H$ is independent of time (because we examine a theoretical situation after accelerating particles and turning off accelerators, so the system is isolated) and $V$ depends on time (because interaction term in general depends on distance between particles) and so will be $H_0$.

So the eigenstates of $H_0$ are dependent on time and I don't know how is their evolution? Of course their evolution as if they are state of a physical system is irrelevant. I want their evolution because of their dependence on $H_0(t)$.

## closed as unclear what you're asking by ACuriousMind♦, user36790, CuriousOne, Gert, garypJun 1 '16 at 0:56

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• I edited your question for grammar, but it is still not clear to me what you are asking. Why do you split a time-independent Hamiltonian into two time-dependent parts? What has this to do with "turning off accelerators"? Why are you interested in eigenstates of $H_0(t)$? It appears you are trying to do scattering theory/perturbation theory but have not properly understood it. – ACuriousMind May 30 '16 at 10:05
• thanks. really I'm trying to understand lorentz invariant theory of scattering in schrodinger picture. (I turn off accelerators long before collision to have an isolated (but moving fast!) system of particles so I can have time independent hamiltonian $H$ for scattering.) I guess understanding eigenstates of time dependent operator $H_0$ (time dependent operator even in schrodinger picture!) will help me understanding S-matrix. – moshtaba May 30 '16 at 10:57

The eigenstates of a time-independent Hamiltonian $$H |\psi_j \rangle = E_j|\psi_j\rangle$$ have the usual rotating-phase time dependence in the Schrödinger picture: $$|\psi_j(t)\rangle = |\psi_j(t_0)\rangle \cdot \exp(E_j(t-t_0)/i\hbar)$$ However, your formula indicates that $H_0$ and $V$ are time-dependent. So if the state vector is an eigenvector of $H_0(t)$ at one moment $t$, it will almost certainly not be an eigenstate of $H_0(t')$ at another moment $t'$ – because the operator is probably changing in a way that doesn't preserve the eigenvectors' being eigenvectors. For that reason, the simple "changing phase" Ansatz isn't a solution to the Schrödinger's equation.
For a generic Hamiltonian $H_0(t)$, there can't exist any state vector $|\psi(t)\rangle$ that simultaneously solves Schrödinger's equation (with the Hamilonian $H$ or even with $H_0(t)$); and that remains an eigenstate of $H_0(t)$ at each moment of time.
Because you say that $H$ is time-independent and $H_0,V$ seem to be time-dependent, it seems that all the formulae you are describing are in the Heisenberg picture, not Schrödinger's picture. In the Heisenberg picture, the state vector is independent of time (a constant function of $t$). In the Schrödinger's picture, the dependence of $H_0$ and $V$ on time would have to be "explicit" (the dynamical dependence is encoded in the evolution of the state vector in this picture) and in that case, it would be unreasonable for the explicitly time-dependent parts of $H_0,V$ to cancel in $H$.
In the Heisenberg picture, the dynamical time dependence is included in the evolution of the operators, $$H_0(t) = \exp(-Ht/i\hbar) H_0(0) \exp(Ht/i\hbar)$$ and similarly for $V$ instead of $H_0$ (and for any operator).
• about last line of the paragraph "Because you say ..." I think (am I wrong?) this experiment have time independent $H$ (because system is isolated). we can write $H=H_0+V$, now because $V$ is time-dependent ($V$ depends on distance between particles that depends on time) $H_0$ trivially become time dependent so that $H=H_0 +V$ remains true. – moshtaba May 30 '16 at 11:16
• Sorry, in Schrödinger's picture, if you add an explicitly time-dependent perturbation $V(t)$, e.g. a variable external field, then be sure that the total Hamiltonian $H(t)$ will be time-dependent due to this explicit dependence, too. The total Hamiltonian is only time-independent for closed systems without any explicit dependence. – Luboš Motl May 30 '16 at 11:20
• thanks, Let we accelerated two particles from left and right and then turn off accelerators, and examine the system of only this two particles in schrodinger picture . then $H$ is independent of time, is't it? if I split it into $H_0$ and $V$ then this two operators are time-independent?! what about intensity in interaction that is time dependent (distance dependent) that is coded in $V$?! thanks. – moshtaba May 30 '16 at 11:36
• If (and when, e.g. for those values of $t$ when) $V$ explicitly depends on time, $H$ depends on time, too. I don't understand why we need 6 messages to communicate this self-evident point. In your original question, you said that $H$ was time-independent while $V$ was time dependent. I wrote that this is either implausible, or it's what happens in the Heisenberg picture. – Luboš Motl May 30 '16 at 12:05


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Question

# To calculate: The points needed on the final to avetage the points to 75. Given information: Grades in three tests in College Algebra are 87, 59 and 73. The average after final is 75.

Upper level algebra
To calculate:
The points needed on the final to avetage the points to 75.
Given information:
Grades in three tests in College Algebra are 87, 59 and 73.
The average after final is 75.

2020-10-19
Calculation:
Let the grade in the final test be x
Average of n numbers $$\displaystyle{a}_{{{1}}}+{a}_{{{2}}}+{a}_{{{3}}}+\ldots+{a}_{{{n}}}$$ is given by $$\displaystyle{A}={\frac{{{a}_{{{1}}}+{a}_{{{2}}}+{a}_{{{3}}}+\ldots+{a}_{{{n}}}}}{{{n}}}}$$
Plugging the values,
$$\displaystyle{A}={\frac{{{87}+{59}+{73}+{x}}}{{{4}}}}$$
Given average is 75
Therefore,
$$\displaystyle{75}={\frac{{{87}+{59}+{73}+{x}}}{{{4}}}}$$
Multiplying both sides by 4
$$\displaystyle{75}\cdot{4}={\frac{{{87}+{59}+{73}+{x}}}{{{4}}}}\cdot{4}$$
$$\displaystyle{300}={219}+{x}$$
Subtracting 219 from both the sides,
$$\displaystyle{300}—{219}={219}+{x}-{219}$$
$$\displaystyle{81}={x}$$
$$\displaystyle{x}={81}$$
Therefore the score required in final test to get an average of 75 is 81.