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# zbMATH — the first resource for mathematics

Free resolutions of simplicial posets. (English) Zbl 0882.06004
A simplicial poset is a finite poset with minimal element in which every interval is isomorphic to a Boolean lattice. This notion generalizes the face poset of a simplicial complex. Indeed, any simplicial poset may be realized as the face poset of a regular cell complex in which every cell is isomorphic to a simplex. R. Stanley defined a ring $$A_P$$ associated with a simplicial poset $$P$$ which generalizes the face ring (or Stanley-Reisner ring) of a simplicial complex. In the paper under review, the author extends several important calculations from face rings of simplicial complexes to face rings of simplicial posets. The Betti polynomial of module $$M$$ is the generating function for the ranks of the terms in a minimal free resolution of $$M$$. The first main result of the paper calculates the Betti polynomial of $$A_P$$ as a module over $$k[V]$$, the polynomial ring on the set $$V$$ of vertices of $$P$$. The result is expressed in terms of ordinary (topological) Betti numbers of subcomplexes of the realization of $$P$$. The proof follows the general outline of the proof in the simplicial complex case [M. Hochster, “Cohen-Macauley rings, combinatorics, and simplicial complexes”, in: Ring Theory II, Proc. Second Oklahoma Conference 1975, 171-223 (1977; Zbl 0351.13009)], but uses a refinement of the usual $${\mathbb{Z}}^n$$-grading of the Koszul complex ($$n=|V|$$). A similar, but more complicated analysis leads to the second main result, a calculation of the cohomology $$H^*(A_P):=H^*({\mathcal K}({\mathbf x}^\infty,A_P))$$ of a certain chain complex $${\mathcal K}({\mathbf x}^\infty,A_P)$$ determined by combinatorially defined local cohomology can be used to compute depth of modules. Thus the author is able to prove that (i) the depth of $$A_P$$ is a topological invariant, that is, depends only on the realization of $$P$$ as a topological space, and (ii) the depth of $$A_P$$ is equal to $$1+m$$, where $$m$$ is maximal such that the $$m$$-skeleton of the realization of $$P$$ is a Cohen-Macauley complex. These also generalize known properties of simplicial complexes. The article ends with a detailed example.

##### MSC:
 06A11 Algebraic aspects of posets
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##### References:
 [1] Björner, A., Posets, regular CW complexes and Bruhat order, Europ. J. combin., 5, 7-16, (1984) · Zbl 0538.06001 [2] Björner, A.; Hibi, T., Betti numbers of Buchsbaum complexes, Math. scand., 67, 193-196, (1990) · Zbl 0727.55011 [3] A. Duval, 1991, Simplicial Posets:f, Massachusetts Institute of Technology [4] Garsia, A.; Stanton, D., Group actions on stanley – reisner rings and invariants of permutation groups, Adv. in math., 51, 107-201, (1984) · Zbl 0561.06002 [5] Hibi, T., Quotient algebras of stanley – reisner rings and local cohomology, J. algebra, 140, 336-343, (1991) · Zbl 0761.55015 [6] M. Hochster, 1977, Cohen-Macaulay rings, combinatorics, and simplicial complexes, Ring Theory II: Proc. of the Second Oklahoma Conference, 171, 223, Dekker, New York [7] Lang, S., Algebra, (1984), Addison-Wesley Menlo Park [8] Munkres, J., Elements of algebraic topology, (1984), Benjamin/Cummings Menlo Park · Zbl 0673.55001 [9] Munkres, J., Topological results in combinatorics, Mich. math. J., 31, 113-128, (1984) · Zbl 0585.57014 [10] Reisner, G., Cohen – macaulay quotients of polynomial rings, Adv. in math., 21, 30-49, (1976) · Zbl 0345.13017 [11] Smith, D., On the cohen – macaulay property in commutative algebra and simplicial topology, Pac. J. math., 141, 165-196, (1990) · Zbl 0686.13008 [12] Spanier, E., Algebraic topology, (1966), McGraw-Hill New York · Zbl 0145.43303 [13] Stanley, R., Combinatorics and commutative algebra, Progress in mathematics, 41, (1995), Birkhäuser Boston [14] Stanley, R., Enumerative combinatorics, (1986), Wadsworth & Brooks/Cole Monterey [15] Stanley, R., fh, J. pure appl. algebra, 71, 319-331, (1991)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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## Mathematics Formulas 10th Class Function,Ncert Solutions Up Board Class 10th,1990 Ranger Bass Boat For Sale Pdf - PDF Books

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Mathematics Formulas 10th Class Function

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# Background

Information on thresholds and integration time can be found here Google sheet
Old version

An excellent explanation about timing for different detectors PDF

This section describes Synchrotron-Radiation (SR) studies carried out for the EPIC experiment. Two types of events will be described below. On one hand we have physical events, which correspond to what we expect to measure in the lab. On the other hand we have technical hepmc events, which correspond to all the information stored in a hepmc file in between lines that begin with the letter "E". We will refer to these as real and technical events, respectively.

The Synrad+ simulations provide a series of single-photon technical events in hepmc files. Each photon comes from a different vertex and has, besides the photon momentum vector and vertex coordinates, a weight that maps a given photon to a flux (photons/sec). The energy spectrum of these photons is shown below.

An event generator was constructed by creating a histogram with a photon per bin, and the bin content corresponding to the weight of that photon. To generate an event, the user begins by predefining a time integration window ${\displaystyle T}$ within which SR photons will be collected. Subsequently, photons are sampled from the aforementioned histogram until the sum of all inverse weights is greater than the predefined time integration window. That is, we continue sample photons as long as:

${\displaystyle \sum _{k=1}^{N}{\frac {1}{w_{i}}}

The figure below shows an example event for a 100-ns time integration window. The base of the arrow represents the vertex from which a given photon emerges and enters the detector, the length of the arrow corresponds to the relative size of the momentum component being represented, and the color scale indicates the total momentum of the photon, with darker colors corresponding to lower momenta.

Example 100-ns-wide SR event

The code that does this sampling can be found here. The output corresponds to a hepmc file with technical events matching real events for the given integration window. At the moment, the sampling is done based on single photons generated for an electron beam of energy ${\displaystyle E_{e}=10\ {\rm {GeV}}}$ and a current of 2.5 A. These hepmc files can subsequently be propagated through Geant (e.g. in DD4HEP) to determine the number of hits recorded in different subdetectors.

After generating 400k 100-ns-wide events (with this time integration window, events have on average 250 photons) and passing them through Geant in DD4HEP with the following command:

npsim --runType batch --numberOfEvents -1 --compactFile \${DETECTOR_PATH}/epic.xml --inputFiles input.hepmc --outputFile output_file.edm4hep.root

we get a root file with recorded hits, which can be used to estimate the expected number of hits per event in different subdetectors. See image below. Subdetectors not included in that plot did not register any SR hits from the 400k events propagated.

Hit frequency for different subdetectors.

This plot was generated with the EPIC "brycecanyon" detector configuration and with a 5 µm gold layer inside the beampipe. Below, we can see a comparison between this configuration and the case with no gold coating inside the beampipe, for the three innermost silicon layers:

Scatter plot of SR hits in the three innermost silicon layers of the EPIC detector. These hits are collected from 100-ns-wide SR events. Left: no gold coating in the beampipe. Right: 5 µm gold coating.

The gold coating reduces the expected hits in these layers by two orders of magnitude.

### Electron Beam - Gas

Study for ATHENA proposal is documented in https://wiki.bnl.gov/athena/index.php?title=Beam_backgrounds#Electron_Beam

Input hepmc for the simulation is:

Macros which were used to run the simulation are located in:

Output file from the ATHENA simulation:

The large hadronic cross section of the p/A + H2restgas interactions is a concern. Secondary interactions of the particles produced in hadron beam gas events with aperture limitations, i.e. magnets, beam pipe, masks are also one of the main sources of neutrons that thermalize within the detector hall. In the following we will discuss the results on the occupancy / rates from the different ATHENA subdetectors due to proton beam gas events.

We use Charles Hetzel’s vacuum simulation after 10000Ahr (All pumps on option), the range of the beam gas we consider is from -5.5m to 5m. We use the Pythia8 fixed target events including beam effects ( cross angle, crab cavity, beam energy spread, angular beam divergence, bunch length) for our simulation. The cross section for proton H2restgas interactions is 78.54mb (Pythia8). The background collision rate from proton beam gas is 31.45kHz (275GeV), 30.74kHz (100GeV), 30.96kHz (41GeV).

 Here we show the vertex Z distribution for the proton beam gas background collision.

We use the "ddsim" in the EPIC default configuration for the full simulations. In the following plot, we show the hit rate in each sub detectors.

In the following table we show the numbers of the hits per second for sub detector from dd4hep (no threshold appled):

 VertexBarrel 259000 DIRCBar 2.39E+03 MRICH 1.55E+05 EcalEndcapP 2.18E+07 DRICH 3400 TrackerEndcap 1.40+E6 MPGDTrackerBarrel 8.87E+04 EcalEndcapN 2.66E+06 EcalBareel 2.43e+07 EcalBareelScFi 1.32E+07 HcalBarrel 6.16E+06 HcalEndcapP 1.02E+08 HcalEndcapN 390000

In the following table we show the numbers of the hits per second for sub detector after the threshold applied:

 VertexBarrel 259000 DIRCBar 2.39E+03 MRICH 5.17E+04 EcalEndcapP 3.15E+05 DRICH 1130 TrackerEndcap 1.40E+06 MPGDTrackerBarrel 8.86E+04 EcalEndcapN 4.39E+05 EcalBareel 2.26e+07 EcalBareelScFi 5.77E+04 HcalBarrel 1.35E+06 HcalEndcapP 1.53E+07 HcalEndcapN 39800

For Ecal_Barrel, the readout unit is 0.5mmX0.5mm pixel. For Ecal_endcap, there is no segmentation, the readout for the hits is 1 fiber. For HCal, the readout unit is 100mmX100mm for barrel and endcap, the hits for Hcal are per layer.

Position of hits is shown in following series of plots. The position is given in transverse x and y coordinates and in radius in the transverse plane and z coordinate with energy threshold applied.

Tracking detectors, VertexBarrel
Electromagnetic calorimeters, EcalBarrelSciGlass+EcalEndcapP + EcalEndcapN +EcalEndcapPInsert
Hadron calorimeters, HcalBarrel + HcalEndcapN + HcalEndcapP


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# Tonight I Visit The The Bowlless Bowl

#### TENYEARS

The barrenness usually comes after one finds that one was in the matrix to begin with. It is not a bad thing. It is a cleansing which occurs so one can see/experience more clearly. Did not see the new movie yet but I will.

My mind was thrown open, had another vision come true and floored many people bigtime including myself. This is a rush. So I was thinking once again, I think I am going to start busting up all the fish bowls? You think they would award the Nobel prize for proving the existence of the reality unseen? I can, the question is where and who and how should it be presented? Writing papers to be looked at by fish is not the way to go so don't go there. What should be done any suggestions? Leave the bowl or stand by the boarderless boarder if you are going to answer, otherwise don't silence is golden.

Related General Discussion News on Phys.org

Tonight I Visit The The Bowlless Bowl
I preferred the hubless wheel which had many spokes coming out of the hub.

#### Iacchus32

Originally posted by TENYEARS
The barrenness usually comes after one finds that one was in the matrix to begin with. It is not a bad thing. It is a cleansing which occurs so one can see/experience more clearly. Did not see the new movie yet but I will.

My mind was thrown open, had another vision come true and floored many people bigtime including myself. This is a rush. So I was thinking once again, I think I am going to start busting up all the fish bowls? You think they would award the Nobel prize for proving the existence of the reality unseen? I can, the question is where and who and how should it be presented? Writing papers to be looked at by fish is not the way to go so don't go there. What should be done any suggestions? Leave the bowl or stand by the boarderless boarder if you are going to answer, otherwise don't silence is golden.
If what you say is true about the visions I'm sure word will get around. As for ascertaining the existence of God, this is already possible (more so on a personal level), although it would be nice if someone could spell it out in general terms so science could get the picture. And then maybe we could all strive for a common goal, like cleaning up the planet or something?

Have you considered a publicist?

#### Dissident Dan

I dub this thread a matrixism.

#### TENYEARS

IC, if you only knew my friend if you only knew. As for writing, I have written down many individually, but that does not matter. You see only one has to be understood as real and everything else falls away. I know the universe is infinite and there are probably and infinite number earths. Some where you may be able to perform an event unconciously and it may repeat 1,000,000 times and yet is may have no relevance to reality. The very first one of these experinces I had does. This I do not believe. This I know.

Some once said to be consistency, that is the key. I laughed to myself when the same person read to me consistency is a facet of small minds reading the quote at me as they saw the word consistency and thought the quote was going their way before they read the entire quote. It was pretty funny. There are no small minds only minds that do not realize what they are part of.

If any of you believe in yourselves to the point where you have a real answer to this post please do. You can never have enough ideas for a red brick. Sometimes it is the idea that is not even used that makes all the difference. If not, maybe nothing will happen, or not.

#### zk4586

As for ascertaining the existence of God, this is already possible (more so on a personal level)
more so on a personal level? Can you offer any non-subjective proof for the existence of God?

#### Mentat

Originally posted by TENYEARS
The barrenness usually comes after one finds that one was in the matrix to begin with. It is not a bad thing. It is a cleansing which occurs so one can see/experience more clearly. Did not see the new movie yet but I will.
Well, the next movie was even better, in my opinion. However, are you really prepared to accept some screen-writer's science fiction idea as an actual account of reality? I can assure you, the screen-writer himself doesn't believe in this fabricated reality, and neither do the actors, who played in the movie.

My mind was thrown open, had another vision come true and floored many people bigtime including myself.
I'm wondering when you are going to start actually telling us a vision, instead of just telling us that you've had one.

This is a rush. So I was thinking once again, I think I am going to start busting up all the fish bowls?
Obviously you cannot do this. It is not within your power. If one doesn't wish to leave their fish-bowl, they do not have to - in the end, the only person one can truly convince is oneself.

I can, the question is where and who and how should it be presented?
Perhaps you should try it out on us here at the PFs first, and then work on springing it on the world.

Writing papers to be looked at by fish is not the way to go so don't go there.
But think of this: Those you are proposing to preach to are (for the most part) all fish. You are trying to break a few fish free of their "universes", but you must first deign to the level of a fish. Otherwise your wasting your time, looking down on other fish, while they can't (and don't care to) see you at all.

Leave the bowl or stand by the boarderless boarder
If the boarder is truly boarderless, then it is only a matter of time before the fish "fall out", along with the water that was their feeling of security and acceptance.

otherwise don't silence is golden.

#### Iacchus32

Originally posted by TENYEARS
If any of you believe in yourselves to the point where you have a real answer to this post please do. You can never have enough ideas for a red brick. Sometimes it is the idea that is not even used that makes all the difference. If not, maybe nothing will happen, or not.
How about the schmuck of schmucks, James Randi, and his $1 Million Paranormal Challenge? Of course you could probably waiver the million bucks (if your claim is legitimate) just to show what an ass he is for making such a foolish proposition in the first place ... while also making the waiver contingent upon him publishing the results. At this point you could probably do without the money, as I'm sure the notoriety will follow. http://www.randi.org/research/index.html #### Iacchus32 Remote Viewing Originally posted by zk4586 more so on a personal level? Can you offer any non-subjective proof for the existence of God? Did you ever see the Nova program about the CIA conducting tests regarding Remote Viewing? A very interesting program! I believe the website below was created by one of the original test subjects. Of course this may be the actual key right here, if only the rest of the scientific community cared to take a look? http://www.probablefuture.com/ #### zk4586 ...yes, if only. #### Iacchus32 Academy of Remote Viewing From the URLAT, http://www.probablefuture.com/matrix.htm ... Most human entities spend their "waking" lives hypnotized by their sensory world, which gives them the illusion of a material reality "out there." In reality, space and time are really nonexistent both at the level of Pure Aware Consciousness, and also at the level of the unaware "blind parts" that experience for It the illusion of creative thought within an illusory space/time construct called Creation. Mankind rarely realizes that life is but a sensory illusion that gives experience and learning to Thought, and that the only reality is Thought Itself, which is forever evolving. Consciousness can be defined as Awareness projected onto a vibratory Dream. Period. For Space/Time Creation it is but pure Dark lesser vibratory (conscious and subconscious) thought interacting more or less intensely with the inner Holographic Light/Thought Reality originating through the facets of The MATRIX ("Deep" of Genesis 1: 2). Oh my God! ... Is that you Lifegazer!? Last edited: #### Dissident Dan LoL: "Being means Be-In-GOD" #### TENYEARS IC, Randi is not interested in truth and I would not give him the rights to my visions, experiences and a method I thought of to provide a form a proof for what I speak and a bit more. The planet is my concern and its future. I hope there is one. Mentat, you want a vision, I will not post it here. I will send it to you on a personal message. On these conditions you can tell no one the content, only that if the paramters which you read are in accordance of what you percieve to be some sort of viable proof of a vision. You may never speak of it or show it to anyone regardless of what you think. If you ask for it I will send it but you are bound by your word. Remember it's your word if you ask. I will give it to anyone who abides by this and only this. #### LogicalAtheist Originally posted by TENYEARS IC, Randi is not interested in truth and I would not give him the rights to my visions, experiences and a method I thought of to provide a form a proof for what I speak and a bit more. The planet is my concern and its future. I hope there is one. Mentat, you want a vision, I will not post it here. I will send it to you on a personal message. On these conditions you can tell no one the content, only that if the paramters which you read are in accordance of what you percieve to be some sort of viable proof of a vision. You may never speak of it or show it to anyone regardless of what you think. If you ask for it I will send it but you are bound by your word. Remember it's your word if you ask. I will give it to anyone who abides by this and only this. Uh - Man whatever your "vision" is, I'm sure it's already been falsified. I doubt you're doing any thinking that no one else has done. #### zk4586 What's with all the James Randi bashing? Since when did he become the scapegoat for your own failed attempts to objectify God? #### Iacchus32 Grim Reaper Originally posted by zk4586 What's with all the James Randi bashing? Since when did he become the scapegoat for your own failed attempts to objectify God? Because Randi is so materialistically set in his views that you might as well be talking to a block of concrete. He couldn't possibly conceive of anything "spiritual" behind the nature of a cockroach. And yet it wouldn't be so bad if he wasn't so smug and arrogant about it, while standing up and mocking the whole of idea of religion. Tell me this is not what he's doing when dressed up in priestly garb (or as a monk) and goes around spouting off the virtues of Atheism? While it's this very notion of being so matter-of-fact (self-assured), yet consolidated by arrogance, that will become the great millstone that hangs around his neck when the time comes. For which reason Randi reminds me of the "spirit of death." Honest. And I can picture him standing there in his robe and hood, with sickle in hand, and say yep, that's him, the "grim reaper." And indeed, this may be the very reason why he's here, to consolidate the "spirit of death" with death ... And hey, I'm not even a religious person when I say this! #### TENYEARS I had many they came true and I have had more and they came true they are not mine the reality exists I just happen to see it. I have an idea no that would not be fair would it. That would be stealing if I knew I was right and took your money anyway would'n't it? Or is this an exuse I am making to hide what you belive to be a fear which is mine of not being right which would be my fear because humans are relative and cannot go beyond their little chemical bodies right. Or is this little scam of a paragraph nothing more than a mirror that looks into a mirror into a mirror...... Know this as an attempt to get you tired or to make you think that there is some dept there at all. Tired yet? But you an ZK are both kids without even a paper route so I would probably be taking your allowance money. Zk, who said I failed? 1987 son it was more than a relative marker in the movement of infinity for me. All markers were called in on that day. IC, you seem to know a lot about this character. I just saw his site and watched a video of him speaking. I know he is not interested in answers. Tucker had a car that worked, it was better and the big three had him shut down. There is something far bigger here than the big three or randi or goverenments. It is the human condition to take the path of least resistance which means it will wait until there is now choice and by then it is usually to late. #### Iacchus32 Originally posted by TENYEARS Zk, who said I failed? 1987 son it was more than a relative marker in the movement of infinity for me. All markers were called in on that day. Yep, 87' was a vintage year alright ... the year that I died and gave birth to myself. http://www.dionysus.org/x0501.html #### TENYEARS Iacchus32, already knew there was some experience there. I have read things you have posted somewhere and certain things are not parrotable in the context you said them. #### Iacchus32 Originally posted by TENYEARS Iacchus32, already knew there was some experience there. I have read things you have posted somewhere and certain things are not parrotable in the context you said them. Parrotable? What does that mean? Do you mean not repeatable? #### Fliption #### TENYEARS What I ment was I found the spirit motivating some of your words and even if they were read, they could not have been put in that context and in that way at that time by logic alone. Logic falls short of the spirit. The spirit is the next level or the only level. #### Zero Originally posted by Iacchus32 How about the schmuck of schmucks, James Randi, and his$1 Million Paranormal Challenge? Of course you could probably waiver the million bucks (if your claim is legitimate) just to show what an ass he is for making such a foolish proposition in the first place ... while also making the waiver contingent upon him publishing the results. At this point you could probably do without the money, as I'm sure the notoriety will follow.

http://www.randi.org/research/index.html
LOL, uh huh..how dare he ask for PROOF?!?! Who does he think he is, not just taking people's words that they aren't having psychotic delusions, or scamming people?!?

#### Iacchus32

Originally posted by Fliption
Of course it'll cost them $98 to do so. Actually, based upon the two or three pages that I visited, there's a great deal of information that can be gleaned without having to cough up the$98.00. In fact I don't recall seeing that on any of the pages I visited?

#### Iacchus32

Originally posted by Zero
LOL, uh huh..how dare he ask for PROOF?!?! Who does he think he is, not just taking people's words that they aren't having psychotic delusions, or scamming people?!?
Did you know I had posted this before you brought up James Randi in the https://www.physicsforums.com/showthread.php?s=&threadid=2448" This is why I thought maybe why you brought it up there? If not, it's kind of funny how we were thinking along the same wavelength?

Anyway, why should I put my "faith" in James Randi? Why should I "follow" him? And what is it about me, that's "skeptical" about him? Why should I buy what he's selling -- which, is "James Randi?" ... And hence the old addage, "Buyer beware."

Last edited by a moderator:


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## Introduction

Irrigated agriculture plays a crucial role in meeting global food demand. Although it uses only 20% of cultivated lands, nearly 50% of agricultural production1 relies on irrigation. The expansion of irrigated agriculture over the last 50 years is a significant contributor to increases in global food production that meet the needs of a growing population2. This historical trend is particularly important in light of the projected increases in food demand due to population growth and meat-intensive diets3,4.

Surface water resources account for 62% of the total worldwide water supply for irrigated agriculture. In many parts of the world, surface water supply depends heavily on snow accumulation and thawing regimes5,6,7. Most areas with latitudinal parallels over 40 degrees receive a majority of their precipitation in the form of snow8. There are also many mountainous regions in other latitudes that provide water for downstream agricultural regions9,10 including the Hindu Kush and Himalayas, the Karakoram, the Tien Shan massifs, the Andes, the mountains of the Middle East, and the Atlas Mountains8. Mountainous regions of the western United States (US), such as the Rockies, Cascades, Sierra Nevada mountains, are important snow-dependent regions5,11,12 that significantly influence national food production. In this study, we focus on the Yakima River Basin (YRB; Supplementary Fig. 1), an agriculturally important and heavily irrigated snow-dependent river basin13,14 in the Pacific Northwest of the US. Background on the YRB, including its regional context, and water rights structure can be found in the method section.

Climate change affects irrigated agriculture through two key mechanisms. First, the crops respond directly to climate change through biophysical mechanisms. Crop growth rate, productivity, and water demand will be affected by changes in atmospheric CO2 concentration, temperature, and precipitation (although for irrigated crops, precipitation plays a reduced role in affecting yield). While increasing CO2 concentrations increase yields (especially for crops with a C3 photosynthetic pathway), warming can deteriorate the productivity of some major crop types (especially annuals, such as corn, wheat, and potatoes) because of an acceleration in growth rate that also reduces the time available for photosynthesis and biomass accumulation15,16,17. There are many strategies that can be used to improve crop performance (average yield)18,19 including improvement in biotic resistance, abiotic resistance, and change in physical characteristics of crops. There is also a growing focus on slow-maturing crop varieties to reduce the impacts of temperature induced accelerated crop growth16,18,20. Although a broad body of literature has investigated this issue (readers can also see Supplementary Notes 3), the effectiveness of the yield-improving biophysical crop adaptations are not fully understood. The second mechanism where climate change impacts irrigated agriculture is the expected increase in the severity and frequency of droughts in snow-dependent areas12,13,14. In these basins, agriculture receives the majority of its summertime water supply from the melting of accumulated snowpacks. A warmer climate is expected to reduce snowfall and increase rainfall, reducing snowpack volume. In addition, higher temperatures accelerate the melting of snowpacks, further shifting water availability away from irrigation months6,7,21,22. This can lead to curtailment (or interruption) of water rights for irrigation, thus producing an indirect water scarcity effect from climate change on crop yields.

The emerging and potential future changes in the seasonality and magnitude of streamflow regimes in snow-dependent regions are unprecedented in their historical observations11,13. Therefore, characterizing candidate shifts in water availability requires modeling frameworks that account for regulated flows through managed storage facilities, institutional water rights, and important regulations23. All of these factors potentially limit the validity of many of the simulation-based assessments that do not account for these region-specific details.

Our research question is two-fold: first, What are the direct and indirect impacts of climate on irrigated agriculture in the YRB, and second, To what extent does farmer adaptation (through planting different crop varieties) aid adaptation against the negative effects of climate change? To address the first question, we investigate how the expected magnitude and interannual variability of yield in irrigated agriculture respond to the compounding (and sometimes competing) effects of changes in water availability, temperature, and atmospheric CO2 concentrations, and embedded within the water management and regulatory context of the YRB, a snowmelt-dependent watershed representative of many in the western US. To address the second, we explore how adaptive actions through crop variety improvements affect productivity in the YRB. In examining the impacts of both climate change and on farmer adaptation, we quantify the impacts to both expected mean yields, but also interannual yield variability, given the significant implications to crop insurance programs of the losses of revenue24,25 associated with extreme drought events. Our results show that climate change can impact the YRB by deteriorating its snow-driven water availability and accelerating crop biophysical processes leading to reduced average annual yields. We also show that in the absence of significant, simultaneous improvements in crop varieties, water governance institutions and the infrastructure of the YRB, adaptation efforts could face a significant tradeoff between improved average yields and increasingly severe revenue volatility in the region.

## Results

### Increasing water stress

Irrigation unmet demand (UD, defined in the method section), an institutionally consistent drought indicator for the YRB, is strongly sensitive to climate change-induced changes in precipitation and temperature, showing transitions towards higher water stress (Fig. 1, panel a). This increase in UD is mainly controlled by temperature (Fig. 1, panel b) because the YRB, as a snow-dependent basin, receives 75–80% of its precipitation in the form of snow, and irrigation water availability in summer months depends heavily on the timing of snowmelt. Moreover, the active capacity of five major reservoirs in the YRB covers about 40% of total irrigation demand26. Therefore, the buffering capacity of this supply is not sufficient to compensate for this shift in streamflow timing. Moreover, as past studies12,27 have shown, in the Pacific Northwest, climate change does not produce a clear direction of change for precipitation, there is significant disagreement in the projections attained across the ensemble of available general circulation models (GCMs)27,28. Analysis of simulated future drought frequency (Fig. 1, panels c and d) also indicates that the probability of significant drought (UD > 30%) could significantly increase from 23% over the historical (1980–2010) period to ~40% [with a range of 13–63% across 5 GCMs  × 2 RCPs] over 2030–2060. During the same period (2030–2060), the average severity of significant droughts (UD > 30%) is projected to increase from historical UD of 40–46% of 6% relative to the historical average [with a UD projection range of 34–47% across the GCM model ensemble considered here]. Additionally, There there is the potential for unprecedented droughts in the YRB system (droughts stronger than with UD > 65%), which have not been recorded in the observed past. Our results show that for both time periods (2030–2060) and (2060–2090) there are even possibilities for experiencing very extreme droughts approaching the maximum value of water stress measure (UD = 100%).

### Agricultural productivity, yield, and interannual yield variability

The ten most popular crops in the YRB are categorized into the following three groups (Fig. 2): first, annual crops (winter wheat, spring wheat, corn, and potatoes), second multiple-cutting crops (alfalfa and pasture), and third, tree fruits (grapes, apples, cherries, and pears). In terms of average agricultural productivity (Fig. 2, panel a and Supplementary Tables 3), all crop types show vulnerability to climate change with significant potential reductions in their average annual yields. For example, during the 2060–2090 period (RCP 8.5), average annual yield of potato, alfalfa, and apple could undergo a reduction of 46%, 22%, and 48%, respectively. This happens primarily due to the effects of more frequent and more severe droughts where shortages in available irrigation water occur. A portion of this reduction in the annual crop group can be attributed to a further reduction in maximum yield, which is defined as yield under fully irrigated condition (Supplementary Fig. 4, panel a and Supplementary Notes 5). The reduction in average yield of annual crops can cause damage to regional agricultural economies and even global food security because the deterioration of water availability and agricultural productivity follows the same trend across many snow-dependent regions of the world5,6,10.

During the 2030–2060 period (Supplementary Fig. 4), the maximum yield of annual crops such as spring wheat, corn, and potatoes show reductions of 9%, 23%, and 9%, respectively. These reductions result when higher temperatures accelerate the growth rates of the crops; in turn, causing earlier maturation and less opportunity for biomass accumulation16,29. However, we also show that the maximum yield of alfalfa could potentially increase in the future (26% during the 2030–2060 period). Alfalfa is a multiple-cutting crop, consequently faster growing cycles can increase the number of cuttings30 and enhance seasonal biomass accumulation. The minimum yield (under the no irrigation condition) increases for all crop types (Supplementary Fig. 4, panel b). For example over the 2030–2060 period, the minimum yield for spring wheat, corn, and potatoes increases by 17%, 25%, and 21%, respectively. Increased precipitation and CO2 concentration are the primary drivers of this improvement in non-irrigated yield31,32 and out-compete the yield-reducing impact of a shortened growing period. Consequently, the yield gap, which is defined here as the gap between the actual yield under water-stress conditions and the fully irrigated attainable yield, could decrease in the future. The reduction in yield gap has at least two significant implications: first, implications for Changes in the Marginal Benefit of Water (MBW): The shortened gap between fully irrigated and non-irrigated conditions also leads to lower absolute crop yield loss during drought years (Supplementary Fig. 3 and Supplementary Notes 4). The MBW in agriculture is defined as the magnitude of the improvement in yield in response to an extra unit of water31. The MBW has significant implications for investment decisions about new irrigation systems and other infrastructure32. A greater MBW indicates greater benefits for farmers who increase their water availability, thus making this investment more viable. Although we do not explicitly simulate MBWs, our results suggest that MBW will likely be lower in the future in YRB. When the yield gap is smaller (e.g., status quo crops under climate change scenarios), the MBW becomes smaller because an extra unit of water does not affect the productivity as much as in earlier periods. Therefore, annual crops undergo a reduction in MBW (e.g., 45% and 16% percent reductions in yield gap for corn and potatoes, respectively), but the MBW of multiple-cutting crops increases (e.g., an 11% increase in the alfalfa yield gap); second, implications for Changes in the Interannual Variability of Yield: Although often overlooked in impact assessments of climate change, the interannual variability of yield is also important to consider; we show that although droughts are expected to intensify in the future, interannual variabilities of yield do not necessarily increase (Fig. 2, Panel b and Supplementary Table 3). Note that in our analysis, we use standard deviations (SDs) of annual series of yield to represent interannual variability of agricultural productivity. For example, throughout the 2030–2060 period, the reductions of SD for potatoes, corn, spring wheat, and alfalfa are 15%, 22%, 1%, and 19%, respectively. The primary explanation for these reductions in interannual variability is that maximum yield will decrease for many crop types (Supplementary Fig. 4, Panel a), including wheat, corn, and potatoes. Change in interannual variability of yield has direct implications for the parties involved in agricultural insurance programs, including insurance firms, insurance-buying farmers, and governmental insurance programs. Lower interannual variability of yield will result in higher accuracy in prediction of yield loss and, at the same time, fewer drought-related payouts to farmers24,33.

### Improved crop varieties

As our future potential yield simulations indicate, the attainable yield of some important food-security crops could strongly decline (Fig. 2). This trend aligns with several previous studies’ findings and arguments16,19,29,33,34 that the use of current crop varieties can significantly reduce the productivity of some important crop types and potentially threaten global food security. In response to this issue, improvement in crop seeds has been proposed as a primary means of adapting to the adverse impacts of climate change on yields19,20. Improved varieties are expected to have longer growing periods to allow more time for photosynthesis and biomass accumulation33. In this study, we explore an adaptation scenario that considers seed-modification, and as demonstrated in Fig. 3 (Panels a and b), we show that during the 2030–2060 period, average yields of potatoes, spring wheat, and corn do improve with expected increases of 14%, 20%, and 31%, respectively (Supplementary Tables 2). However, for the same period (Supplementary Tables 3), the SD of yield is also shown to increase by 52%, 130%, and 123% for potatoes, spring wheat, and corn, respectively. Although these modified crops can compensate for the reduction in average annual potential yields, they remain vulnerable to deficits in irrigation availability during increasingly frequent and severe drought periods. In other words, average yield would increase, but at the same time, volatility in agricultural production and farmers’ revenue increases substantially (Fig. 3, Panel d). We also show that the probability of low-yield years (defined as years that produce under 60% of fully irrigated yield) may substantially increase with climate change (Fig. 3, Panel c). Although this trend is stronger under the RCP 8.5 scenario (Fig. 3, Panel c), the trend remains significant for a more optimistic RCP 4.5 future as well. Although we illustrate these trends using potatoes, similarresults occur for other crop types including multiple-cutting crops, tree fruits, and other annual crops (Supplementary Fig. 513). Higher variability in yield under improved crop varieties has uneven implications for insurance-buying farmers and governmental programs that subsidize crop insurance. Although more productive crop varieties enhance the volatility of the yield production, they will still increase average productivity, which expands the yield gap and increases the MBW. The greater MBW will make investment in agricultural water-supply projects (e.g., dams, water transfer projects, and irrigation systems) more likely.

### Insurance-related implications of changes in interannual variability

Our results on the relationship between climate change and yield variability has important implications for crop insurance programs24. At least two groups have a large stake in insurance programs: farmers and insurers. In many drought-sensitive areas, farmers sign up for insurance programs to reduce their losses when water scarcity causes low yields. Our results show that the current seed varieties for some important crops (e.g., wheat, corn, and potatoes), the potential for experiencing a very bad year (less than 60% of fully irrigated yield) declines. For farmers, overall agricultural revenue could be lower, but lower yield variability should mean reduced insurance premiums. Some governments, such as the U.S. federal government, subsidize insurance for certain crops35. Implications for these governments and other insurers, however, will likely be positive (in terms of the cost of such insurance programs), simply because the deviation from the normal condition and the perception of risk36 is lower. Therefore, we highlight that, in the absence of adaptation of seed varieties, the scenarios imply reductions in the cost of these insurance programs. Crop variety improvement, however, will reverse this because farmers could inadvertently become more prone to extreme low-yield years, and higher yield variabilities will likely enhance the necessity and cost of insurance programs for farmers24,25,37. In the US where the federal government subsidizes insurance primia for many crop types36,38, higher variability will likely increase governmental expenditure on insurance programs. Also, our results in this study imply that the relationship between yield loss and drought magnitude (e.g., Supplementary Fig. 3) will be fundamentally transformed under different climate change and crop variety improvement scenarios. These non-stationarities indicate the importance of actively reconsidering the underlying assumptions of index-based insurance programs37.

When contemplating the mean-variability tradeoff, and its winners and losers, it is important to consider who pays for the potential losses. In the US, there are post-Dustbowl programs that make the US federal government pay for the insurance of commodity crops (e.g., wheat, corn);36,39 therefore, the general public pays for the losses. However, there are many farmers in the US that purchase their own insurance35. In the case of insurance-buying farmers, because they pay for their own insurance, more volatility could increase their insurance premiums and reduce their annual revenue. It is also expected that the effects of this volatility would eventually be reflected in what consumers pay for their food35.

Our results imply that, in snow-dominated regions, two simultaneous strategies must be pursued to ensure a sustainable regional agricultural economy and productivity. First, water availability needs to be enhanced through improvements in water-related infrastructures and institutions. Examples of such strategies may include building water storage and transfer facilities, and adaptation of water rights and regulations. Second, improvement in potential productivity of crops. The results of our study emphasize the need for seed improvement strategies to carefully consider increasing sources of variability related to changes in temperature and irrigation water availability. To achieve that goal, it seems necessary to preserve the current diversity of seed varieties that have undergone different climatic conditions during several thousands of years40. Agrobiodiversity increases our ability to find and inject climate change-resilient varieties into the breeding process. In that regard, establishing efficient seed-sharing programs and banks41, promoting intellectual property rights platforms and agreements42, and embarking on deeper agriculture-related social and cultural studies43 seems to be the way forward. Also, strategies such as breeding under elevated temperature conditions hold promise for achieving yield improvement objectives44. In addition, faster plant-breeding processes can provide the research community with a higher degree of flexibility to create varieties that are resilient to elevated temperatures19,44,45,46. Although these production goals might be achievable, creating them could involve crucial hurdles. For example, past works19,44 have argued that, in order to efficiently create these high temperature-resilient varieties, the breeding process of these crops must become significantly faster. Currently, it takes about 30 years to create a new variety through conventional breeding; this will prevent us from making efficient and immediate adjustments in the breeding process. Moreover, based on the current trajectories of change, we might experience shortages of food before a breeding program can find an appropriate variety44. Moreover, recent research45 exploring different breeding experiments for wheat claim that our current genetic potential in production of temperature-resilient varieties might be limited due to the underlying gene availability. However, there are promising successes in developing temperature-resilient varieties16,47. Supplementary Fig. 2 approximately demonstrates how different water stress and crop variety scenarios may affect agricultural productivity (also, refer to Supplementary Notes 1 for more information). This figure underlines that, when crop varieties are improved without an improvement in the water system, there will be higher volatility in the overall agricultural production systems.

### Transferability of results to other snow-dominated regions of the world

The YRB is a heavily irrigated snow-dominated system, and our study (in agreement with prior studies12,13,21) shows that the system will undergo higher water stress as higher temperatures reduce snowpack. The other key pathway that negatively affects the YRB’s irrigated agriculture is the temperature-dependent reduction in potential yield44,48. We argue that, because both of these effects have been induced by higher temperatures, and higher temperatures are the most robust climate change effect that is expected to affect most of the planet49,50, our results could be of interest broadly to snow-dominated agricultural systems globally that must also manage the joint stressors institutionally constrained water availability and increasing temperatures.

## Discussion

In this study, we do not take into account other factors that might change in the future. These factors include a change in cropping pattern, shift in the geographical location of agricultural areas, and change in irrigation systems. Therefore, for example, 2060 does not realistically represent all of the factors that might be different in that year. Additionally, we do not take into account any possibility of change in current water rights and regulations. More specifically, scenarios that consider any possibility of imposing relinquishment are not included in our study. For example, under current western water law doctrine, long-term reduction in irrigation demand and diversion will be penalized by loss of water rights. However, relinquishment of unused water rights has not been historically practiced in the YRB. Finally, although the biophysical model used in this study to simulate crop heat stress, it does not capture crop mortality. Future work that captures a more refined simulation of crop heat stress and its dependence on the availability of irrigation water51 (used for cooling in addition to crop water demand) is warranted. Moreover, although beyond the scope of this work, a more realistic simulation of crop heat stress and its close dependence on the availability of irrigation water may be relevant to other studies. Such a simulation tool requires finer temporal resolution (hourly and shorter), tight coupling of an agro-hydrologic model with a regional atmospheric model, and a boundary-layer model that can accurately represent plant-level microclimates. Research initiatives in these directions can promote our understanding of crop response to higher temperatures, which is one of the most contentious aspects of future food production projections15. Additionally, in this study, we only consider one scenario of yield improvement (slower maturing varieties); however, there are many pathways that can alter the properties of crop production. For example, crops with shorter growing periods can be used as a drought escape strategy. They tend to reduce the risk of facing extreme water and temperature events and promise more reliable production. However, fast-growing varieties have less opportunity for yield accumulation and produce a lower amount of biomass and yield48,52. In other words, crops with shorter growing periods can introduce their own mean-variability tradeoff into the agricultural systems. A more in-depth exploration of possible seed-related adaptation strategies would be a beneficial extension of this study. Lastly, In this study, we do not take into account other factors that might change in the future. These factors include alterations in cropping pattern, shifts in the geographical location of agricultural areas, and changes in irrigation systems. Changes such as these could cause intended and unintended consequences that could propagate into different sectors and affect several aspects of the entire system. Therefore, our projections regarding 2060, for example, do not represent all of the candidate factors that might be shape yield dynamics in that year. However, the overall conclusions of this study emphasizing the need to simultaneously consider adaptation of seed varieties and their dependence on drought-related water institutions are broadly applicable across adaptation scenarios and different snow-dependent agricultural systems.

In summary, this study explores the effects of climate change on irrigated agriculture in the snow-dependent regions of the YRB. Modeling results show that there could be a significant increase of water stress on irrigated agriculture. This water stress and the shortening of growing seasons will reduce the overall productivity of agriculture. We argue that this overall reduction in productivity can substantially harm individual farmers, the regional economy, and, given the importance of irrigated agriculture in worldwide food production, even global food security. We also show that, although the uncertainty of water supply for irrigated agriculture increases in snow-dependent basins such as the YRB, the interannual variability of yield significantly decreases. We found this due to a concomitant reduction of the gap between fully irrigated crops and non-irrigated crops. Smaller yield gaps occur because the potential yield of some annual crops that are important to food security (such as winter wheat, corn, and potatoes) decreases. Put simply, we conclude that an increase in drought frequency and severity might not directly translate into more volatility in annual agricultural revenues. Furthermore, we explored a seed-modification adaptation, and we show that although it has the potential to improve overall agricultural productivity, the adaptation could lead to a significant increase in the revenue volatility of the food production system. This higher interannual variability can lead to challenges for insurers, governmental insurance subsidy programs, and insurance-buying farmers. Finally, we conclude that in order to ensure sustainable food productivity in irrigated regions, adaptation strategies should address both biophysical crop and water-supply considerations.

## Method

### Simulation tools

We used a physically-based coupled agro-hydrologic platform that establishes connections between crop growth and development, land-surface hydrology, and river-system processes (Fig. 4). This framework has two main components: VIC-CropSyst53, which simulates agricultural and hydrological processes, and YAK-RW54 which is a river system (i.e., water management) model that simulates the operation of dams and prioritizes the allocation of water among different water-dependent sectors. The process-oriented and distributed nature of VIC-CropSyst makes our platform a powerful tool for capturing the effects of spatial variations in climate, soil, crop type, irrigation system, and topography. The capabilities of YAK-RW, on the other hand, allow us to reasonably simulate impacts of water rights and the operation of water facilities. A comprehensive description of all of the components of the simulation tools used in this study can be found in Malek et al.14.

Data: Simulation period and spatial resolution: The resolution of our simulation is 1/16 degrees. The historical period is from 1980 to 2010, and the future period is from 2030 to 2090. The following describes all of our model inputs.

Climate inputs: In this study, we provided minimum and maximum temperature, precipitation, and wind speed. The resolution of our meteorological data was 1/16 degrees. For the future period, we used outputs of five general circulation models (GCMs) forced by two representative concentration pathways (RCPs), RCP 4.5 and RCP 8.5, which are related to the radiative forcing of 4.5 and 8.5 $$\frac{{w}}{{m^2}}$$, respectively55. We used the method introduced by Brekke et al.56. to select these five GCMs from eighteen that were available. The GCMs used in this study include GFDL-ESM2G57, HadGEM2-ES365, HadGEM2-CC36558, INMCM459, and CanESM260.

Soil inputs: We used the same soil file as Malek et al.53. who used the State Soil Geographic dataset61 to develop soil inputs for the VIC-CropSyst.

Land cover information: We used the vegetation parameters file (VPF) and crop parameter file (CPF) used by Malek et al.14. The CPF includes information on crop acreage, planting date, irrigation system, crop type (perennial or annual), and proration rate. VPF provides the vegetation class, acreage, root depth, and root distribution in each soil layer.

Crop inputs: Some key crop-specific parameters of the VIC-CropSyst can be found in Supplementary Note 2 and Supplementary Table 1. These parameters are important for the simulation of agricultural processes such as growth, transpiration, biomass, and yield production. More comprehensive information on the crop input parameters can be found in the permanent repository of this paper; readers can also refer to Stockle et al.62 for more discussion of these parameters. Supplementary Note 3 also provides a thorough discussion of the assumptions involved in the development of the improved crop varieties considered in this study. Finally, Supplementary Note 6 and Supplementary Fig. 14 present a sensitivity analysis of these yield improvement assumptions.

### Yakima river basin

The YRB (Supplementary Fig. 1) is an intensively irrigated agricultural basin in central Washington State and spans an area of 16,000 square kilometers63. The basin-wide annual precipitation is about 680 mm, with the upstream mountainous regions receiving 2500 mm and the downstream regions receiving less than 200 mm. The YRB is a snowmelt-dominated basin64,65, and snow makes up 60–80% of its total precipitation66, Accumulated snow melts during the warm months (March to September) and feeds the Yakima River. There are five major dams in the YRB (Supplementary Fig. 1), which are capable of storing around 30% of the annual flow;67 they regulate streamflow and play a significant role in meeting irrigation demands. About 10% of employment in the area directly involves the agricultural sector, and irrigated agriculture generates more than \$1 billion in annual revenue63. Yakima County is the state’s largest producer of many agricultural commodities (e.g., apples, grapes, and cherries).

### Water rights and water stress index in the Yakima river basin

Two types of water rights are provided by the U.S. Bureau of Reclamation’s Yakima Basin Project: (1) proratable and (2) non-proratable26. The irrigation water rights of proratable farmers can be curtailed during drought years when there is not enough water to meet agricultural demand. Non-proratable farmers, on the other hand, will always receive their full water rights, even in significant drought years. In YRB, a specific definition of water availability called proration rate is used to indicate the severity of a drought. In essence, proration rate is the ratio of water availability over water demand. More information on calculations of district-level proration rate can be found in Malek et al.14,68. In this study we use a more-universal term of unmet demand to represent drought severity. Fraction of unmet demand (UD) is defined as:

$${\mathrm{UD}} = 1 - \frac{{{\mathrm{Water}}\;{\mathrm{supply}}}}{{{\mathrm{Water}}\;{\mathrm{demand}}}}$$
(1)

Where, Higher UD indicate more water stress for farmers. Finally, in calculation of future basin-wide UD we took the reduction of irrigation of demand into account.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.


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# Arby's and Combinatorics

1. Jul 4, 2007

Arby's has this deal out now: 5 for $5.95. You can pick from 8 choices to fill 5 spots. The menu says "over 790 possible combinations" , so one can assume that the number of choices is between 790 and 800. What is the equation to figure out how many choices you have and what is the exact number of possible combinations? 2. Jul 4, 2007 ### Werg22 There are well over 800 possibilities. 3. Jul 4, 2007 ### d_leet How much is well over 800? And how do you figure that out? 4. Jul 4, 2007 ### Werg22 Anyway to question is posed the number of possibilities is over 800; if one selection is independent of the others, there are 8^5 possibilities and if it is, there are 8*7*6*5*4 possibilities. 5. Jul 4, 2007 ### Integral Staff Emeritus Wiki has the answer. In this case, order does not matter and and you can order the same thing more then once. (2 Arby sandwichs and 3 frys) edit: I get 792 6. Jul 4, 2007 ### cristo Staff Emeritus The answer is not "way more than 790," but is actually 792. Check the "combination with repetition" section of this page of wikipedia. 7. Jul 4, 2007 ### d_leet All selections are independant, but 8^5 doesn't take into account any duplications of choices this is what I thought at first as well and realized it made no sense. 8. Jul 4, 2007 ### Werg22 Oops, sorry, I just realized. 9. Jul 4, 2007 ### sk381 Another related question.. Say I have 3 items A,B,C and I wish to select 3 at a time (order does not matter and an object can be chosen more than once)... using the forumla for Combination with repetition, I should have (3+3-1)!/3!*2!=10 combinations If I list all the possible combinations, I have: AAA BBB CCC AAB BBC AAC BCC ABB ACC But this is only 9.....what combination am I missing? 10. Jul 4, 2007 ### sk381 sorry.. didn't look at the most obvious choice.. ABC!!! 11. Jul 20, 2007 ### matticus lol wow i was just about to post a topic about this and i searched google first to see if anyone had. i was curious as to why they said over 790 choices, as that would seem to imply it's between 790 and 800 (or else they would have said over 800 choices). there are 8 choices, and 5 slots to put them in. that should be 8^5 if i remember from discrete math. 8^5 is a lot larger than 790 though. 12. Jul 20, 2007 ### cristo Staff Emeritus Read post #5 or #6. 13. Jul 22, 2007 ### cepheid Staff Emeritus Er, yeah, because the order of the elements doesn't matter. You're just choosing, not permuting. If you order two medium drinks, two curly fries and a roast beef sandwich, that's the same as ordering a roast beef sandwich, an order of curly fries, a medium drink, another order of curly fries, and a medium drink. You've put in the same food order, you just haven't requested the items in the same sequence. The two examples are not distinct combinations of five items. What Cristo said, if you want to see the mathematics. By the way, here in Canada, you still pay$6, but you only get to choose four items! I don't understand why that is. The menu says 330 possible combinations, which leads me to believe that there are still 8 items to choose from (if you use the formula provided in the Wikipedia link, it works out exactly).

That leads me to my next question. Does anyone know how to "derive" or at least intuitively explain this formula for combination with repetition? Because, all three of the other combinatorial formulas make perfect sense to me (and if you don't believe me, I'd be happy to explain why I think they make sense intuitively). But combination with repetitition...looking at the formula, I can't seem to figure out how one might arrive at it.


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# Math Help - Integrals over independent random variables

1. ## Integrals over independent random variables

Suppose for $i\in[0,1]$, $U(i)$ is a uniformly distributed independent random variable.

Am I right in thinking that for any continuous function $f$:

$\int_0^1{f(U(i))di}=\int_0^1{E[f(U(i))]di}$

where $E$ is the expectation operator? (That is if these integrals are actually defined?) Is there a reference for this?

I imagine it's a straight forward application of the MCT and the LLN, but I'm just a little bit worried by the fact that Brownian motion is often referred to as an integral of white noise, though perhaps more strictly it's an integral with respect to white noise, which is why that's different?

Tom

2. Originally Posted by cfp
Suppose for $i\in[0,1]$, $U(i)$ is a uniformly distributed independent random variable.

Am I right in thinking that for any continuous function $f$:

$\int_0^1{f(U(i))di}=\int_0^1{E[f(U(i))]di}$

where $E$ is the expectation operator? (That is if these integrals are actually defined?) Is there a reference for this?

I imagine it's a straight forward application of the MCT and the LLN, but I'm just a little bit worried by the fact that Brownian motion is often referred to as an integral of white noise, though perhaps more strictly it's an integral with respect to white noise, which is why that's different?

Tom
What do you think $U(i)$ denotes? Or more precisely what do you think $f(U(i))$ denotes?

CB

3. There are a continuum of independent uniform random variables indexed by $i\in[0,1]$. For each such $i$, $U(i)$ is the realisation of that random variable, that is to say, a sample from it. (So a value in $[0,1]$.)

(If you want to be precise the state space $\Omega$ is the set of all functions from $[0,1]$ to $[0,1]$, and each random variable is a functional $p_i$ on this space defined by $p_i(\omega)=\omega(i)$. If $\omega\in\Omega$ is the realised state then $U(i)=p_i(\omega)=\omega(i)$.)

I perhaps should have been clearer in my original message. It's traditional in the field I work in to blur the distinction between random variables and samples from them, for notational convenience.

4. Originally Posted by cfp
There are a continuum of independent uniform random variables indexed by $i\in[0,1]$. For each such $i$, $U(i)$ is the realisation of that random variable, that is to say, a sample from it. (So a value in $[0,1]$.)

(If you want to be precise the state space $\Omega$ is the set of all functions from $[0,1]$ to $[0,1]$, and each random variable is a functional $p_i$ on this space defined by $p_i(\omega)=\omega(i)$. If $\omega\in\Omega$ is the realised state then $U(i)=p_i(\omega)=\omega(i)$.)

I perhaps should have been clearer in my original message. It's traditional in the field I work in to blur the distinction between random variables and samples from them, for notational convenience.
You have to specify the set for which your random variable is uniformly distributed on, so here you want $U_i(0,1)$. Without specifying that $U_i$ is uniform over $[0,1]$ (or $(0,1)$ ) you have not given a statement of the problem.

CB

5. Ahh I misunderstood your original point. But yeah, standard uniforms are fine (i.e. on $[0,1]$), though it's trivial to see that if my claim holds for any univariate probability distribution admitting a density (not least any uniform), then it holds for all univariate probability distributions which admit a density.

Which brings us back to the original question, is my claim that:

$\int_0^1{f(U(i))di}=\int_0^1{E[f(U(i))]di}$

true?

Thanks,

Tom

6. Originally Posted by cfp
Ahh I misunderstood your original point. But yeah, standard uniforms are fine (i.e. on $[0,1]$), though it's trivial to see that if my claim holds for any univariate probability distribution admitting a density (not least any uniform), then it holds for all univariate probability distributions which admit a density.

Which brings us back to the original question, is my claim that:

$\int_0^1{f(U(i))di}=\int_0^1{E[f(U(i))]di}$

true?

Thanks,

Tom
To me both sides look like they are equal to $E(f(x)),\ X \sim U(0,1)$ , but I'm no expert on stochastic calculus of any kind.

CB

7. Gah, I've actually just noticed that there was a mistake in my description of the problem. $f$ is also a function of $i$, so the question is actually whether:

$\int_0^1{f(i,U(i))di}=\int_0^1{E[f(i,U(i))]di}$


text:
# What is the chance a team will have at least 10 more wins than losses at any point in a 100 game season? They have a 50% chance of winning each game.

More generally: Each game, $n = 1,2,...,N$, a team has probability, $p = 0.5$, of winning. Their standing $x$ is given by $x(n) = x(n-1)\pm1$ depending on whether they win ($+1$) or lose ($-1$). Their standing starts at $x(n=0)=0$. What is the chance the team will have at least $x_w$ more wins than losses at any point during a $N$ game season? So, what is the probability $x(n) \ge x_w$ after any $n$ number of games between $0$ and $N$.

From the binomial distribution, we could find the probability of 10 more wins than loses at the end of the season. As the number of games gets much larger than 100, could we not approximate with a normal distribution as well? However, I am interested in the probability they will have at least that difference in wins vs. loses at any point in the season, whether it be game 10, game 42 or game 100.

Would I find this by

1. finding the binomial distribution of wins vs losses after each game,
2. calculating, from the "area under the curve," the probability of less than 10 more wins than losses,
3. taking the product of these probabilities from n = 1 to n = N, and finally
4. subtract this value from 1?

This question could be rephrased as, "at least 10 more heads than tails after any number of coin tosses up to 100 tosses." It could also relate to other processes described by binomial or normal distributions (noise, particle diffusion).

Sorry in advance for anything unclear, especially my notation in the first paragraph. Obviously, I am not a mathematician. Feel free to ask me to clarify. Thank you!

• The question is clear, and important. It is a "gambler's ruin" problem, for which there is a large literature. Your 3.) is not correct, the events are not independent. The usual approaches do not use a normal approximation. – André Nicolas Jul 14 '16 at 0:22
• Thank you very much! From this I found exactly what I was looking for! mathpages.com/HOME/kmath683/kmath683.htm – James Jul 14 '16 at 3:52
• You are welcome. Because in 3.) you multiplied, I interpreted your question as asking for the probability of being $10$ or more ahead at some unspecified time $\le 100$. If you make an explicit specification of the time, then the answer is easily obtained from the binomial distribution, and if $n$ is not too small you can use the normal approximation. That is not what the "gambler's ruin" calculation does. – André Nicolas Jul 14 '16 at 4:01
• I meant, "What is the probability they will have that difference x_w at some point before the season ends." Even if they finish at 0.500 with # wins = # losses, they would still meet my criterion if at, say, game 52, they had 31 wins and 21 losses. In terms of gambler's ruin, wouldn't that be similar to the limit of the probability of losing (reach 0) or winning (reach N) within 100 iterations as N approaches infinity? They would start with $10, and in this case, I'd be interested in 10 more losses than wins. – James Jul 14 '16 at 4:14 • Then you want gambler's ruin against a opponent with infinite wealth, and if you reach$-10$you are broke. You do not want the limit, it is a bit more complicated than that, but you can find an answer from the recurrence. Another approach (similar, but matrix-heavy) is Markov chains. – André Nicolas Jul 14 '16 at 4:19 ## 1 Answer You can describe the number of wins after$n$games with$X_n \sim Bin(0.5, n)$, i.e. a binomial distribution with probability of success in a single event$p = 0.5$. The number of losses is always going to be$n - X_n$(assuming no draws). So you want to find the probability$P(X_n - (n - X_n) \ge 10) = P(2X_n - n \ge 10) = P(X_n \ge \frac{n + 10}{2})$which just involves a standard calculation of binomial probabilities (noting that when$n$is odd, it's equivalent to$P(X_n \ge \frac{n + 11}{2})\$).

• Thank you! Would this give, for example, the probability of 10 or more wins than losses after 23 games as P(X_23 >= 17) = 0.0173? – James Jul 14 '16 at 2:34
• Looks good to me! – ConMan Jul 14 '16 at 3:47


text:
# How do I find the value of log 100?

$\log 100 = 2$

#### Explanation:

One way we can approach log problems is to remember that

${a}^{b} = c \iff {\log}_{a} c = b$

In our question, since the value of $a$ in the right hand side log isn't specifically listed, it's assumed to be 10. So what we have is:

${a}^{b} = c \iff {\log}_{10} \left(100\right) = b$

${10}^{b} = 100 \iff {\log}_{10} \left(100\right) = b$

By observation of the left hand side, we can see that $b = 2$


text:
# Band structure of overdoped cuprate superconductors: Density functional theory matching experiments

Kramer, K P; Horio, M; Tsirkin, S S; Sassa, Y; Hauser, K; Matt, C E; Sutter, D; Chikina, A; Schröter, N B M; Krieger, J A; Schmitt, T; Strocov, V N; Plumb, N C; Shi, M; Pyon, S; Takayama, T; Takagi, H; Adachi, T; Ohgi, T; Kawamata, T; Koike, Y; Kondo, T; Lipscombe, O J; Hayden, S M; Ishikado, M; Eisaki, H; Neupert, T; Chang, J (2019). Band structure of overdoped cuprate superconductors: Density functional theory matching experiments. Physical review. B, 99(22):224509.

## Abstract

A comprehensive angle-resolved photoemission spectroscopy study of the band structure in single-layer cuprates is presented with the aim of uncovering universal trends across different materials. Five different hole- and electron-overdoped cuprate superconductors (${\mathrm{La}}_{1.59}{\mathrm{Eu}}_{0.2}{\mathrm{Sr}}_{0.21}{\mathrm{CuO}}_{4}$, ${\mathrm{La}}_{1.77}{\mathrm{Sr}}_{0.23}{\mathrm{CuO}}_{4}$, ${\mathrm{Bi}}_{1.74}{\mathrm{Pb}}_{0.38}{\mathrm{Sr}}_{1.88}{\mathrm{CuO}}_{6+{\delta}}$, ${\mathrm{Tl}}_{2}{\mathrm{Ba}}_{2}{\mathrm{CuO}}_{6+{\delta}}$, and ${\mathrm{Pr}}_{1.15}{\mathrm{La}}_{0.7}{\mathrm{Ce}}_{0.15}{\mathrm{CuO}}_{4}$) have been studied with special focus on the bands with a predominately $d$-orbital character. Using a light polarization analysis, the ${e}_{g}$ and ${t}_{2g}$ bands are identified across these materials. A clear correlation between the ${d}_{3{z}^{2}{-}{r}^{2}}$ band energy and the apical oxygen distance ${d}_{\mathrm{A}}$ is demonstrated. Moreover, the compound dependence of the ${d}_{{x}^{2}{-}{y}^{2}}$ band bottom and the ${t}_{2g}$ band top is revealed. A direct comparison to density functional theory (DFT) calculations employing hybrid exchange-correlation functionals demonstrates excellent agreement. We thus conclude that the DFT methodology can be used to describe the global band structure of overdoped single-layer cuprates on both the hole- and electron-doped side." name="description" /><meta content="A comprehensive angle-resolved photoemission spectroscopy study of the band structure in single-layer cuprates is presented with the aim of uncovering universal trends across different materials. Five different hole- and electron-overdoped cuprate superconductors (${\mathrm{La}}_{1.59}{\mathrm{Eu}}_{0.2}{\mathrm{Sr}}_{0.21}{\mathrm{CuO}}_{4}$, ${\mathrm{La}}_{1.77}{\mathrm{Sr}}_{0.23}{\mathrm{CuO}}_{4}$, ${\mathrm{Bi}}_{1.74}{\mathrm{Pb}}_{0.38}{\mathrm{Sr}}_{1.88}{\mathrm{CuO}}_{6+{\delta}}$, ${\mathrm{Tl}}_{2}{\mathrm{Ba}}_{2}{\mathrm{CuO}}_{6+{\delta}}$, and ${\mathrm{Pr}}_{1.15}{\mathrm{La}}_{0.7}{\mathrm{Ce}}_{0.15}{\mathrm{CuO}}_{4}$) have been studied with special focus on the bands with a predominately $d$-orbital character. Using a light polarization analysis, the ${e}_{g}$ and ${t}_{2g}$ bands are identified across these materials. A clear correlation between the ${d}_{3{z}^{2}{-}{r}^{2}}$ band energy and the apical oxygen distance ${d}_{\mathrm{A}}$ is demonstrated. Moreover, the compound dependence of the ${d}_{{x}^{2}{-}{y}^{2}}$ band bottom and the ${t}_{2g}$ band top is revealed. A direct comparison to density functional theory (DFT) calculations employing hybrid exchange-correlation functionals demonstrates excellent agreement. We thus conclude that the DFT methodology can be used to describe the global band structure of overdoped single-layer cuprates on both the hole- and electron-doped side.

## Abstract

A comprehensive angle-resolved photoemission spectroscopy study of the band structure in single-layer cuprates is presented with the aim of uncovering universal trends across different materials. Five different hole- and electron-overdoped cuprate superconductors (${\mathrm{La}}_{1.59}{\mathrm{Eu}}_{0.2}{\mathrm{Sr}}_{0.21}{\mathrm{CuO}}_{4}$, ${\mathrm{La}}_{1.77}{\mathrm{Sr}}_{0.23}{\mathrm{CuO}}_{4}$, ${\mathrm{Bi}}_{1.74}{\mathrm{Pb}}_{0.38}{\mathrm{Sr}}_{1.88}{\mathrm{CuO}}_{6+{\delta}}$, ${\mathrm{Tl}}_{2}{\mathrm{Ba}}_{2}{\mathrm{CuO}}_{6+{\delta}}$, and ${\mathrm{Pr}}_{1.15}{\mathrm{La}}_{0.7}{\mathrm{Ce}}_{0.15}{\mathrm{CuO}}_{4}$) have been studied with special focus on the bands with a predominately $d$-orbital character. Using a light polarization analysis, the ${e}_{g}$ and ${t}_{2g}$ bands are identified across these materials. A clear correlation between the ${d}_{3{z}^{2}{-}{r}^{2}}$ band energy and the apical oxygen distance ${d}_{\mathrm{A}}$ is demonstrated. Moreover, the compound dependence of the ${d}_{{x}^{2}{-}{y}^{2}}$ band bottom and the ${t}_{2g}$ band top is revealed. A direct comparison to density functional theory (DFT) calculations employing hybrid exchange-correlation functionals demonstrates excellent agreement. We thus conclude that the DFT methodology can be used to describe the global band structure of overdoped single-layer cuprates on both the hole- and electron-doped side." name="description" /><meta content="A comprehensive angle-resolved photoemission spectroscopy study of the band structure in single-layer cuprates is presented with the aim of uncovering universal trends across different materials. Five different hole- and electron-overdoped cuprate superconductors (${\mathrm{La}}_{1.59}{\mathrm{Eu}}_{0.2}{\mathrm{Sr}}_{0.21}{\mathrm{CuO}}_{4}$, ${\mathrm{La}}_{1.77}{\mathrm{Sr}}_{0.23}{\mathrm{CuO}}_{4}$, ${\mathrm{Bi}}_{1.74}{\mathrm{Pb}}_{0.38}{\mathrm{Sr}}_{1.88}{\mathrm{CuO}}_{6+{\delta}}$, ${\mathrm{Tl}}_{2}{\mathrm{Ba}}_{2}{\mathrm{CuO}}_{6+{\delta}}$, and ${\mathrm{Pr}}_{1.15}{\mathrm{La}}_{0.7}{\mathrm{Ce}}_{0.15}{\mathrm{CuO}}_{4}$) have been studied with special focus on the bands with a predominately $d$-orbital character. Using a light polarization analysis, the ${e}_{g}$ and ${t}_{2g}$ bands are identified across these materials. A clear correlation between the ${d}_{3{z}^{2}{-}{r}^{2}}$ band energy and the apical oxygen distance ${d}_{\mathrm{A}}$ is demonstrated. Moreover, the compound dependence of the ${d}_{{x}^{2}{-}{y}^{2}}$ band bottom and the ${t}_{2g}$ band top is revealed. A direct comparison to density functional theory (DFT) calculations employing hybrid exchange-correlation functionals demonstrates excellent agreement. We thus conclude that the DFT methodology can be used to describe the global band structure of overdoped single-layer cuprates on both the hole- and electron-doped side.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Physics Institute 530 Physics Physical Sciences > Electronic, Optical and Magnetic Materials Physical Sciences > Condensed Matter Physics English 17 June 2019 07 Aug 2019 09:41 29 Jul 2020 11:01 American Physical Society 2469-9950 © 2019 American Physical Society Green https://doi.org/10.1103/physrevb.99.224509

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