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# To get to the reunion, we drove a total of 504 miles in 9 hours. find our average rate of speed. (distance is the product

###### Question:

To get to the reunion, we drove a total of 504 miles in 9 hours. find our average rate of speed. (distance is the product of rate and time).

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# (25 pts. )Solved (4-x)(-Y)Solve the initial value problem dx 3(2) - -1. (4 -x)(1 -y)...

## Question

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(25 pts. ) Solve d (4-x)(-Y) Solve the initial value problem dx 3(2) - -1. (4 -x)(1 -y)

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# Predicting the weather in two months

On Nov 1, I looked at the Dallas monthly weather averages (pasted below) and tried to predict the average temperature for Jan 2020. It's hard to see any trend, so I would simply predict 46.2 +/- 3.5 °F.

What else should I have measured to predict this significantly better? I was thinking about looking at solar irradiance data (with sunspot solar cycles), sea currents, or sea temperatures...what data would improve my prediction by more than 1°F?

I'm hoping for an answer like, "Well, if you go to website X, their measurement of Y indicates that temperatures will likely deviate by Z" with perhaps some science showing that this is a good way to predict long-term weather.

Year  Jan   Feb   Mar   Apr   May   Jun   Jul   Aug   Sep   Oct   Nov   Dec   Total
-----------------------------------------------------------------------------------
2019  45.8  50.2  55.0  66.0  73.4  79.9  84.6  87.4  85.5  65.5
2018  45.8  51.1  63.3  61.6  79.0  85.7  88.8  85.2  78.1  66.2  52.5  48.4  67.1
2017  51.2  60.6  65.7  69.3  75.4  82.5  86.6  84.4  80.6  69.6  62.4  49.7  69.8
2016  47.0  55.2  61.2  68.1  72.5  84.0  87.4  85.8  81.5  74.1  63.5  49.7  69.2
2015  44.5  45.7  56.1  65.8  70.9  82.1  87.1  87.3  82.7  71.2  58.7  53.7  67.2
2014  45.3  47.0  55.1  66.3  74.4  82.4  83.8  86.2  80.3  71.6  51.5  50.1  66.2
2013  49.1  52.0  56.4  63.0  72.3  82.6  84.5  87.1  82.4  68.2  53.5  43.1  66.2
2012  50.4  52.5  64.3  70.3  77.9  84.3  87.7  86.5  80.0  67.0  59.7  51.2  69.3
2011  42.8  49.5  61.3  70.8  72.8  86.8  91.4  93.4  80.0  68.2  57.9  47.6  68.5
2010  44.3  41.7  55.8  66.7  76.9  86.5  85.9  89.8  79.8  68.8  58.5  49.3  67.0
2009  48.1  55.8  58.9  64.8  73.3  83.8  86.6  85.3  76.2  62.7  59.5  42.7  66.5
2008  46.9  54.0  61.0  66.5  77.0  86.5  89.0  86.7  78.5  69.6  59.7  49.0  68.7
2007  42.3  49.4  63.9  62.4  74.6  81.4  83.8  87.8  81.6  72.1  61.3  49.8  67.5
2006  55.1  49.4  62.2  72.0  78.0  83.7  87.7  89.8  77.6  68.1  57.5  50.0  69.3
2005  49.5  52.6  56.7  65.5  73.5  83.6  85.1  86.8  83.7  68.8  60.8  48.1  67.9
2004  48.5  45.8  61.8  66.2  74.4  79.4  83.5  81.5  78.4  72.1  57.1  48.9  66.5
2003  43.7  45.6  56.2  66.9  75.5  79.0  86.3  86.7  74.5  68.7  59.4  49.6  66.0
2002  47.6  47.0  54.8  68.4  72.0  80.1  83.2  84.7  79.6  64.9  53.8  47.7  65.3
2001  42.7  50.2  51.8  67.8  74.2  80.3  86.7  84.9  74.7  65.0  59.7  49.5  65.6
2000  50.6  57.3  61.0  65.3  76.6  80.7  87.3  90.2  80.4  69.7  49.8  39.4  67.4
1999  48.6  55.5  56.4  67.9  73.8  82.1  86.1  90.2  79.0  69.2  62.9  51.5  68.6
1998  48.4  51.1  54.9  63.8  78.5  85.5  91.6  87.8  83.6  69.5  57.6  47.0  68.3
1997  44.0  49.5  58.3  60.4  70.1  78.8  84.9  83.1  80.2  67.2  52.1  45.6  64.5
1996  43.1  52.1  53.3  64.2  79.7  82.5  86.1  82.5  74.5  66.9  54.9  49.2  65.8
1995  48.1  52.5  56.9  64.2  73.2  79.9  85.5  85.5  76.6  67.8  54.8  47.4  66.0
1994  45.7  48.6  59.0  65.9  71.5  84.1  83.9  84.0  76.3  67.3  57.9  49.0  66.1
1993  45.1  49.0  56.1  63.3  71.9  81.7  87.3  87.5  78.2  63.8  51.6  49.5  65.4
1992  46.9  54.4  59.0  66.0  71.1  79.4  84.3  80.2  77.7  69.5  52.7  49.9  65.9
1991  42.8  53.7  59.8  67.4  75.4  81.0  85.0  82.5  75.2  68.1  51.7  50.3  66.1
1990  51.8  53.9  57.7  64.0  73.4  84.0  82.5  84.6  80.0  66.4  59.8  44.0  66.8
1989  50.0  42.2  56.7  66.4  74.3  77.9  82.8  82.3  74.7  69.0  58.2  39.0  64.5
1988  42.2  47.1  56.0  64.5  72.8  80.4  85.3  87.9  79.2  65.7  58.1  49.1  65.7
1987  44.5  50.8  53.9  65.0  75.1  79.6  83.4  86.5  77.1  66.5  55.7  47.3  65.5
1986  48.8  51.2  60.2  67.2  71.5  80.8  86.4  83.4  80.2  65.7  52.4  46.1  66.2
1985  37.8  45.0  60.8  67.2  74.0  80.2  84.4  87.6  77.7  67.6  56.3  42.3  65.1
1984  39.3  50.9  56.3  63.7  73.7  82.5  85.5  85.8  76.1  67.0  54.6  52.6  65.7
1983  43.4  48.5  54.5  60.6  69.5  77.3  83.6  84.9  77.1  67.8  57.3  34.8  63.3
1982  44.6  44.5  59.8  62.5  72.5  79.2  84.6  86.7  78.1  67.0  55.6  49.2  65.4
1981  44.6  48.9  55.7  69.2  70.5  80.3  85.9  83.4  76.2  66.1  57.5  47.3  65.5
1980  45.5  46.6  54.2  63.1  75.0  87.0  92.0  88.5  80.3  65.4  54.9  49.4  66.8

• I don't think it can be done with any useful confidence values. Weather is very chaotic, and small inputs could have drastic results. Daily temperature forecasts are often out by more than a degree F. – Friddy Nov 12 at 22:00
• @bobuhito why not just use a climate model ? – gansub Nov 13 at 0:50
• I'm extremely skeptical that weather can be forecast with any usability at that range due to the uncertainties and errors that propagate and accumulate rapidly over time. Even the long-term climate outlook (i.e. generally warmer/colder, wetter/drier than average) is difficult and problematic. Anecdotally, I've tracked one of the major forecast corporations' very long-term predictions several times, and each time the quality rapidly degrades past ~10 days out, with cooling and warming trends becoming completely out of phase with reality - i.e., worse than just assuming the climatic average. – dplmmr Nov 14 at 0:18
• @gansub: Not really all that good beyond a week or so, really. It's those darned butterflies in the Amazon, you know :-) And a climate forecast won't tell you what the weather is going to be like. E.g. looking at El Nino will give you some idea about the climate in California next winter, but it's not much use in deciding whether to plan a skiing trip for President's Day weekend next year. – jamesqf Nov 14 at 18:28
• @jamesqf that was the point i was trying to make. Beyond two weeks nobody runs weather models. It is climate models such as cwrf to get likely trends. Yes you cant plan a ski trip that far out. – gansub Nov 15 at 0:32


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Enterprise Multiples

.

EV/SALES

Equational Form

$\frac {EV}{Sales} \,= \,\frac{ROIC\, -\, g}{ROIC\,(WACC\,-\,g)}\,(1\,-\,T)\,(M)$

g solution

$g\,=\,\frac{ROIC\,[(T'\, x\, EBIT)\,-\,(EV\, x\, WACC)]}{(T'\,x\,EBIT)\,-\,(EV\, x\, ROIC)}$

Inputs

 $EV\,=\,Mkt\,Cap\,Equity\,+\,Mkt\,Value\,Debt\,-\,Cash$ $EBIT \,=\,Earnings\,\,Before\,\,Interest\,\,\&\,\,Taxes$ $T\,=\, Avg\, \,Tax\,\, Rate\,\, on\,\, EBIT$ $T'\, =\, 1\,-\,T$ $NOPLAT \,=\,EBIT\,(1\,-\,T)$ $IC\, =\, Fixed\, Assets\, +\, \Delta\, NWC$ $ROIC\,=\,\frac{NOPLAT}{IC}$ $WACC\,=\,\frac{E}{V}\,R_{E}\, +\, \frac{P}{V}\,R_{P}\,+\,\frac{D}{V}\,R_{D}\,(1\,-\,T)$ $Sales\,=\, Total\,\, Operating\,\, Revenue$ $M\,=\,\frac{EBIT}{Sales}$

Description 1

• EV/sales is a crude measure, but least susceptible to accounting differences; it is equivalent to its equity counterpart, price to sales, where a company has no debt.
• EV/sales is useful when accounting differences among comparables are extreme, or where profit or cash flow figures are unrepresentative or negative. It is frequently used for unprofitable or cyclical firms where there are problems in measuring profit or cash flow further down the P&L. As a proxy for cash flow, sales has the virtue of being stable and relatively unaffected by accounting policies.
• EV/sales is also useful in identifying restructuring potential. Net margin is a key driver of this measure; low profitability (low net margin) would result in a low value for a given level of sales.
• Be careful that the sales figure is representative; generally EV/sales should not be used for companies with variable, periodic sales, such as property developers.

Caveats There are three caveats in using this multiple:

1. Sales volatility: EV/sales is frequently applied to technology firms, which are likely to have negative cash flow and/or earnings while they are in their initial growth phase. But these companies frequently have highly volatile sales as well.
2. Revenue recognition policies: Sales are not unaffected by accounting policies.
3. Sales can be substantially affected by different interpretations of accounting standards in such areas as:
— Use of gross versus net revenue in recording sales on agency transactions
— Treatment of sales where a customer has the right of return
— Long-term contracts accounted for under percentage of completion or completed contract methods

1 Suozzo et al (2001) Valuation Multiples: A Primer; NYU Sterns School of Business; UBS Warburg; Gobal Equity Research; pg. 28-29

EV/EBITDA

Equational Form

$\frac {EV}{EBITDA} \,= \,\frac{ROIC\, -\, g}{ROIC\,(WACC\,-\,g)}\,(1\,-\,T)\,(1\,-\,D)$

g solution

$g\,=\,\frac{ROIC\,[(T'\, x\, EBIT)\,-\,(EV\, x\, WACC)]}{(T'\,x\,EBIT)\,-\,(EV\, x\, ROIC)}$

Inputs

 $EV\,=\,Mkt\,Cap\,Equity\,+\,Mkt\,Value\,Debt\,-\,Cash$ $EBIT \,=\,Earnings\,\,Before\,\,Interest\,\,\&\,\,Taxes$ $EBITDA\,=\,EBIT\, +\, Depreciation\,\,\&\,\,Amortization$ $T\,=\, Avg\, \,Tax\,\, Rate\,\, on\,\, EBIT$ $T'\, =\, 1\,-\,T$ $NOPLAT \,=\,EBIT\,(1\,-\,T)$ $IC\, =\, Fixed\, Assets\, +\, \Delta\, NWC$ $ROIC\,=\,\frac{NOPLAT}{IC}$ $WACC\,=\,\frac{E}{V}\,R_{E}\, +\, \frac{P}{V}\,R_{P}\,+\,\frac{D}{V}\,R_{D}\,(1\,-\,T)$ $D\,=\,\frac{(D\,+\,A)}{EBITDA}$ $D'\, =\, (1\,-\,D)$

Description 1

• EBITDA is a proxy for operating cash flow, and EV/EBITDA – probably the most popular EV multiple – is a price to cash flow multiple. Its popularity stems from the fact that it is unaffected by differences in depreciation policy and appears unaffected by differences in capital structure.
• However, while EBITDA is closer to cash flow than other profit measures it is not a true cash flow, as it does not incorporate either asset depreciation or capital expenditure. Also, EBITDA is a pretax measure, whereas management can potentially add value through skilled tax management.
• EV/EBITDA is affected by a firm’s level of capital intensity (measured as depreciation as a percentage of EBITDA). All things being equal, higher capital intensity results in a lower EV/EBITDA multiple.
• EV/EBITDA is most useful in comparing companies with a selected peer group that has a comparable level of capital intensity; comparisons of a stock with a sector average can also be useful as long as there is not a large variation in capital intensity within the sector.
• On the other hand, comparisons of a stock with the market, comparisons across sectors and comparisons of a sector with the market are unlikely to be meaningful.

1 Suozzo et al (2001) Valuation Multiples: A Primer; NYU Sterns School of Business; UBS Warburg; Gobal Equity Research; pg. 29-31

EV/NOPLAT

Equational Form

$\frac {EV}{NOPLAT} \,= \,\frac{ROIC\, -\, g}{ROIC\,(WACC\,-\,g)}$

g solution

$g\,=\,\frac{ROIC\,\,x\big[NOPLAT\,\,-\,\,(EV\,\,x\,\,WACC)\big]}{NOPLAT\,\,-\,\,(EV\,\,x\,\,ROIC)}$

Inputs

 $EV\,=\,Mkt\,Cap\,Equity\,+\,Mkt\,Value\,Debt\,-\,Cash$ $EBIT \,=\,Earnings\,\,Before\,\,Interest\,\,\&\,\,Taxes$ $T\,=\, Avg\, \,Tax\,\, Rate\,\, on\,\, EBIT$ $T'\, =\, 1\,-\,T$ $NOPLAT \,=\,EBIT\,(1\,-\,T)$ $IC\, =\, Fixed\, Assets\, +\, \Delta\, NWC$ $ROIC\,=\,\frac{NOPLAT}{IC}$ $WACC\,=\,\frac{E}{V}\,R_{E}\, +\, \frac{P}{V}\,R_{P}\,+\,\frac{D}{V}\,R_{D}\,(1\,-\,T)$

Description 1

• NOPLAT is a more sophisticated and complete form of EBIT that allows for differences in tax efficiency and effective tax rates. If the company were all equity-financed, NOPLAT would equal earnings.
• However, the calculation of NOPLAT introduces a measure of subjectivity. This makes it harder to compare to other parties’ calculations of NOPLAT.
• The NOPLAT multiple is effectively a degeared PE and is a perfectly reasonable statistic to use provided it is measured on a consistent basis across companies.
• EV/NOPLAT can be used as a substitute for EV/EBIT.

1 Suozzo et al (2001) Valuation Multiples: A Primer; NYU Sterns School of Business; UBS Warburg; Gobal Equity Research; pg. 32

EV/FCFOPS

$\frac {EV}{FCF_{OPS}} \,= \,\frac{ROIC\, -\, g}{ROIC\,(WACC\,-\,g)}\,(1\,-\,T)$

g solution

$g\,=\,\frac{ROIC\,[(T'\, x\, EBIT)\,-\,(EV\, x\, WACC)]}{(T'\,x\,EBIT)\,-\,(EV\, x\, ROIC)}$

Inputs

 $EV\,=\,Mkt\,Cap\,Equity\,+\,Mkt\,Value\,Debt\,-\,Cash$ $EBIT \,=\,Earnings\,\,Before\,\,Interest\,\,\&\,\,Taxes$ $FCF_{OPS} = EBIT$ $T\,=\, Avg\, \,Tax\,\, Rate\,\, on\,\, EBIT$ $NOPLAT \,=\,EBIT\,(1\,-\,T)$ $IC\, =\, Fixed\, Assets\, +\, \Delta\, NWC$ $ROIC\,=\,\frac{NOPLAT}{IC}$ $WACC\,=\,\frac{E}{V}\,R_{E}\, +\, \frac{P}{V}\,R_{P}\,+\,\frac{D}{V}\,R_{D}\,(1\,-\,T)$ $NWC\,=\,Current\,Assets\,-\,Current\,Liabilities$ $T'\, =\, 1\,-\,T$

Description 1

• Core EV/FCFOPS (Free Cash Flow from Operations). ROIC is calculated using FCFOPS in the numerator.
• FCFOPS is core EBITDA less charges for capital usage and for the effect of inflation on working capital needs. FCFOPS should be measured before goodwill amortization. A calculation of operating free cash flow is shown below.
• EV/FCFOPS is a price to cash flow measure similar to EV/EBIT. FCFOPS is not a true cash flow, however, as it does not include actual capital expenditure or change in working capital; it is a normalized EBIT or a smoothed cash flow.
• FCFOPS is more comparable than EBITDA and less susceptible to accounting distortions than EBIT, and is therefore a more suitable basis for valuation multiples. FCFOPS does, however, add another layer of subjectivity via the calculation of maintenance capital spending and net working capital inflation.
• EV/FCFOPS is preferable to EV/EBITDA for comparing companies within a sector, or for comparing companies across sectors or markets where companies have widely varying degrees of capital intensity.
• Operating free cash flow is a smoothed measure of free cash flow to the firm, for which historical figures can be highly volatile. Because of this, multiples using FCFOPS can be easier to interpret than those using free cash flow.
• This multiple cannot be used when current cash flow is negative. Use normalized FCFOPS or a forward-priced multiple instead.

1 Suozzo et al (2001) Valuation Multiples: A Primer; NYU Sterns School of Business; UBS Warburg; Gobal Equity Research; pg. 32

EV/EBIT

Equational Form

$\frac {EV}{EBIT} \,= \,\frac{ROIC\, -\, g}{ROIC\,(WACC\,-\,g)}\,(1\,-\,T)$

g solution

$g\,=\,\frac{ROIC\,[(T'\, x\, EBIT)\,-\,(EV\, x\, WACC)]}{(T'\,x\,EBIT)\,-\,(EV\, x\, ROIC)}$

Inputs

 $EV\,=\,Mkt\,Cap\,Equity\,+\,Mkt\,Value\,Debt\,-\,Cash$ $EBIT \,=\,Earnings\,\,Before\,\,Interest\,\,\&\,\,Taxes$ $T\,=\, Avg\, \,Tax\,\, Rate\,\, on\,\, EBIT$ $T'\, =\, 1\,-\,T$ $NOPLAT \,=\,EBIT\,(1\,-\,T)$ $IC\, =\, Fixed\, Assets\, +\, \Delta\, NWC$ $ROIC\,=\,\frac{NOPLAT}{IC}$ $WACC\,=\,\frac{E}{V}\,R_{E}\, +\, \frac{P}{V}\,R_{P}\,+\,\frac{D}{V}\,R_{D}\,(1\,-\,T)$

Description 1

• EBIT is a post-goodwill figure. However, we believe that goodwill amortization is not an economic charge and should properly be added back to operating profit.
• EBIT is a better measure of ‘free’ (post-maintenance capital spending) cash flow than EBITDA, and is more comparable where capital intensities differ.
• EBIT is, however, affected by accounting policy differences for depreciation.
• EV/EBIT is most useful where there are relatively small differences in accounting treatment of depreciation among comparables.
• Alternatively, you can normalise depreciation.
• Note that the goodwill adjustment does not apply to financial statements reported under US GAAP. Recently-published Financial Accounting Statement 142, Goodwill and Other Intangible Assets, stipulates that effective 1 January 2002 (for calendar year companies), goodwill is no longer to be amortized.

1 Suozzo et al (2001) Valuation Multiples: A Primer; NYU Sterns School of Business; UBS Warburg; Gobal Equity Research; pg. 31

EV/IC

Equational Form

$\frac {EV}{IC} \,= \,\frac{ROIC\, -\, g}{WACC\,-\,g}$

g solution

$g\,=\,\frac{NOPLAT\,-(EV\,x\,WACC)}{IC\,-\,EV}$

Inputs

 $EV\,=\,Mkt\,Cap\,Equity\,+\,Mkt\,Value\,Debt\,-\,Cash$ $IC\, =\, Fixed\, Assets\, +\, \Delta\, NWC$ $EBIT \,=\,Earnings\,\,Before\,\,Interest\,\,and\,\,Taxes$ $NWC\,=\,Current\,Assets\,-\,Current\,Liabilities$ $T\,=\, Avg\, \,Tax\,\, Rate\,\, on\,\, EBIT$ $NOPLAT \,=\,EBIT\,(1\,-\,T)$ $ROIC\,=\,\frac{NOPLAT}{IC}$ $WACC\,=\,\frac{E}{V}\,R_{E}\, +\, \frac{P}{V}\,R_{P}\,+\,\frac{D}{V}\,R_{D}\,(1\,-\,T)$

Description 1

• Invested capital, measured from the asset side of the balance sheet, is the sum of net tangible and intangible assets and net working capital. Invested capital excludes non-core investments and other assets, and to ensure comparability these must also be excluded from enterprise value.
• Alternative calculations are possible:
1. Total capital employed: Total capital employed is invested capital plus investments and associates. Return on total capital employed differs from return on invested capital in that the numerator for return on total capital employed includes income from non-core operating investments and other assets. To ensure comparability, enterprise value must also include the value of these assets.
2. Adjusted capital employed: This is total capital employed adjusted to bring book capital closer to economic capital. These adjustments (to assets; from Stern, Stewart) include adding back accumulated goodwill amortization, LIFO reserve and cumulative write-offs of special items, among others.
3. Replacement capital employed: Capital employed at replacement value, intended to better approximate the cost of replacing the capital base at current prices and therefore the return on assets at market value. This can be roughly approximated by adding back accumulated depreciation in addition to making the other adjustments noted above.
• The enterprise equivalent of price to book value, EV/invested capital is equal to EV/NOPLAT × ROIC (whereas price to book is equal to PE × ROE).
• Note that invested capital measures only operating assets whereas net book value includes the value of the firm’s cash holdings. Net income includes return on cash, but NOPLAT does not. NOPLAT is in this respect a better indication of core profitability than net income.
• This is a useful measure for sectors where tangible assets are key. Because of its close linkage to return on capital, it is useful to view this measure together with return on capital.
• Items included in invested capital or total capital employed must be consistent with items included in enterprise value.

1 Suozzo et al (2001) Valuation Multiples: A Primer; NYU Sterns School of Business; UBS Warburg; Gobal Equity Research; pg. 34-35


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# Derivative of [s=ut+.5at^2]

by Fourthkind
Tags: 2d motion, classical physics, derivative, projectile motion, vector motion
 P: 5 Hey guys, I'm currently in year 11 (Australia) and my physics class has recently started projectile motion. I noticed in class that the elementary differential calculus I've been learning in math could be applied to the questions we are working on. I'd been itching to try it out and today, after school, I did a question and (naturally) was able to solve it using what I had learned. In the particular question I solved, it was necessary to find the derivative of the formula: f(t) = ut + .5at2 , where f(t) = SHorizontal Which I found to be: f'(t) = u + at Now, I realized that this was part of a formula that I had previously learnt: v = u + at Here the formula equates to velocity, but in the prior it equated to the slope of the tangent. Now I'm left wondering why this repetition exists and what (if any) the connection is. Thanks for any help.
 HW Helper Thanks P: 4,235 Velocity is defined as the rate of change of position with respect to time. If you were to plot position as a function of time, velocity would be the slope of the position-time curve. Similarly, acceleration is the rate of change of velocity with respect to time. If you plotted velocity as a function of time, acceleration would be the slope of the velocity-time curve.
 P: 5 Ah, yes. I realized before I fell asleep that the slope of the tangent at any point along that function with t on the x-axis is the velocity at that point in time (obviously). Thanks for the help.

 Related Discussions Advanced Physics Homework 0 Calculus & Beyond Homework 7 Calculus & Beyond Homework 10 Precalculus Mathematics Homework 10 Advanced Physics Homework 3


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Wednesday, July 16, 2014

Super Mathio

Hedge wrote this fun post on Mario Brothers and math - specifically parabolas - that got me to tweet: I also now want a video game where you jump by clicking the vertex to make a precise parabola based on where you are. She asked the good question: Can you make that?

Should have thought that one through.

Fortunately, though, it's Twitter, so brilliant people to the rescue.

Desmos tweeted:

Pretty sweet! Find it at Desmos:

Beyond my current Desmos levels, but pretty amazing. Great image use.

Then Andrew Knauft tweeted:
Find it at OpenProcessing.

You click to do the jump. Beyond my current Processing skills, but excellent.

So I did -of course- have to try it in GeoGebra. I'm pretty happy with the result. The math wasn't too hard, though the scripting the buttons and resetting graphics is always tricky for me.

It's at GeoGebraTube for you to play with. There's a data mode that let's you try to calculate the best high point first - which is where I would want to go with it. It's an insufficient data situation as I purposely left off the coordinates of the bonus box.

There are some things to notice about parabolas as you play it, and for deeper work it would be interesting to think about how to add scoring.

GeoGebra Note: Andrew Knauft also helped with this! You define piecewise functions in GeoGebra using the If command, which also has an else variant. I.e. either If[<condition>,<then>] or If[<condition>,<then>,<else>]. I was getting errors with the inequalities to make the towers, and Andrew figured out that GGB didn't like an inequality with the else form. Redefined them as Ifs alone, and I was good to go.

This was very fun. I'm curious to know if or how you would use it, or what features you might add. As always, if you have an idea for dynamicizing, let me know!

Friday, July 11, 2014

Complex Instruction

Complex instruction isn't real.

OK. I think that's enough of that. (Complex instruction is an actual instructional approach for differentiation, check it out.) This post is part my general thinking about complex numbers and, in other part, a very specific visualization.

I. In General
This summer teaching the math-history-themed capstone class for our seniors, one of the themes that came out was how complex numbers are the natural setting for much of mathematics. In my own experience, I didn't really appreciate that until grad school, and our majors are graduating without enough experience to feel like they are really numbers.

My perspective: I think counting numbers feel like numbers to us because:
• we have an intuition for them as quantities,
• we can represent them in multiple ways, we can compare them fluidly,
• we are comfortable with what operations mean,
• and we have multiple ways to compute with them.
The more of those aspects are broken down, the less they feel like numbers to us. We see that with students at large whole numbers, fractions, decimals, radicals, etc. I've surveyed about 50 math major seniors and inservice math teachers now, and they are significantly less comfortable complex numbers than radicals and transcendentals. When I shared some of those results on Twitter, several mathematicians wrote back that this means the students have not developed an appreciation for how weird the real numbers are. Part joke, all truth.

Part of the power of being a mathematician is being able to make this transition for wide classes of objects: functions, matrices, operators, manifolds, algebras, etc.

Complex numbers came up in math history in the solving of quadratics with the great Islamic mathematicians; the solving of cubics with Cardano, Bombelli and Tartaglia; the great acceptance of them under Euler; and the deep understanding of them that followed Gauss. The students saw the connections amongst the lack of acceptance for irrationals, negatives and complex, but - in general - saw their discomfort with complex as justifiable, while rejecting irrationals or negatives was just a bit silly. The exception to this was the few students who had taken a second semester of abstract algebra and saw C as a beautifully complete field.

I think the problem might be that algebra is before we see our students, and the focus on calculus and linear algebra in our program has to support the engineering students as well as serve math majors. Algebra and linear algebra seem like the natural places to start seeing complex numbers as natural and necessary.

It was probably Michael Pershan who really got me thinking about this, as he started sharing his usual deep reflections on the topic a couple of years ago.  Bonus: Mike Lawler has a couple extensions (one and two) which my students this summer found interesting and valuable.

II. The Fundamental Theorem of ... Algebra

One of the morals of the capstone class was that if mathematicians labeled a theorem as Fundamental, it's worth your focus and understanding.

If you have time, please watch at least the beginning of this terrific Numberphile video with David Eisenbud, director of MSRI at Berkeley.

Marvelous. One of the things that so infatuated me with my advisor, Nigel Higson, was his ability to motivate and moralize the mathematics. The dog on the leash is the nice metaphor here. I think the power of this representation is not in the static image he draws, but his ability to manipulate and use this imagery to connect ideas. For most of history, this has been a significant barrier in mathematics.

III. The Visual
Of course this has to end up in GeoGebra...

The ability of GeoGebra to have two graphics windows allows us to dynamicize Eisenbud's picture. Here's the GeoGebraTube page for this sketch. In works online (the student page), but is better if you can download and open it. GeoGebra does not compute in complex numbers natively - yet - so this sketch has a lot of workarounds to treat points as complex numbers. This post isn't about how to make the ssketch, though, it's about using it.

GeoGebra tip: when a point or slider is selected, you can move it with the arrow keys. Hold down shift while clicking an arrow for smaller changes.

The sketch has one window for input to a polynomial, and the second window for the output. For example:
To get that feel for what Eisenbud was doing, I turned on the trace for the points. Complex valued functions and analysis is very much about paths and what happens to them under transformation.

We can use this sketch to find real and complex roots. When you click circular path, it plots all the points in the input plane with the same magnitude as A, the blue point. The output plane shows the result. So when the orange transform of that circle goes through the origin, there's a root with that magnitude. The green slider rotates around the circle, allowing you to findthe exact value. Above is a real root for that function, $$f(x)=x^3+2x^2+x+1$$, near $$x=-1.76$$. Where are the complex roots?

One more example: Find the first root.

So, since this is a real coefficient function, the conjugate must be a root. I made this cubic to have the root $$x=2$$, so I show that root also. The transformation of the rectilinear path by the function is quite fascinating, no?

Studying different kind of roots can be pretty interesting. What do you think is going on here?

We can investigate the winding number phenomena as well:

There are so many things to notice once we have a visual. It supports more people in developing the kind of intuition and acceptance that might some day see accepted status for our complex brothers and sisters. After all, the reals are complex, too.

Seriously, I'd like to think that we're coming into an age where technology makes these ideas more accessible to a wider number of people. Can you think of other features or visualizations that we should try to make? What are the key concepts of complex numbers that need the support of a dynamic visualization?

Notes:
• GeoGebra's gif export only does the active graphics window, so these are done with the RecordIt app for Mac - highly recommended.
• It's a little weird that the function has real coefficients but complex constant, but it seemed worth it to get at some of the points that Dr. Eisenbud raises in Numberphile video.
• The work arounds consist of making separate real and imaginary functions from the input function. It looks like this:


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# Mixture of Experts¶

A Mixture of Experts layer has the following form:

$$y = \sum_{i=1}^n G(x)_i E_i(x)$$

where $$y$$ is the output of the layer, $$x$$ is the input, $$G$$ is a gating network, and $$E_1, \dots, E_n$$ are the expert networks.

A natural choice is “Softmax Gating”, which is to set $$G(x) = \mathrm{softmax}(Wx)$$ where $$W$$ is a trainable weight matrix. However, one motivation for introducing expert networks is to have a network with a large number of parameters, but where only a few experts are involved in the computation for each input. The softmax involves all the experts in the computation for each input albeit with different weights placed on each expert.

Noisy Top-K Gating, an alternative to softmax gating, is introduced in Shazeer et al 2017:

$$G(x) = \mathrm{softmax}(\mathrm{KeepTopK}(H(x), k))$$

$$H(x)_i = (Wx)_i + \mathrm{StandardNormal()} \cdot \mathrm{softplus}((Wx)_i)$$

$$\mathrm{KeepTopK}(v, k)_i = \begin{cases}v_i & \text{if } v_i \text{ is in the top } k \text{ elements of } v \\\\ -\infty & \text{otherwise } \end{cases}$$

If we ignore the random noise term in $$H(x)_i$$, which helps with load balancing between experts (see Appendix A of Shazeer et al 2017), then this is just like “Softmax Gating” except that we only allow $$k$$ non-zero entries of $$G(x)$$. In other words, only $$k$$ experts are involved in the computation for each input and which experts are determined based on the input using trainable weights.

One potential issue with Noisy Top-K Gating is that it introduces “some theoretically scary discontinuities in the output of gating function”, which Shazeer et al 2017 does to be a problem in practice. More recent work (Hazimeh et al 2021) has tried to address this issue by introducing a differentiable mechanism for gating (“DSelect-k”). I haven’t looked closely at that paper, but note that “During inference, DSelect-k only needs to evaluate a subset of the experts, which can lead to computational savings. However, DSelect-k supports conditional training only partially. At the start of training, it uses all the available experts, so conditional training is not possible.”

Fedus et al 2021 introduces the Switch Transformer, which uses a variant of the Mixture of Experts layer with Noisy Top-K Gating, K = 1 and an expert placed on each device: “We design models based off T5-Base and T5-Large (Raffel et al., 2019) to obtain up to 7x increases in pre-training speed with the same computational resources. These improvements extend into multilingual settings where we measure gains over the mT5-Base version across all 101 languages. Finally, we advance the current scale of language models by pre-training up to trillion parameter models on the ‘Colossal Clean Crawled Corpus’, and achieve a 4x speedup over the T5-XXL model.”

They also perform some analysis of the scaling properties of experts in the style of the scaling laws work and include a section on “Future Work”, which discusses some of the research challenges that remain for these models, e.g., “1. A significant challenge is further improving training stability for the largest models…2. Generally we find that improved pre-training quality leads to better downstream results (Appendix E), though we sometimes encounter striking anomalies…3. Perform a comprehensive study of scaling relationships to guide the design of architectures blending data, model and expert-parallelism…”.


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# Characterizing ecosystem phenological diversity and its macroecology with snow cover phenology

## Abstract

One critical challenge of exploring flora phenology is on characterizing ecosystem phenological diversity (EPD), and thus how EPD’s performance is influenced by climate changes has also been an open macro-ecological question. To fill these two gaps, we proposed an innovative method for reflecting EPD, by taking the advantage of the often-classified inverse factor of spatial resolution discrepancy between the used remote sensing datasets of vegetation phenological dates (green-up and brown-up) and snow cover phenological dates (SPDs) (onset and end) around the Arctic, and further, we examined the cross response/feedbacks of the two kinds of EPDs to the two categories of SPDs. We found that the circumpolar green-up and brown-up EPDs both were shrinking, driven more by the delaying of the onset SPDs than the advancing of the end SPDs; North America and North Eurasia performed with inconsistent EPD response/feedbacks to the related SPD anomalies; and further, the EPD-SPD response/feedbacks in some locations exhibited the time-lag effect, e.g., the green-up EPDs made the strongest response to the onset SPDs of two years earlier. Overall, the validated method and the new findings are of implications for improving the phenology modules in Earth system models, and the contributions of the present study have enlightening significance for kicking off the new EPD branch in macrosystem phenological ecology.

## Introduction

Following phenophase1 and phenological shift2, phenological diversity that is defined as the variety of phenological dates of life3 now is turning into another ‘hot-spot’ phenological trait, highlighted by the communities concerning biotic phenology3,4,5. Examining the performance, feature, and evolution of phenological diversity, as an important indicator of biodiversity6, is of extensive significance for understanding of how species, communities, and ecosystems may make response/feedbacks to climate changes7. To characterize this phenological trait, people proposed specific parameters such as phenological synchrony4 or asynchrony5, which are quantified as the consistency or inconsistency between the phenological dates of varying biological functional groups4,5, respectively. With such parameters used as the indicators, it was discovered that the phenological differences among ecotypes are not related to small genetic differences but simply phenotypic adaptations to different climatic conditions8. People also realized that exploring the rules of phenological diversity offers a major avenue to advance the adaptations for ecotypes like winegrapes in agriculture to severe climate anomalies9. Further, it was noticed that phenological synchrony has some consequences for ecological interactions and biotic population dynamics10, while phenological asynchrony arising from climate warming has the high potential of affecting parasite transmission, with non-linear impacts on disease burden11. These pioneering studies all suggested that investigating phenological diversity is of considerable implications for biology and ecology.

However, study on phenological diversity so far is still in its infancy. That is, the limited studies on phenological diversity are yet struggling at the initial phase of stepping forward from species to communities12. Their common scheme is to test if multiple species, hypothetically representing a community, can sustain the effect of phenological synchrony or not13. Some evidences from in-situ observations and controlling experiments alluded that maintenance of this synchrony in the context of climate changing is common14,15,16, while others demonstrated that the degrees of the changes in synchrony vary between populations17,18 or, at least, are larger than expected19,20. This contradiction has triggered a puzzle – whether shifts toward phenological asynchrony are widespread21,22,23. In other words, the evidences to date have not drawn a clear picture about how prevalent and large the shifts in phenological synchrony from species, taxonomic groups, to communities have been in response to recent climate changes7. In turn, this has set up almost no ecological theoretical bases for upscaling the phenological diversities of species and communities to infer the global performance of macro-scale ecosystems (macroecosystems), as marked by the gray-colored ‘broken road’ in Fig. 1. Hence, it is in intense demand to develop efficient methods for reflecting the situations of ecosystem-level phenological diversity.

Another evidence for this ‘infant’ point is that previous considerations of environmental forces on phenological synchrony were primarily restricted to temperature24,25,26. In fact, it was also found that snow is another key environmental factor capable of influencing ecosystem functions27. Steltzer et al. earlier noticed that while the timing of snowmelt in spring, at least partially, depends on spring air temperature, snowmelt and temperature often act independently from one another to control the flowering timing28. Iler et al. were also aware that some species seem to be approaching their limits of phenological shifts in response to snowmelt rather than temperature29. Winkler et al. further observed that the near-term, community-wide changes in phenology responding to the variations in snowmelt timing are likely to be contingent on the legacy of past climates on community structures30. These endeavors exposed more ecological roles of snow cover phenology27 on plant seasonal timing, but their inferences were still aimed at community levels (Fig. 1, with the mathematical scheme of the representative up-scaling method7 shown). So, the knowledge about the macroecological effect31 of snow cover phenological anomalies on ecosystem phenological diversity (EPD) is still almost void.

As pointed out in Fig. 1, a ‘big’ challenge of this field in the next stage is to improve our ability to understand such eco-effects and project the full consequences of various environmental drivers on larger-scale community- and ecosystem-level phenological diversities7. One of the reasons of deeming it ‘big’ is that these two levels tend to show complex phenological combinations and interactions22, particularly for the latter ones. This is illustrated by the fact that species-specific climate response within ecological communities proved to be capable of disrupting the synchrony of co-evolved mutualisms, which are generally based on the shared timing of seasonal bio-events24. This kind of disruption effects may become stronger at the ecosystem levels, investigating whose underlying ecological processes, consequently, facilitates bettering Earth system models in simulating global changes32. However, simultaneously collecting the datasets of phenological diversity at the species and macroecosystem scales based on those traditional in-situ phenological observations or small-scale experiment approaches1,7 is a difficult, actually almost impossible, task in practice. This, substantially, restricts quantitative assessments of EPDs from the beginning, also as the root cause of the last ‘knowledge-void’ problem.

To address these two disciplinary-foundation gaps, we creatively assumed the newly-published time-series data of plant phenological product (1999–2013)33 and the global data of snow cover phenological dates (SPDs) (2001–2014)34, together as an alternative strategy for quantitatively characterizing the EPDs (green-up and brown-up) and SPDs (onset and end) over the Northern Hemisphere (>45°N) (NH), to examine: (i) whether the Arctic circumpolar EPDs performed with some trends during the study period, and (ii) whether their developments can be attributed to the interannual SPD variations. Note that the ‘interannual’ here considered the year-long phase from snow cover onset to ecosystem brown up of the next year (OB) as one ‘integral year’ for exploring the response, while ecosystem green up to snow cover end of the next year (GE) for feedbacks. Deriving both EPDs’ and SPDs’ trends and their relationships facilitates handling more mysteries about EPD, which is critical for comprehensive predictions of future shifts in EPD due to climate changes.

## Results

### EPDs’ and SPDs’ trends

The two histograms of the statistically significant green-up and brown-up EPD trends (1999–2013) over the NH (SI Appendix, Fig. S1A,B, respectively) were derived (Fig. 2A,C). For the whole NH, these key EPD features both seemed to be shrinking (the average green-up EPD trend = −0.21 days/decade and the average brown-up EPD trend = −0.17 days/decade). For more details, the histograms of those statistically significant green-up and brown-up EPD trends over the North America (>45°N) (NA) (SI Appendix, Fig. S2A,C) and North Eurasia (>45°N) (NE) (SI Appendix, Fig. S2B,D) separately were also derived. For the NE, the degradations in green-up and brown-up EPDs were both stronger (the average green-up EPD trend = −0.26 days/decade and the average brown-up EPD trend = −0.21 days/decade) than those of the NH; but for the NA, its degradation in brown-up EPD was weaker, even negligible (the average brown-up EPD trend = −0.02 days/decade), while its green-up EPD development was in a totally negligible but still positive tendency (the average green-up EPD trend = 0.0001 days/decade). In summary, no obvious decreasing or increasing in EPD over the NA was observed, but an apparent decreasing in EPD over the NE was detected.

### EPDs’ response/feedbacks to SPDs

The simultaneous statistics of the responses and the delayed responses of the green-up EPDs to the onset SPDs (SI Appendix, Fig. S5A,B) and end SPDs (Fig. 3A,B) over the NH, NA, and NE were derived (Fig. 4A). In terms of Rmean (the mean of the coefficients of the statistically-significant correlations calculated by following the method in sub-Section ‘Analyses’), it could be inferred that the circumpolar green-up EPDs (>45°N) were briefly affected negatively by snow cover onsets in a one-OB-year-delayed way (denoted as DO-D, hereafter). The responses over the NE worked in the same pattern (DO-D, with the subscripts ‘-D’ marked to indicate the time-lag effect hereafter), whereas the EPDs over the NA dominantly were influenced negatively by snow cover onsets in a within-the-same-OB-year way (DO).

Figure 4B shows the statistics of the responses and the delayed responses of the brown-up EPDs to the onset SPDs (SI Appendix, Fig. S6A,B) and end SPDs (SI Appendix, Fig. S7A,B) over the NH, NA, and NE. By following Rmean, we found that the brown-up EPDs over the whole NH were driven positively by snow cover onsets in a one-OB-year-delayed way (DO-D). However, neither the NA nor the NE acted in the same manner. The brown-up EPDs over the NA were relatively more affected negatively by snow cover onsets, and the brown-up EPDs over the NE were more driven positively by snow cover onsets, both in a within-the-same-OB-year way (DO).

The synchronous statistics of the feedbacks and the delayed feedbacks of the green-up EPDs to the onset SPDs (SI Appendix, Fig. S5C,D) and end SPDs (Fig. 3C,D) over the NH, NA, and NE were generated (Fig. 4C). On account of Rmean, it was concluded that the Arctic circumpolar green-up EPDs (>45°N) dominantly influenced snow cover ends negatively in a within-the-same-GE-year way (DE). The feedbacks over the NA behaved in the same mode (DE), whereas the green-up EPDs over the NE wholly exerted a positive effect on snow cover ends in a one-GE-year-delayed way (DE-D).

Figure 4D presents the statistics of the feedbacks and the delayed feedbacks of the brown-up EPDs to the onset SPDs (SI Appendix, Fig. S6C,D) and end SPDs (SI Appendix, Fig. S7C,D) over the NH, NA, and NE. Based on the index Rmean, it was derived that the Arctic circumpolar brown-up EPDs (>45°N) primarily made a positive feedback to snow cover onsets in a one-GE-year-delayed way (DO-D). The dominant feedbacks over the NE behaved similarly, but the brown-up EPDs over the NA dominantly interfered with snow cover ends negatively in a within-the-same-GE-year way (DE).

The specific parameters relating to all of the above-listed cases, performing with the optimal correlations statistically, are listed in Table 1. For green-up EPD’s response and brown-up EPD’s feedbacks, the NE acted like the NH, while the NA did the best with other SPDs in different ways. For green-up EPD’s feedbacks, the NA acted more like the NH, also in terms of the ratio between the covers of negative and positive trends. For brown-up EPD’s response, neither the NA nor the NE acted similarly as the NH. Moreover, the findings about the distinctive area ratios between NA and NE for some EPD-SPD response/feedback cases project a new challenge following this study, i.e., what is the mechanism of ecosystem-snow interactions causing such differences.

Further, the cases with the subscripts ‘-D’ indicate that the time-lag effect existed between EPDs and SPDs in their response/feedbacks. For example, the green-up EPDs showed the strongest response to the onset SPDs of two years earlier in some locations, and this has validated the hypothesis of the time-lag effect existing. The specific situations in details would become more complicated when the scattered spatial distributions of the grids with the best correlations derived were examined (Fig. 3; SI Appendix, Figs S5S7).

Overall, the green-up and brown-up EPDs for the NH, NA, and NE were shrinking in a whole sense, and this was triggered more by the delaying of snow cover onsets, instead of the advancing of snow cover ends; the green-up EPDs over the NH, NA, and NE consistently implemented their feedbacks more closely to the end SPDs in the last decade, whereas the brown-up EPDs performed with divergent feedbacks to either of the SPDs; finally, the EPD-SPD response/feedbacks in some cases also exhibited the time-lag effect, e.g., the green-up EPDs showing the strongest response to the onset SPDs of two years earlier.

## Discussion

Compared to the currently-mainstreamed researches on attempting to attribute flora phenological changes to long-term temperature trends based on meta-analyses7 of species-level observations for deriving community-level phenologicial knowledge12,13,14,15,16, this study directly concentrating on macrosystem phenological ecology by innovatively using the derivations from different remote sensing data33,34 is relatively rare, particularly accounting for snow cover phenology27 as another essential climate force4,28,30. Those macroecological findings can help to fill the gap (Fig. 1) toward understanding of how climate changes influence Arctic circumpolar ecosystem phenology and how the changes of ecosystem phenological traits adjust the modes of climate developments. Overall, this work has opened an enlightening way for studying the EPD branch of macrosystem phenological ecology.

As previous studies showed that the difficulty in attributing changes in phenological synchrony to temperature change can be a function of both methodology and biology7, our study mirrored this difficulty from the same two aspects as well. In methodology, the applied means of correlation analysis is still featured with uncertainties, as illustrated by the inconsistency between the derived SPD traits over the NH, NA, and NE (Table 1). Generally, the EPD change and response/feedback performance for the whole NH shall follow the mode of NA or NE that plays a leading role, but an exceptional case exists (see the row of ‘brown-up SPD response’ in Table 1). This inconsistency presents the shortage of the assumed method when it meets the extreme scenario with approximate area proportions for the positive and negative correlation cases. In light of the suspect that the high uncertainty might be briefly related to the short temporal length of the time series of the analyzed data7, the authors suggested to develop more fundamental EPD data of longer time series than the used one for retrieving more reliable EPD values and reducing the uncertainties.

In biology, the complexity in the processes of snow variations influencing EPD trends is also cued by the results. The cases of the derived statistically significant EPD trends approaching 0 are quite few (see Fig. 2A,C), implying that the interannual variations of green-up and brown-up EPD values were intense. This suggested that apart from the interannual variations of snow cover timing, EPD may also be affected by other environmental factors. In fact, even for the snow factor, it may be affected by other environmental factors36, as evidenced by the similar scenarios that few cases of statistically significant snow phenophase trends approaching 0 were extracted (Fig. 2B,D). Substantially, it is challenging to link EPD variations directly to SPD changes because snow is a complex phenological cue37. In the case of forest ecosystems, snow cover can affect land surface temperature, solar irradiance, and evapotranspiration, and these environmental factors can also reversely reshape the morphology of snow cover distribution, all together regulating the timing and magnitude of phenological shifts38,39. These all suggest that more comprehensive analyses, with the potential environmental factors40 as many as possible considered, need to be carried out later.

Although the used methods take on the above-listed uncertainties, this study is of fundamental implications for filling the targeted gaps between ecosystem-level phenological developments and climate changes and further projecting the future of biosphere under global change. In other words, although the underlying mechanism of species and communities behaving with such global EPD performance after adapting to environmental changes is unclear, the EPD-SPD response/feedback rules statistically derived in this study can cue a new framework of Arctic circumpolar ecosystem processes involving inter-annual phenological variations of snow cover accumulations to improve the performance of phenology modules in current Earth system models27.

In fact, it is a so difficult task to investigate the underlying mechanism of SPD influencing EPD, which, substantially, relates to an integration of complex macro-ecological interactions rather than a plus of traditional physiological responses30,39. As the previous studies told, the changes in phenological diversity can have fitness consequences41 and affect ecosystem-level properties like primary productivity42 and pollination43, but such consequences do not always emerge44 or seem to be scale-dependent45. It is also unclear the extent to which species will be able to evolutionarily adapt to restoring phenolgical synchrony46. This means that it is inappropriate to give some simple hypothetic explanations to the derived rules. Thereby, the next challenge in this direction is to further handle the gap between the remote sensing-based inferences on EPD changes at the regional/continental scales and the previous findings about phenological diversity performance derived by upscaling from species to communities12,13,14,15,16.

In summary, an innovative method was proposed for breaking through the conventional gap of characterizing ecosystem phenological diversity. The validated method and those macroecological findings are of implications for improving the phenology modules in various Earth system models, and the contributions in this study can mean a new era of the novel EPD branch in macrosystem phenological ecology.

## Methods

The rationale of operating this study was directly using the remote sensing-derived datasets in macrosystems ecology47 to handle the basic challenges long faced by traditional phenological ecology. The proposed method of characterizing EPD was based on the discrepancy between the spatial resolutions of the two used datasets33,34. As we know, snow cover onsets (ends) tend to simultaneously cover massive regions27, while ecosystem green-up (brown-up) dates48 are distinctive for different communities and species at finer scales7. The 0.5° spatial resolution of the snow dataset34 can reflect the trait of SPDs in the former scenario, while the 4 km (~0.0357°) resolution of the plant phenological date product33 can mirror the feature of flora phenological dates in the latter one. Given that one grid in the SPD data34 corresponds to ~14 × 14 grids in the EPD-source data33, the EPD characterization was operated by unifying the resolution of the EPD-source data to the SPD data, with each grid of the derived EPD data defined as the standard deviation of phenological dates within its corresponding 14 × 14 grids, analogous to the definition of phenological synchrony4 and asynchrony5. Then, their linear correlations with the effect of characterizing the EPD-SPD response/feedbacks were examined, in the within-the-same-OB-year, one-OB-year-delayed, within-the-same-GE-year, and one-GE-year-delayed manner, respectively. This innovative remote sensing-based solution strategy is theoretically reasonable, with the high potentials of handling the problem of EPD data shortage that lead to the ‘big challenges’ targeted in this study.

### EPD data

The EPD-source data was the Arctic circumpolar vegetation dynamics product for global change study33. Specifically, based on the SPOT VGT data, Gonsamo and Chen derived the first set of phenology index-based plant dynamics product, comprising green-up and brown-up phenological dates, for the NH during 1999–201333. With the factors like snow27 possibly influencing the derivation of the phenological dates considered, the spatial resolution of the product was unified into 4 × 4 km (0.03571428 × 0.03571428°). The validation based on the data from the flux tower sites of deciduous broadleaf forests, evergreen needle-leaf forests, mixed forests, and wetlands over the NA and NE suggested good agreements between the phenological dates from this Arctic circumpolar vegetation dynamics data and the reliable estimates from CO2 flux measurements. The validation proved that the circumpolar vegetation dynamics product is an improvement over the operational global MODIS Combined Land Cover Dynamics MCD12Q2 product for the Arctic circumpolar region33. For derivation of the EPDs required in this study, the standard deviations of the phenological start and end dates for the matrix of 14 × 14 grids33 corresponding to any grid in the SPD dataset34 were calculated (Eq. 1) to characterize the green-up and brown-up EPDs, respectively.

$${v}_{{\rm{EPD}}}^{i}=\sqrt{\frac{1}{196}\mathop{\sum }\limits_{j=1}^{14}\mathop{\sum }\limits_{k=1}^{14}{({x}_{j,k}^{i}-\overline{{x}^{i}})}^{2}}$$
(1)

where $$\overline{{x}^{i}}=\frac{1}{196}\mathop{\sum }\limits_{j=1}^{14}\mathop{\sum }\limits_{k=1}^{14}{x}_{j,k}^{i}$$, and $${v}_{{\rm{EPD}}}^{i}$$ denotes the i th SPD grid-related EPD value, derived based on the corresponding phenological date $${x}_{j,k}^{i}$$ set of 14 × 14 grids. Compared to the ranges of the start and end values within each matrix33, assuming standard deviation can theoretically avoid the influences of the outlier values in some 4 × 4 km grids and hence can better quantify the phenological diversities at the ecosystem scales. The theoretical foundation of proposing such a measure is that the used EPD-source data33, substantially, is an EPD product at the finer scale.

### SPD data

The SPD data was extracted from the snow seasonal timing data in the northern middle and high latitudes (2001–2014)34. This data is an improved merge of five extensively-used snow datasets, including the reanalyzed dataset of daily snow depths generated by the Canada Meteorological Center, the binary daily snow cover mask derived from both the Interactive Multi-sensor Snow and Ice Mapping System and the Northern Hemisphere Weekly Snow Cover and Sea Ice Extent, the 8-Day Level 3 snow cover fraction product derived from the MODIS data, and the snow water equivalent data derived from the Near-real-time Ice and Snow Extent data. A multi-data approach was employed in developing the combined snow cover phenology matrix that can integrate snow cover timing information probed from multiple sources of snow observations. In this data, the snow cover accumulation season is defined to be from November of previous year to February of current year, and the snow cover melting season is from March to June of current year34. For daily snow observations, the snow cover onset date is defined as the first five consecutive days on which snow was observed to cover the ground surface in accumulation season, and the snow cover end date is defined as the last five consecutive days when snow cover was noticed in the melting season34. Specifically for the derivation of the SPDs used in this study, the Arctic circumpolar region higher than 45°N in latitude was simply extracted from the SPD-source data.

### Analyses

This study used partial-correlation analysis to probe the response/feedback effects of the green-up and brown-up EPDs to the onset and end SPDs around the Arctic. This approach can exclude the confounding effect of other climate variables (e.g., rainfall, temperature, and solar radiation) and the covariate effect between snow cover onsets and ends32. Specifically, the correlation degrees between the derived EPDs (green-up and brown-up) and SPDs (onset and end) were explored, and further, the explorations were extended to consider the effects of delayed response/feedbacks. To characterize this time-lag effect, the year-long period briefly from snow cover onset to brown-up of the next year as one ‘effective’ OB year was defined for investigating the effect of response, and the year-long period briefly from green-up to snow cover end of the next year as one ‘effective’ GE year was defined for probing the feedback effect. The correlations between the EPDs and SPDs both from the same OB year were classified as exploring the response in the within-the-same-OB-year way, and the correlations between the EPDs and the SPDs from the earlier OB year as investigating the response in the one-OB-year-delayed way; the correlations between the EPDs and SPDs both from the same GE year as examining the feedbacks in the within-the-same-GE-year way, and the correlations between the EPDs and the SPDs from the later GE year as probing the potential feedbacks in the one-GE-year-delayed way. The latter scenarios in these response/feedback explorations could reflect the time-lag effect. Since the checking of the time-lag effect was operated onto two explicit time series of the used datasets, the autocorrelation phenomenon often met in exploring the relationships between the time series of multiple kinds of features27 could be neglected here. Note that in subject to the short term of the assumed EPD and SPD data, only the effect of one year delaying was taken into account. Correspondingly, all of the partial correlation analyses were implemented at the significance level of p < 0.1 (indicated by *) in order to extract more grids with distinguishable response/feedbacks.

## References

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28. Steltzer, H. H., Landry, C. C., Painter, T. H. T., Anderson, J. J. & Ayres, E. E. Biological consequences of earlier snowmelt from desert dust deposition in alpine landscapes. Proc. Natl. Acad. Sci. USA 106, 11629–11634 (2009).

29. Iler, A. M., Høye, T. T., Inouye, D. W. & Schmidt, N. M. Nonlinear flowering responses to climate: are species approaching their limits of phenological change? Phil. Trans. R. Soc. B 368, 20120489 (2013).

30. Winkler, D. E., Butz, R. J., Germino, M. J., Reinhardt, K. & Kueppers, L. M. Snowmelt timing regulates community composition, phenology, and physiological performance of Alpine plants. Front. Plant Sci. 9, 1140 (2018).

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32. Piao, S. et al. Leaf onset in the northern hemisphere triggered by daytime temperature. Nat. Commun. 6, 6911 (2016).

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34. Chen, X., Liang, S., Cao, Y., He, T. & Wang, D. Observed contrast changes in snow cover phenology in northern middle and high latitudes from 2001-2014. Sci. Rep. 5, 16820 (2016).

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38. Trujillo, E. et al. Elevation-dependent influence of snow accumulation on forest greening. Nat. Geosci. 5, 705–709 (2012).

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40. Wang, X. et al. Snow cover phenology affects alpine vegetation growth dynamics on the Tibetan Plateau: Satellite observed evidence, impacts of different biomes, and climate drivers. Agri. Forest Meteorol. 256, 61–74 (2018).

41. Doiron, M., Gauthier, G. & Lévesque, E. Trophic mismatch and its effects on the growth of young in an Arctic herbivore. Glob. Chang. Biol. 21, 4364–4376 (2015).

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43. Burkle, L. A., Marlin, J. C. & Knight, T. M. Plant-pollinator interactions over 120 years: Loss of species, co-occurrence, and function. Science 339, 1611–1615 (2013).

44. Burthe, S. et al. Phenological trends and trophic mismatch across multiple levels of a North Sea pelagic food web. Mar. Ecol. Prog. Ser. 454, 119–133 (2012).

45. Senner, N. R., Stager, M. & Sandercock, B. K. Ecological mismatches are moderated by local conditions for two populations of a long-distance migratory bird. Oikos 126, 61–72 (2017).

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## Acknowledgements

Thank Dr. Alemu Gonsamo for publishing the circumpolar vegetation dynamics product and Dr. Xiaona Chen for distributing the snow cover phenology data. This work was financially supported by the National Natural Science Foundation of China (Grant No. 31670718 and 31870531).

## Author information

Authors

### Contributions

Y.L. designed the study and analyzed the data. Y.L. and J.H. both contributed to paper writing.

### Corresponding author

Correspondence to Yi Lin.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Lin, Y., Hyyppä, J. Characterizing ecosystem phenological diversity and its macroecology with snow cover phenology. Sci Rep 9, 15074 (2019). https://doi.org/10.1038/s41598-019-51602-1

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s41598-019-51602-1


text:
Here is the comparison for the average temperature in a cosmological simulation.All codes are using the 4th order scheme for the potential gradient.

Cholla is using PPMP HLLC and SIMPLE, $\beta=0$.

Now using Cooling and UV Background:

## Power Spectrum

Here is the power spectrum for Cholla and Enzo without using Cooling/UV:

### $\eta=0.030$ and $\beta=0.2$:

These where the parameters that best matched Ramses: Gas is much colder.

Now using Cooling and UV Background:


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d Word Problem 19 | Job Exam Rare Mathematics (JEMrare) - Solved MCQs

# Word Problem 19

Problem 19: A man buys milk at Rs 2 per kg. Mixing it with water he sells the mixture at the same price. If his profit is 11-1/9 %, the quantity of water added to each kg of milk is ;

a) 1/3 kg                b) 1/9 kg                     c) 1/5 kg           d) 1/2 kg

Solution:

We are given the following information;

Cost of milk = Rs 2 per kg

Sale price of mixture = Rs 2 per kg

Profit of mixture sale = 11-1/9 %

Quantity of water added per kg of milk = ?

We suppose

cost price of mixture = Rs Y per kg

We know Profit is given by the formula;

Profit = SP – CP

Where SP = sale price

CP= cost price

and Profit = 11-1/9 % of CP

or

$Profit = \frac{11 \frac{1}{9} }{100}\times Y$

or

$Profit = \frac{100Y}{900}$

Putting value of profit, SP and CP in above formula, we get;

$\frac{100Y}{900}=2-Y$

We get;

$\frac{1000Y}{900}=2$

and finally CP Y is given as;

$Y=\frac{1800}{1000}=\frac{18}{10}=\frac{9}{5}=1.8 Per Kg$

Now assuming that water is free of cost and use blending formula as below;

(Note:

We subtract both entities

prices from mixture

price and put result across

crossed lines in blending formula !)

So we got;

Milk : Water = 1.8 : 0.2 = 9 : 1

or

Milk : Water = 1:1/9

or

we can say that for each 1 kg of milk 1/9 kg of water is added

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<meta http-equiv="refresh" content="1; url=/nojavascript/"> Everyday Stoichiometry ( Study Aids ) | Chemistry | CK-12 Foundation

# Everyday Stoichiometry

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Everyday Stoichiometry

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### Everyday Stoichiometry

Stoichiometry is how a chemist is able to keep track of the chemicals being used in a reaction.  This enables them to be able to know how much of each chemical is being consumed, how much is needed, and what the limiting reagent is.

1 bag of lettuce(L), 4 carrots(C), 1 cup almonds(A), 3 tomatoes(T), and 2 avocado (Av).
Then this would be the equation for this salad:

L $+$ 4C $+$ A $+$ 3T $+$ 2Av $\rightarrow$ LC4AT3Av2.

That is the equation only for one (large) salad. However, you are making salad for your school reunion and need 3 of these (large) salads. How many bags of lettuce, and how many carrots will you need? What conversion factors will you use? Lastly, is 5 tomatoes enough?

For the explanation of how to go about the questions and a better understanding of everyday stoichiometry, click here.

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# Tag Info

2

$^1_1H$ is a special case as the proton does not have any other nucleons to bind onto. I suppose that you could call the binding energy per nucleon zero which means that you require no energy to split up the nucleus of $^1_1H$ into its constituent parts? Note also that the binding energy per nucleon is not necessarily the full measure of whether a nucleus ...

3

[The figure shown in the OP question above ...] is experimental data for the ratio $R =$ [...] as a function of the centre of mass energy $\sqrt{s}$ The so called cross section ratio $$R[~\sqrt{s}~] = \frac{\sigma^{(0)}[~e^+~e^- \rightarrow \text{hadrons}, \sqrt{s}~]}{\sigma^{(0)}[~e^+~e^- \rightarrow \mu^+~\mu^-, \sqrt{s}~]}$$ ?? Actually, ...

2

'Crossover diagrams' exist when the final products are indistinguishable. When using Feynman diagrams for computations, you need to add all distinct diagrams that depict the same process, i.e., have the same external lines. Both the crossed and uncrossed diagrams for the electron-electron process have the same final products, since you cannot tell the ...

Top 50 recent answers are included


text:
Intermediate Algebra (6th Edition)

The solution is $\Big(-\infty,-\dfrac{8}{3}\Big]\cup(2,\infty)$
$\dfrac{3}{8}x+1\le0$ or $-2x\lt-4$ Solve the first inequality: $\dfrac{3}{8}x+1\le0$ Take $1$ to the right side: $\dfrac{3}{8}x\le-1$ Take $8$ to multiply the right side: $3x\le-8$ Take $3$ to divide the right side: $x\le-\dfrac{8}{3}$ Expressing the solution in interval notation: $\Big(-\infty,-\dfrac{8}{3}\Big]$ Solve the second inequality: $-2x\lt-4$ Take $2x$ to the rigth side and $4$ to the left side: $4\lt2x$ Rearrange: $2x\gt4$ Take $2$ to divide the right side: $x\gt\dfrac{4}{2}$ $x\gt2$ Expressing the solution in interval notation: $(2,\infty)$ Since the compound inequality is formed by the word "or", the solution is composed by the numbers that satisfy each inequality: The solution is $\Big(-\infty,-\dfrac{8}{3}\Big]\cup(2,\infty)$