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# Partial equivalence relation

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In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) $R$ on a set $X$ is a relation that is symmetric and transitive. In other words, it holds for all $a, b, c \in X$ that:

1. if $a R b$, then $b R a$ (symmetry)
2. if $a R b$ and $b R c$, then $a R c$ (transitivity)

If $R$ is also reflexive, then $R$ is an equivalence relation.

## Properties and applications

In a set-theoretic context, there is a simple structure to the general PER $R$ on $X$: it is an equivalence relation on the subset $Y = \{ x \in X | x\,R\,x\} \subseteq X$. ($Y$ is the subset of $X$ such that in the complement of $Y$ ($X\setminus Y$) no element is related by $R$ to any other.) By construction, $R$ is reflexive on $Y$ and therefore an equivalence relation on $Y$. Notice that $R$ is actually only true on elements of $Y$: if $x R y$, then $y R x$ by symmetry, so $x R x$ and $y R y$ by transitivity. Conversely, given a subset Y of X, any equivalence relation on Y is automatically a PER on X.

PERs are therefore[why?] used mainly in computer science, type theory and constructive mathematics, particularly to define setoids, sometimes called partial setoids. The action of forming one from a type and a PER is analogous to the operations of subset and quotient in classical set-theoretic mathematics.

Every partial equivalence relation is a difunctional relation, but the converse does not hold.

The algebraic notion of congruence can also be generalized to partial equivalences, yielding the notion of subcongruence, i.e. a homomorphic relation that is symmetric and transitive, but not necessarily reflexive.[1]

## Examples

A simple example of a PER that is not an equivalence relation is the empty relation $R=\emptyset$ (unless $X=\emptyset$, in which case the empty relation is an equivalence relation (and is the only relation on $X$)).

### Euclidean parallelism

In the Euclidean plane, two lines m and n are parallel lines when mn = ∅. The symmetry of this relation is obvious and the transitivity can be proven in the Euclidean plane, thus Euclidean parallelism is a partial equivalence relation. Nevertheless, mathematicians developing affine geometry prefer the facility of an equivalence relation and therefore sometimes revise the definition of parallelism to allow a line to be parallel to itself, making the new relation of "affine parallelism" that is a reflexive relation.

### Kernels of partial functions

For another example of a PER, consider a set $A$ and a partial function $f$ that is defined on some elements of $A$ but not all. Then the relation $\approx$ defined by

$x \approx y$ if and only if $f$ is defined at $x$, $f$ is defined at $y$, and $f(x) = f(y)$

is a partial equivalence relation but not an equivalence relation. It possesses the symmetry and transitivity properties, but it is not reflexive since if $f(x)$ is not defined then $x \not\approx x$ — in fact, for such an $x$ there is no $y \in A$ such that $x \approx y$. (It follows immediately that the subset of $A$ for which $\approx$ is an equivalence relation is precisely the subset on which $f$ is defined.)

### Functions respecting equivalence relations

Let X and Y be sets equipped with equivalence relations (or PERs) $\approx_X, \approx_Y$. For $f,g : X \to Y$, define $f \approx g$ to mean:

$\forall x_0 \; x_1, \quad x_0 \approx_X x_1 \Rightarrow f(x_0) \approx_Y g(x_1)$

then $f \approx f$ means that f induces a well-defined function of the quotients $X / \approx_X \; \to \; Y / \approx_Y$. Thus, the PER $\approx$ captures both the idea of definedness on the quotients and of two functions inducing the same function on the quotient.

## References

1. ^ J. Lambek (1996). "The Butterfly and the Serpent". In Aldo Ursini, Paulo Agliano. Logic and Algebra. CRC Press. pp. 161–180. ISBN 978-0-8247-9606-8.

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Difference between CPM and RPM for RNA-seq reada quantification
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Entering edit mode
3.8 years ago
Emilio Marmol ▴ 140

Hello everyone, I have a dataset of differentially expressed miRNAs.

I aligned my microRNA reads (10 millions per sample aprox) against reference genome, obtaining an average of 7.5 million aligned read per sample, and used FeatureCounts from RSubRead to count them asigning to reference miRNA GTF in miRBase, obtaining an average of 60% of asignment. Then, I used DESeq2 to perform differential expression analysis.

As said in Mullokandov et al. (2013), functional miRNAs should only be considered when expressed above 100-1000 RPM on average. I calculated CPM from FeatureCounts counts file and obtained average expression per group (2 groups contast) and expression per sample.

I would like to know if there is any difference between the concept of RPM and CPM, as I think the are quite the same, but not sure about that. I assumed miRNAs with a cpm expression below 100 to be spurious and noise, but I would like to be sure about that.

Any suggestions and clarifications?

Thanks

RNA-Seq miRNAs CPM RPM • 7.3k views
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CPM cutoff of 100 sounds like a lot. But I have limited experience in miRNA sequence analysis. Here is one post for CPM cutoff for miRNA data.

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I understand what that post says, but this is not refereing to biologically active miRNAs but to initially threshold filtering for eliminating reads with zeros and very small counts.

100-1000 or more RPM (or CPM but I am not sure XD) threshold is considered for withing the cell miRNA capability to drive a significant response in mRNA targeting and repressing gene expression and not just slight and spurious reactions. It is a matter of selecting really interesting DE miRNAs for discussion rather than for statistical analyses, in which I already applied such pre-filtering anyway.

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Following sentence was not clear

I assumed miRNAs with a cpm expression below 100 to be spurious and noise

Gave an impression that you are looking for cutoff for expression and not function.

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well, I assumed less than 100 CPMs to be a very low expression level so that the miRNA could not drive a significant function in repressing gene expression. If you have differential expression but with single or very scarce miRNA molecules, the effect is not expected to be such a thing to take into consideration. Maybe for profiling in cancer or so could be useful, but not for infering metabolic changes.

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No CPM, aka TPM, and F(R)PKM are not the same!! I think it is time for some StatQuest. You may also want to google for blogs and videos of Lior Pachter, one of the leading RNA-seq analysis guys on that matter.

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CPM is not TPM...

Question is the difference between counts per million and reads per million.

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I'm not asking for RPKM, TPM or FPKM sorry...

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Hi v82masae,

I have few queries regarding your miRNA requirement.

• You might have used Illumina sequencing platform to sequence miRNA. After adapter trimming what was the length of the small RNA you kept?
• Did you compare your sequenced miRNA against miRBase database?
• What tools did you use to generate differentially expressed miRNAs?
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Entering edit mode

II did miRNA sequencing yes, with HiSeq 2500 Illumina, single-end.

I used Cutadapt for trimming adapters with an acceptance window of 15-30 nt (quite big, as reported windows are normally 18-25, but I wanted to retain big fragments for contrasting against circRNAs and precursor forms). I tried a smaller window and the output did not differ quite much, just a few thousand reads if not less.

After trimming, I aligned the reads against reference genome with about 80% mapping, and used FeatureCounts for counting reads in BAM files with an average 60% of successful assignment to GTF miRNA annotation file downloaded from miRBase v21 yes.

For differential expression I used DESeq2 and EdgeR, both giving quite similar results and kept DESeq2 just for department papers homogeneity as we are using DESeq2 for the majority of other papers from same project.

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Entering edit mode

Thanks, V82masae.

Based on this paper, I used the mirna_soft tool (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4138881/).

Even we sequenced using Illumina, 51 bp single-end. After trimming, I retained the length between 18-35nt. Then I compared the sequenced reads against the miRBase v21 database. For Bovine, I extracted bovine specific miRNA from mature.fa file from miRBase v21.

• Did you do alignment with your species miRNAs or reference genome?

• Did you use bowtie2 aligner?

I got the output file with miRNA id, sequence, count of unique matches and count of multi-matches. For one of the samples, total trimmed reads- 4,679,140. Unique matches to the bovine miRNA sequences (from miRBase) was 800,000 matches. unmatched reads - 3,796,387

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I did alignment from trimmed reads to reference genome yes. As my model organism (pig) had a very poorly annotated miRNAome, I also aligned my reads agains human and mouse genome and did all the pipeline to detect miRNAs differentially expressed that were not described in the miRNA GTD annotation in pigs, considering that within phylogenetically close species, miRNA sequences are commonly conserved.

I tried some de novo miRNA finding algorithms like mirDeep2, but the output did not convince me, too noisy. I opted for finding manually some DE non-annotated miRNAs in human and mouse and looking for their positions in pig assembly, then doing folding to see if secondary structure was equivalent to expected miRNA. Found 12 new miRNAs that way, 7 of them were in fact new miRNAs as they were included in the recently new pig assembly that was released one month ago, and the other 5 were compatible with fragments of circRNAs, but considered as miRNAs in human and mouse, so maybe some fragmentation of circRNA could be involved.

I used bowtie1, it is better suited for small reads than bowtie2, without considering any mismatches in seed (not taking into consideration ismoMIRs), maximum of 20 multimappings and --best --strata mode with one best alignment per read reported.

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Entering edit mode

Hi V82masae,

I am working on both bovine and pig.

What do you think about this workflow, from miRBase I extracted bovine and pig related miRNAs from mature.fa (miRBase) file. Using miRNA_soft tool as per the research article I provided in the previous comment, I counted the reads mapped to bovine and pig miRNA sequences extracted from mature.fa. I got an output file with the following details:

miRNA id, miRNA sequence, Unique matches and Multi-matches.

I got similar output for all the samples. Then again using miRNA_soft tool, I compared the counts generated in the previous step to generate differential miRNA expression profiling.

I am a naive person in miRNA sequencing. Do you recommend this workflow?

What is miRNA GTD?

Can you share the pig reference genome genbank id or URL?

Why didn't you compare your sequenced reads with miRBase database instead you compared it with the pig, mouse and human reference genomes?

0
Entering edit mode
3.8 years ago
mforde84 ★ 1.3k
reads per million  = counts per feature / (total reads / 1000000)
counts per million = (counts per feature / total reads) * 1000000

0
Entering edit mode

Ain't these equations same, just different representations?

0
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Watch the StatQuest video I linked. You will see that the difference in the order of the operations is quiet remarkable on the result.

0
Entering edit mode

Order of operations are different for TPM and RPKM. But the question is what is the difference between CPM and RPM. I think there is none and the equations above also do not appear different from each other.

0
Entering edit mode

counts per feature / (total reads / 1000000) == (counts per feature / total reads) * 1000000

0
Entering edit mode

Well, for me those two equations are the same. Let's say I have a total of 5 million reads per sample i, and 1000 counts per miRNA j, so as you wrote:

RPM = 1000 / (5000000 / 1000000) = 1000 / 5 = 200 CPM = (1000 / 5000000) * 1000000 = 0.0002 * 1000000 = 200

I used the RPM formula you posted for calculations, but contrasting it with cpm function in EdgeR package, the output is the same, so that's why I got confused about terminology. I initially tried counting reads in fastq files (like those are reads, not counts, and then looking for sequences matching the miRNAs of interest, but then I realised that this would make the alignment and counting procedure spurious, so that all statistics were based on counting by assigning to miRNA locations, not read numbers itself. Soooo, I just kept CPM considering as the same of RPM, but I am not sure about that, and that's why I asked...


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# Two-column longtable or tabular

I'm using longtable to store a glossary-like list of abbreviations used in my document, and as it only has two columns, but is two pages long, I was wondering if there was a way to make it wrap into columns, or an alternative package that would make the creation of a glossary simpler.

I've read over the documentation of nomencl, but it doesn't look to be quite what I'm after, as I don't want an index, but a list of abbreviations and their meaning. And I don't need hyperlinks to their occurrences or even page references. Just a list of the abbreviations and their definitions.

The table I'm presently using is simple:

\begin{center}\small
\begin{longtable}{>{\scshape}fm{0.10\textwidth}-m{0.50\textwidth}}
abbr & Abbreviation \tabularnewline
abbr & Abbreviation \tabularnewline
...
\end{longtable}
\end{center}

I've set up column formatting marks so the left-hand column is always in small-caps, and the column-width setting is what looks nice-ish on an A4 page with a nice amount of space between the abbreviation and its definition. But the right side of the page is devoid of content, and I would prefer to realign this table so instead of stretching over two pages, it's reduced to just one.

EDIT Using the tabbing environment gives me the desired table split over two columns, but takes away the formatting control I previously had. Can I have the best of both worlds? I really don't want to add a \textsc{abbr} to all 100 lines or so.

EDIT 2 I don't want this table to span columns, I want it to wrap around columns. So it runs to the bottom of the first column, and then continues down the next, before moving to the next page if necessary.

longtable changes the output routine and so it does not work in two column modes. Try supertabular or xtab instead.

• xtab doesn't work. I'm getting a weird error now, apparently a missing number somewhere. And once I did figure out what it was doing, it didn't wrap the table over two columns. Feb 8 '11 at 5:19

Aha: I figured it out, and a table/tabular/etc was not the solution. Instead, if I use a definition list inside a multicol environment, and then use the enumitem package to customise the list, I get exactly what I want.

[EDIT] SOLUTION

This is just a minimal working example that does what I wanted:

\usepackage{multicol}
\usepackage{enumitem}

\begin{document}
\begin{multicols*}{2}
\begin{description}[font=\normalfont\scshape,labelindent=12pt,leftmargin=60pt,style=sameline]
\item[1] First person
\item[2] Second person
\item[3] Third person
\item[abl] Ablative
\item[acc] Accusative
\end{description}
\end{multicols*}
\end{document}

For the effect to show, you need a longer list, but it will create columns with (in this case) an abbreviation and a definition. It doesn't do indices, however.

• I know this post was published a loooong time ago, but I'd like to hear from you some details about your proposed solution. I've just ran into the same problem, and it sounds tough for me to figure out how to solve such a problem. Thanks in advance. Apr 17 '12 at 21:17
• this is not a solution. The answer is you can't (the other answer should be selected as valid) Oct 14 '14 at 18:20


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# Also in the Article

Segment length
This protocol is extracted from research article:
Solvent-dependent segmental dynamics in intrinsically disordered proteins

Procedure

We used the definition of angle order parameters of Hyberts et al. (35) in which N is the number of frames in a simulation and α is the time-dependent angle between the Cα-Cα vectors associated with two residues. If $Sseg2>0.8$, the two residues are considered to belong to the same dynamic segment.

Note: The content above has been extracted from a research article, so it may not display correctly.

Q&A


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# zbMATH — the first resource for mathematics

WPO, COV and IIA bargaining solutions for non-convex bargaining problems. (English) Zbl 1268.91079
Summary: We characterize all $$n$$-person multi-valued bargaining solutions, defined on the domain of all finite bargaining problems, and satisfying weak Pareto optimality (WPO), covariance (COV), and independence of irrelevant alternatives (IIA). We show that these solutions are obtained by iteratively maximizing nonsymmetric Nash products and determining the final set of points by so-called LDR decompositions. If, next, we assume the (set-theoretic) axiom of determinacy, then this class coincides with the class of iterated Nash bargaining solutions; but if we assume the axiom of choice then we are able to construct an additional large set of discontinuous and even nonmeasurable solutions. We show, however, that none of these nonmeasurable solutions can be defined in terms of set theoretic formulae. We next show that a number of existing results in the literature as well as some new results are implied by our approach. These include a characterization of all WPO, COV and IIA solutions – including single-valued ones – on the domain of all compact bargaining problems, and an extension of a theorem of Birkhoff characterizing translation invariant and homogeneous orderings.

##### MSC:
 91B26 Auctions, bargaining, bidding and selling, and other market models
Full Text:
##### References:
 [1] Billingsley P (1968) Convergence of probability measures. Wiley series in probability and mathematical statistics, John Wiley & Sons, New York · Zbl 0172.21201 [2] Birkhoff G (1948) Lattice theory. American Mathematical Society Publications, American Mathematical Society, New York · Zbl 0033.10103 [3] Conley J, Wilkie S (1996) An extension of the Nash bargaining solution to nonconvex problems. Games Econ Behav 13: 26–38 · Zbl 0848.90135 · doi:10.1006/game.1996.0023 [4] d’Aspremont C (1985) Axioms for social welfare orderings. In: Hurwicz L, Schmeidler D, Sonnenschein H (eds) Social goals and social organizations. North-Holland, Amsterdam [5] Debreu G (1964) Continuity properties of Paretian utility. Int Econ Rev 5: 285–293 · Zbl 0138.16301 · doi:10.2307/2525513 [6] Denicolò V, Mariotti M (2000) Nash bargaining theory, nonconvex problems and social welfare orderings. Theory Decis 48: 351–358 · Zbl 0963.91052 · doi:10.1023/A:1005278100070 [7] Fishburn PC (1972) Even-chance lotteries in social choice theory. Theory Decis 3: 18–40 · Zbl 0257.90001 · doi:10.1007/BF00139351 [8] Hamel G (1905) Eine Basis aller Zahlen und die unstetige Lösung der Funktionalgleichung f(x + y) = f(x) + f(y). Math Ann 60: 459–462 · JFM 36.0446.04 · doi:10.1007/BF01457624 [9] Harsanyi JC, Selten R (1972) A generalized Nash solution for two-person bargaining games with incomplete information. Manag Sci 18: 80–106 · Zbl 0262.90087 · doi:10.1287/mnsc.18.5.80 [10] Herrero MJ (1989) The Nash program: non-convex bargaining problems. J Econ Theory 49: 266–277 · Zbl 0683.90104 · doi:10.1016/0022-0531(89)90081-1 [11] Kahneman D, Tversky A (1979) Prospect theory: an analysis of decision under risk. Econometrica 47: 263–291 · Zbl 0411.90012 · doi:10.2307/1914185 [12] Kaneko M (1980) An extension of the Nash bargaining problem and the Nash social welfare function. Theory Decis 12: 135–148 · Zbl 0434.90011 · doi:10.1007/BF00154358 [13] Kaneko M, Nakamura K (1979) The Nash social welfare function. Econometrica 47: 423–435 · Zbl 0431.90091 · doi:10.2307/1914191 [14] Machina MJ (1982) ”Expected utility” analysis without the independence axiom. Econometrica 50: 277–323 · Zbl 0475.90015 · doi:10.2307/1912631 [15] Marek VW, Mycielski J (2001) Foundations of mathematics in the twentieth century. Am Math Month 108: 449–468 · Zbl 0978.03001 · doi:10.2307/2695803 [16] Mariotti M (1998a) Nash bargaining theory when the number of alternatives can be finite. Soc Choice Welf 15: 413–421 · Zbl 1066.91557 · doi:10.1007/s003550050114 [17] Mariotti M (1998b) Extending Nash’s axioms to nonconvex problems. Games Econ Behav 22: 377–383 · Zbl 0914.90275 · doi:10.1006/game.1997.0590 [18] Maschler M, Owen G, Peleg B (1988) Paths leading to the Nash set. In: Roth AE (eds) The shapley value: essays in honor of Lloyd S. Shapley. Cambridge University Press, Cambridge UK , pp 321–330 [19] Mycielski J, Swierczkowski S (1964) On the Lebesgue measurability and the Axiom of Determinateness. Fundam Math 54: 67–71 · Zbl 0147.19503 [20] Nash JF (1950) The bargaining problem. Econometrica 18: 155–162 · Zbl 1202.91122 · doi:10.2307/1907266 [21] Naumova N, Yanovskaya E (2001) Nash social welfare orderings. Math Soc Sci 42: 203–231 · Zbl 1101.91322 · doi:10.1016/S0165-4896(01)00070-1 [22] Podnieks K (2007) http://www.ltn.lv/$$\sim$$podnieks/gt.html [23] Roth AE (1977) Individual rationality and Nash’s solution to the bargaining problem. Math Oper Res 2: 64–65 · Zbl 0413.90089 · doi:10.1287/moor.2.1.64 [24] Shubik M (1982) Game theory in the social sciences: concepts and solutions. MIT Press, Cambridge · Zbl 0903.90179 [25] Starmer C (2000) Developments in non-expected utility theory: the hunt for a descriptive theory of choice under risk. J Econ Lit 38: 332–382 · doi:10.1257/jel.38.2.332 [26] Xu Y, Yoshihara N (2006) Alternative characterizations of three bargaining solutions for nonconvex bargaining problems. Games Econ Behav 57: 86–92 · Zbl 1154.91418 · doi:10.1016/j.geb.2005.09.003 [27] Zame WR (2007) Can intergenerational equity be operationalized. Theor Econ 2: 187–202 [28] Zhou L (1996) The Nash bargaining theory with non-convex problems. Econometrica 65: 681–685 · Zbl 0876.90102 · doi:10.2307/2171759
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
If f(x_n) converges whenever (x_n), then f is continuous.

This is the original question:

Let $$f: X\to Y$$ be a map between metric spaces. Prove that if $$(f(x_n))_{n=0}^\infty$$ converges in $$Y$$ whenever $$(x_n)_{n=0}^\infty$$ converges in $$X$$ then $$f$$ is continuous. [Note: it is not given that $$f(x_n) → f(x)$$ whenever $$x_n → x$$.]

I have spent quite a bit of time on this problem. I finally looked it up and have been reading the solution in the link below:

for every convergent sequence $x_n$, $f(x_n)$ also converges. Does this imply continuity of f?

I am confused by the solution given. I understand that they proved that $$f(x_n)$$ does not converge to $$f(x)$$, but this doesn't prove that $$f(x_n)$$ does not converge at all. If we want to use the contrapositive, shouldn't be have to show that if $$f$$ is discontinuous, then there is some sequence $$(x_n)$$ that converges in $$X$$ but $$f(x_n)$$ does not converge in $$Y$$. That is, we have to show that for some convergent sequence, there is some $$\varepsilon$$ such that for all $$n$$ greater than some $$N$$, $$f(x_n)>\varepsilon$$.

Any clarification or hints would be greatly appreciated.

• A convergent sequence in a metric space is a Cauchy sequence. But the sequence $f(x),f(x_1),f(x),f(x_2),...$ which was given in the accepted answer is clearly not Cauchy, because $d(f(x),f(x_n))\geq\epsilon$ for all $n$. So it doesn't converge at all. – Mark Aug 9 '19 at 21:12

What I proved in that answer was that, if $$f$$ was discontinuous at some point $$x$$, then there is a sequence $$(x_n)_{n\in\mathbb N}$$ of elements of $$X$$ converging to $$x$$ such that the sequence $$\bigl(f(x_n)\bigr)_{n\in\mathbb N}$$ is not convergent. But we are assuming that no such sequence exists.
• That is correct, yes, but that was not the approach that I had in mind. What I thought was that my sequence has a subsequence which converges to $f(x)$ and another subsequence such that each of its terms is at a distance at least equal to $\varepsilon$ (for a fixed number $\varepsilon>0$) from $f(x)$. This later subsequence cannot possibly converge to $f(x)$. But when a sequence converges, each of its subsequences converges to the same limit. – José Carlos Santos Aug 9 '19 at 21:24


text:
# Enzymes that display non Michaelis-Menton behavior:are multi subunitexists in relaxed (R) and tense (T) statescan be activated or inhibited by

###### Question:

Enzymes that display non Michaelis-Menton behavior: are multi subunit exists in relaxed (R) and tense (T) states can be activated or inhibited by allosteric ligands bind ligands at sites other than the active sites All of the above

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text:
# PRODUCED BY ${{\boldsymbol K}^{\pm}}$ BEAM INSPIRE search

VALUE (MeV) EVTS DOCUMENT ID TECN  COMMENT
$\bf{ 82 \pm6}$ OUR AVERAGE
$90$ $\pm12$
 1988 D
LASS 11 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit K}_S^0}$ ${{\mathit \Lambda}}$
$73$ $\pm18$
 1986
SPEC 40 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit K}_S^0}$ ${{\mathit Y}}$
$83$ $\pm15$
 1983 B
OMEG 18.5 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit K}^{-}}{{\mathit K}^{+}}{{\mathit \Lambda}}$
$85$ $\pm16$ 650
 1981 B
HBC 4.2 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \Lambda}}{{\mathit K}^{+}}{{\mathit K}^{-}}$
$80$ ${}^{+14}_{-11}$ 572
 1981
HBC 8.25 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \Lambda}}{{\mathit K}}{{\overline{\mathit K}}}$
$72$ $\pm25$ 166
 1977
OMEG 10 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit K}^{+}}{{\mathit K}^{-}}$( ${{\mathit \Lambda}}$ ,${{\mathit \Sigma}}$)
$69$ $\pm22$ 100
 1972 B
HBC 3.9,4.6 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit K}}{{\overline{\mathit K}}}$( ${{\mathit \Lambda}}$ ,${{\mathit \Sigma}}$)
• • • We do not use the following data for averages, fits, limits, etc. • • •
$92$ ${}^{+25}_{-16}$ 61
 2007
GAMS 32.5 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \eta}}{{\mathit \eta}}$( ${{\mathit \Lambda}}$ $/$ ${{\mathit \Sigma}^{0}}$)
$75$ $\pm20$ 1
 1999
SPEC 40 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit K}_S^0}$ ${{\mathit y}}$
$62$ ${}^{+19}_{-14}$ 123
 1977
HBC 4.15 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit \Lambda}}{{\mathit K}_S^0}$ ${{\mathit K}_S^0}$
$61$ $\pm8$ 120
 1976 C
ASPK 13 ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit K}^{+}}{{\mathit K}^{-}}$( ${{\mathit \Lambda}}$ ,${{\mathit \Sigma}}$)
1  Systematic errors not estimated.
References:
 BINON 2007
PAN 70 1713 Observation of the ${{\mathit f}_{{2}}{(1525)}}$ Resonance in the ${{\mathit \eta}}{{\mathit \eta}}$ System Produced in ${{\mathit K}^{-}}{{\mathit p}}$ Interactions at a Momentum of 32.5 GeV/$\mathit c$
 BARKOV 1999
JETPL 70 248 Discovery of a Narrow Resonance State of the System ${{\mathit K}_S^0}$ ${{\mathit K}_S^0}$ at Mass 1520 MeV
 ASTON 1988D
NP B301 525 A Study of the ${{\mathit K}_S^0}$ ${{\mathit K}_S^0}$ System in the Reaction ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit K}_S^0}$ ${{\mathit \Lambda}}$ at 11 ${\mathrm {GeV/}}\mathit c$
 BOLONKIN 1986
SJNP 43 776 Study of the Reaction ${{\mathit K}^{-}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit K}_S^0}$ ${{\mathit K}_S^0}$ ${{\mathit Y}^{*}}$ at 40 ${\mathrm {GeV/}}\mathit c$
 ARMSTRONG 1983B
NP B224 193 Search for New (${\mathit {\mathit s}}{\mathit {\overline{\mathit s}}}$) States in the ${{\mathit K}^{-}}$ Induced ${{\mathit K}^{+}}{{\mathit K}^{-}}$ System
 AGUILAR-BENITEZ 1981B
ZPHY C8 313 Study of Hypercharge Exchange Reactions of the Type $0{}^{-}{}^{}$ $1/2{}^{+}{}^{}$ $\rightarrow$ $2{}^{+}{}^{}$ $1/2{}^{+}{}^{}$ at 4 ${\mathrm {GeV/}}\mathit c$
 ALHARRAN 1981
NP B191 26 Production of ${{\mathit f}^{\,'}{(1520)}}$ in 8.25 ${\mathrm {GeV/}}\mathit c$ ${{\mathit K}^{-}}{{\mathit p}}$ Interactions
 BARREIRO 1977
NP B121 237 Production and Decay Properties of ${{\mathit f}^{\,'}{(1514)}}$ in ${{\mathit K}^{-}}{{\mathit p}}$ Interactions at 4.2 ${\mathrm {GeV/}}\mathit c$
 EVANGELISTA 1977
NP B127 384 Production of ${{\mathit \phi}}$ and ${{\mathit f}^{\,'}}$ in 10 ${\mathrm {GeV/}}\mathit c$ ${{\mathit K}^{-}}{{\mathit p}}$ Interactions
 BRANDENBURG 1976C
NP B104 413 Analysis of ${{\mathit K}^{-}}{{\mathit K}^{+}}$ and ${{\mathit \pi}^{-}}{{\mathit \pi}^{+}}$ Final States in 13 GeV ${{\mathit K}^{-}}{{\mathit p}}$ Interactions
 AGUILAR-BENITEZ 1972B
PR D6 29 Study of Nonstrange Mesons Produced in ${{\mathit K}^{-}}{{\mathit p}}$ Interactions at 3.9 and 4.6 ${\mathrm {GeV/}}\mathit c$


text:
Clutch Prep is now a part of Pearson
Ch.9: Sampling Distributions: MeanWorksheetSee all chapters

# Large Sample Confidence Interval for a Population Mean

See all sections

Concept #1: Determining the Large Sample Confidence Interval for a Population Mean

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Concept #2: Determining the Large Sample Confidence Interval for a Population Mean: Intro

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Practice: Gas prices are getting more and more expensive. The average gas price, from a random sample of 100 gas stations, was $3.50. It is assumed that gas prices have a standard deviation of$0.04. Construct an 80% confidence interval for the true mean gas price in the United States.

Practice: You want to take a trip to Paris. You randomly select 225 flights to Europe and find a mean and standard deviation of $1500 and$900, respectively. Construct a 95% confidence interval for the average price for a trip to Paris.

Practice: You want to purchase one of the new Altimas. You randomly select 400 dealerships across the United States and find an average and standard deviation of $25,000 and$2500. Construct a 94% confidence interval for the true average price for the new Nissan Altima.

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text:
# TULIPS - Editorial

Author: Malvika Raj Joshi
Tester: Roman Furko
Translators: Vasya Antoniuk (Russian), Team VNOI (Vietnamese) and Hu Zecong (Mandarin)
Editorialist: Kevin Atienza

Hard

### PREREQUISITES:

Centroid decomposition, segment tree, disjoint set union, power-of-two pointers

### PROBLEM:

There are N nodes forming a tree. Each node contains a tulip, and each edge has a given length. If a tulip is picked, a new one will grow in the same spot if it is left undisturbed for X contiguous days.

On day d_j, Cherry visits all node reachable from u_j with edges of length \le k_j. She will pick all the tulips she can find. On day d_1, all nodes have tulips.

For each of the Q days Cherry does this, print the number of tulips she gets.

### QUICK EXPLANATION:

• Construct the “reachability tree” of the given tree. We define the reachability tree as a rooted binary tree with 2N-1 nodes such that each leaf represents a node in the original tree, and each internal node represents a set of nodes in the original tree that are reachable from each other using edges up to a certain length. In other words, each internal node is associated with some value, say k, such that the leaves under it represent a maximal set of nodes in the original tree that are reachable from each other using edges of length \le k. This tree can be constructed in O(N \log N) time.

• Compute the preorder traversal of the leaves of this tree. Now, the nodes in the original tree in this traversal has the property that any visitation by Cherry always affects consecutive nodes in the traversal, making it amenable to range query optimizations.

• Construct a segment tree on top of this traversal that can support the following operations:

• initialize: Initialize all values to 0.
• update: Given a subrange, increase/decrease it by a given amount.
• query: Given a subrange, how many values in it are equal to 0?

The structure may assume that all values will stay nonnegative.

• For every visitation (d_j,u_j,k_j), first find in the preorder traversal the range [l,r] of nodes reachable from u_j using edges of length \le k_j. (This can be done in O(\log N) time using power-of-two pointers.) Next, print the number of values in the segment tree in the subrange [l,r] equal to 0. Next, increase all elements in the subrange [l,r] by 1. Finally, add a future decrement operation that will decrease all elements in the subrange [l,r] by 1 on day d_j+X. (You will need to perform these decrement operations before processing visitations on or after that day. You can keep track of these future decrements using a queue, because d_j+X is increasing.)

(Note: there are many other approaches as well)

# Traversing edges up to a given length

Each visitation has a starting node u and a number k, and performs some operations across all nodes that are reachable from u using edges with weights \le k. We will use the notation R(u,k) to denote the set of nodes reachable from u using edge with weights \le k. Clearly, we can’t iterate through all nodes of R(u,k) for every visitation because that would be slow. So to progress, we must study the R(u,k) sets more closely.

Let’s consider a fixed u, and a varying k. k starts at 0 and continuously increases up to \infty. Let’s see what happens to R(u,k) while this happens. In the beginning (k = 0), no edges are usable, so R(u,k) is simply \{u\}. As we increase k however, more and more edges become usable, giving us access to more nodes, so R(u,k) possibly increases in size. Eventually, k will surpass the largest edge weight in the tree, and R(u,k) becomes the set of all N nodes.

There are two things that we learn from this process:

• As we increase k, \left|R(u,k)\right| never decreases in size.
• Not only that, but we can also say that for k < k', R(u,k) \subset R(u,k')! In other words, once a node becomes reachable from u, it becomes reachable forever.

These are certainly nice facts. Unfortunately these don’t give us the full picture. So let’s look at the full picture. Instead of focusing on a fixed node u, we will focus on all N nodes. Specifically, we will focus on R(u,k) for all nodes u (not just a fixed u), as we increase k.

As before, in the beginning, no edge is usable, so R(u,k) = \{ u \} for all nodes u. Now, as we increase k, one by one each edge becomes usable, effectively uniting the set of reachable nodes it connects. Specifically, once the edge (u,v) becomes usable, R(u,k) and R(v,k) unite and become equal. This happens one by one for each edge, until finally k passes the largest edge weight, and all sets R(u,k) unite into a single set containing all N nodes.

So, what did we learn from this experiment? We learned that the sets R(u,k) behave as if we’re doing union-find. Moreover, upon closer look, we can also see some sort of tree structure behind all this!

Let’s call this tree the reachability tree. We will define the reachability tree as a rooted binary tree with 2N-1 nodes such that:

• Each leaf represents a node in the original tree, and
• Each internal node represents a union operation during our experiment above.

We call this the “reachability tree” because each internal node represents some R(u,k) at some point during our experiment above. More specifically, each internal node is associated with some value, say k, such that its leaves constitute some R(u,k).

Let’s give an example. Consider the following tree:

The reachability tree of this tree is the following:

Note that the original nodes are leaves in this reachability tree. Also, every internal node represents some R(u,k). For example, node f represents the set R(A,16) = R(C,16) = R(D,16) = \{A, C, D\}.

The reachability tree has the following nice property:

Given any u, k, R(u,k) is always the set of leaves of some subtree in the reachability tree. (This includes any k, not just values associated with internal nodes.)

For example, suppose that in the tree above, we start at node F and traverse all edges with weight \le 14. Then R(F,14) = \{F, E, H\}, as shown in the following:

True enough, these nodes can be found as leaves of some subtree in the reachability tree:

This property of the reachability tree is very useful for us, as we intend to operate on R(u,k) sets many times. To see why it’s useful, note that when we perform a preorder traversal of the tree and collect the leaves during this traversal, the set of leaves of any subtree can be found in consecutive positions in the traversal. This allows us to turn any operation on the set R(u,k) into a range query! Thus, this tree can help us immensely if we can construct it.

# Constructing the reachability tree

We can construct the reachability tree by remembering how we discovered it: by uniting sets together, starting from N sets to just one. This is very much like union-find, except that during every union, we also create a new node recording that union in the reachability tree.

That is the main idea of the construction. We’re building the tree from the ground up, starting from the leaves up until the root. We will maintain disjoint sets representing $R(u,k)$s. In addition, each disjoint set’s representative points to its current root node in the reachability tree.

• Initially, all N nodes are on separate sets, each pointing to a new node. These new nodes are the leaves of the reachability tree.
• Next, we iterate through all edges in increasing order of weight. For each edge (u,v) with weight k, we unite the sets containing u and v, and create a new node in the reachability tree whose children are the nodes being pointed at by u's and v's representatives.
• After iterating through all N-1 edges, we now have the reachability tree!

This runs in O(N \log N) time because we sorted the edges. Everything else runs asymptotically faster.

# Performing queries

Next, we need to convert a query (d_j,u_j,k_j) into a range query, with the help of the reachability tree. We will assume that we’ve already performed the preorder traversal on the reachability tree, and collected all leaves in this traversal.

Our goal now is to find the relevant range of nodes [L_j,R_j] corresponding to the set R(u_j,k_j). There are two steps we need to take:

1. Given u_j and k_j, find the subtree that contains all nodes of R(u_j,k_j) (and only those nodes). Let p be the root of this subtree.
2. Given this subtree rooted at p, find the indices of the leftmost and rightmost leaves. (These indices are the L_j and R_j we are looking for.)

Let’s look at the first step. We need to find p. Surely, p is an ancestor of u_j in the reachability tree. More specifically, p is the highest ancestor of u_j whose associated value k is \le k_j. This gives us a way of computing p, but it’s slow (because the reachability tree can be highly imbalanced).

So how do we compute p quickly? It would be nice if we can binary search along all ancestors of u_j. But we can actually do that with the help of power-of-two pointers (also called jump pointers). The idea is, for each node in the reachability tree, to store all its ancestors at power-of-two heights, i.e. for every i \ge 0, we store its $2^i$th ancestor. This structure is usually used to compute certain ancestors of nodes quickly (and is also used in computing lowest common ancestors), but we can also use this structure to compute p quickly. The following pseudocode illustrates the idea:

// in this code, u.jump(i) is the 2^i'th ancestor of u
def find_p(u_j, k_j):
if nodes[ROOT].k <= k_j:
return ROOT

i = 0
while u.jump(i).k <= k_j:
u = u.jump(i++)

while i != 0:
i--
if u.jump(i).k <= k_j:
u = u.jump(i)

return u

The first part simply checks if the root is already a valid p. Otherwise, the first while loop simply finds some ancestor that is not a valid p. The second while loop then finds the smallest ancestor that is not a valid p. So after that loop, the ancestor before that must be the p we are looking for!

Now, for the second part. Given a subtree rooted at p, what is the leftmost and rightmost leaf? This is actually easier: Simply precompute all leftmost and rightmost leaves using a single traversal and dynamic programming! The following recurrences can be used:

\text{leftmost}(u) = \begin{cases} u & \text{if $u$ is a leaf} \\\ \text{leftmost}(\text{left}(u)) & \text{otherwise} \end{cases}
\text{rightmost}(u) = \begin{cases} u & \text{if $u$ is a leaf} \\\ \text{rightmost}(\text{right}(u)) & \text{otherwise} \end{cases}

Now, we can convert (d_j,u_j,k_j) into a range query on the range [L_j,R_j]!

# Range queries

Now that all queries are now range queries, let’s write the current version of the problem:

There are N nodes in a line. Each node contains a tulip. If a tulip is picked, a new one will grow in the same spot if it is left undisturbed for X contiguous days.

On day d_j, Cherry visits all nodes from L_j to R_j and picks all the tulips she can find. On day d_1, all nodes have tulips.

For each of the Q days Cherry does this, print the number of tulips she gets.

This sounds much doable because we’ve essentially gotten rid of the tree and replaced it with range queries. We’ve now essentially flattened the problem. All that remains is to find a way to perform the range queries efficiently!

The idea for our fast solution would be to create an array of N elements, [D_1, \ldots, D_N], where D_i represents the “disturbance” of the i th node. Here, we define the disturbance of a node as the number of times the node has been disturbed in the past X days. This way, a node has an available tulip if and only if its disturbance is 0. Initially of course, D_i = 0 for all i.

Now, when nodes L_j to R_j are visited on day d_j, these things happen:

• Count the number of available tulips in the range. Specifically, count the number of nodes in the range [L_j,R_j] with disturbance equal to 0.
• Disturb these nodes, i.e. increase the disturbance of all nodes from L_j to R_j by 1. However, we must remember to decrease them again after X days, i.e. on day d_j+X.

So all in all, three events happen for each visit (d_j,L_j,R_j):

• On day d_j, we count the number of i s in the range [L_j,R_j] where D_i = 0.
• On day d_j, we increase all disturbances in the range [L_j,R_j] by 1.
• On day d_j+X, we decrease all disturbances in the range [L_j,R_j] by -1.

Thus, to process all visitations, we just need to sort all the events across all visitations altogether. If multiple events happen on the same day, perform decrease operations before queries, and perform queries before increase operations.

But how do we handle range increases/decreases and range queries? At first glance, it seems hard to find any efficient segment tree solution, but things become much easier when you notice that disturbances never drop below 0. So if the disturbance 0 is present in the range at all, then it is surely the minimum disturbance in that range. Thus, to answer the range query "how many disturbances are equal to 0", we can use the following more general range query:

• Given a subrange [L_j,R_j], what is the minimum disturbance in this range, and how many times does it appear?

To answer our original range query, we simply perform this new range query on the range and then check if the minimum disturbance is 0. If it is, then return the count, otherwise, simply return 0.

Thus, all we need to implement is a typical range minimum query structure with range updates, with a slight twist: we also need to count how many times the minimum appears. But our trusty segment tree with lazy propagation can do this job very efficiently! We just need to modify it slightly so that the count can also be returned. To do it, simply add this count in every node of the segment tree, and remember to update it during downward propagations, updates and queries!

For implementation details, see the tester’s or editorialist’s code.

Now, what is the time complexity? Clearly, O(\log N) time is needed for each query/update, so all events run in O(Q \log N) time. Initializing the segment tree requires O(N) time, and computing the sorted order of the events requires O(Q \log Q) time. By including the fact that the reachability tree is constructed in O(N \log N) time, the overall complexity becomes O((N+Q)\log N + Q\log Q).

We can actually remove the O(Q \log Q) bit by noticing that the visits are given in increasing order of d_j already. This means that the only out-of-order events are the decrease events, and even they are already given in increasing order (since d_j+X is increasing). Thus, all we need to do is to merge two sorted list of events together. The merge routine of merge sort does this job in O(Q) time. With this, the overall running time becomes just O((N+Q) \log N).

O((N+Q) \log N)

### AUTHOR’S AND TESTER’S SOLUTIONS:

7 Likes

First of all, really nice question. It took a lot of time for me to get this one, but then again I learned a lot.

Went for the 15 points first. Sort the queries by the maximum edge weight allowed. Sort the edges by their respective edge weights. Put all the nodes in individual sets. Yep, union-find time. Iterate over all the (sorted) edges/queries and combine the sets of the nodes on either ends of your current edge. For a query (d, u, k) once you have processed all edges with weight <=k, the answer is simply the number of elements in the set of u.

Here is the solution that gets you the sweet 15 points.

For a long time I thought I had to go back to the drawing board and figure out something else to proceed. But then came a bright idea. What if I renumber my nodes in a way such that whenever a union operation takes place, the nodes in the new set are always as follows: 1, 2, …, k (for k nodes)

For example, if I have two sets

set1 - (1, 2, 3)

set2 - (1, 2, 3, 4)

My new set will be (1, 2, 3, 1 + len(set1), 2 + len(set1), 3 + len(set1), 4 + len(set1))

union(set1, set2) = (1, 2, 3, 4, 5, 6, 7)

The idea is even though the sets are combined now, I can access set1 via a query on range (1, 3) and set2 with a query on (4, 7).

I initially “numbered” all nodes as 1. For every union I kept track of what the size of the other set was, I also kept track of the max and min element in that set. And at the end I added it all up and changed the node numbering accordingly. Now all of my queries involve nodes in a given range. And I have the max and min, therefore the start and end of that range.

Another iteration of the union-find pass converts our (d, u, k) queries to (d, i, j).

Now we have a few range queries to handle.

1 i, j - return number of elements in range (i, j) with value >= X.

2 i, j - set number of elements in range (i, j) equal to 0

3 Y - increment all then numbers by Y

Couldn’t figure out the segtree approach, went with sqrt-decomp instead (not exactly sqrt, bucket size was smaller since I had sort operations on the edges of the query)

Simple enough there as well, try to keep all the buckets sorted, so you can binary search on them if needed. But mostly there is just one constant value in each bucket which you can make a check against.

Final code with 100 points.

I did not expect it to pass since for each query I might have to sort twice. And in fact, it did not pass for sqrt sized bucket. I changed the bucket size to 100 and got the AC. I wonder if its possible to create cases that stop this approach. (sqrt-decomp part)

Definitely one of the most fun problems I have done.

3 Likes

This question is to the tester. I read your code but I cannot figure out that if you have added all the edges that are there whose length is less than maximum edge length that can be traversed then why you have also added the edges that are not less than the maximum length edges that are there in the query??


text:
The psycheval package is a set of functions that can be useful in psychological evaluations. It accompanies the Individual Psychometrics online textbook.

## Installation

You can install the development version of psycheval like so:

remotes::install_github("wjschne/psycheval")

# Convert a variable to standard scores

Suppose you have a scaled score of 12 (μ = 10, σ = 3) that you want to convert to a standard score (μ = 100, σ = 15).

library(psycheval)
x2standard(12, mu_x = 10, sigma_x = 3)
#> [1] 110

To convert to a z-score (μ = 0, σ = 1):

x2standard(12,
mu_x = 10, sigma_x = 3,
mu_new = 0, sigma_new = 1)
#> [1] 0.67

To convert to T-score (μ = 50, σ = 10):

x2standard(12,
mu_x = 10, sigma_x = 3,
mu_new = 50, sigma_new = 10)
#> [1] 57

# Covariance of weighted sums

Let’s create a covariance matrix of 5 variables that all correlate at .60.

# Create covariance matrix
Sigma <- matrix(0.6, nrow = 5, ncol = 5)
diag(Sigma) <- 1
Sigma
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]  1.0  0.6  0.6  0.6  0.6
#> [2,]  0.6  1.0  0.6  0.6  0.6
#> [3,]  0.6  0.6  1.0  0.6  0.6
#> [4,]  0.6  0.6  0.6  1.0  0.6
#> [5,]  0.6  0.6  0.6  0.6  1.0

Suppose we want to create a sum of the first 2 variables and a sum of the remaining three variables. What would the covariance matrix of the summary variables be?

First we create a weight matrix w with 5 rows and 2 columns consisting entirely of zeros. The first column represents the weight for the first variable. We put ones in the first two rows of the first column and ones in rows 3–5 in the second column.

# Create weight matrix
w <- matrix(0, nrow = 5, ncol = 2)
w[1:2,1] <- 1
w[3:5,2] <- 1
w
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    1    0
#> [3,]    0    1
#> [4,]    0    1
#> [5,]    0    1

The covariance matrix of the sums is:

# covariance matrix of weighted sums
composite_covariance(Sigma, w)
#>      [,1] [,2]
#> [1,]  3.2  3.6
#> [2,]  3.6  6.6

To include the original variables in the covariance matrix, append an identity matrix to w:

# A 5 by 5 idendity matrix
I <- diag(5)
I
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    1    0    0    0    0
#> [2,]    0    1    0    0    0
#> [3,]    0    0    1    0    0
#> [4,]    0    0    0    1    0
#> [5,]    0    0    0    0    1

# Prepend the identity matrix to the w matrix
w_expanded <- cbind(diag(5), w)
w_expanded
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,]    1    0    0    0    0    1    0
#> [2,]    0    1    0    0    0    1    0
#> [3,]    0    0    1    0    0    0    1
#> [4,]    0    0    0    1    0    0    1
#> [5,]    0    0    0    0    1    0    1

# Covariance matrix of original variables and the summary variables
BigSigma <- composite_covariance(Sigma, w_expanded)
BigSigma
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,]  1.0  0.6  0.6  0.6  0.6  1.6  1.8
#> [2,]  0.6  1.0  0.6  0.6  0.6  1.6  1.8
#> [3,]  0.6  0.6  1.0  0.6  0.6  1.2  2.2
#> [4,]  0.6  0.6  0.6  1.0  0.6  1.2  2.2
#> [5,]  0.6  0.6  0.6  0.6  1.0  1.2  2.2
#> [6,]  1.6  1.6  1.2  1.2  1.2  3.2  3.6
#> [7,]  1.8  1.8  2.2  2.2  2.2  3.6  6.6

# Convert to correlations with the cov2cor function
composite_covariance(Sigma, w_expanded, correlation = T)
#>        [,1]   [,2]   [,3]   [,4]   [,5]   [,6]   [,7]
#> [1,] 1.0000 0.6000 0.6000 0.6000 0.6000 0.8944 0.7006
#> [2,] 0.6000 1.0000 0.6000 0.6000 0.6000 0.8944 0.7006
#> [3,] 0.6000 0.6000 1.0000 0.6000 0.6000 0.6708 0.8563
#> [4,] 0.6000 0.6000 0.6000 1.0000 0.6000 0.6708 0.8563
#> [5,] 0.6000 0.6000 0.6000 0.6000 1.0000 0.6708 0.8563
#> [6,] 0.8944 0.8944 0.6708 0.6708 0.6708 1.0000 0.7833
#> [7,] 0.7006 0.7006 0.8563 0.8563 0.8563 0.7833 1.0000

# Compute a multivariate confidence interval conditioned on a set of observed scores

library(readr)
library(ggplot2)
library(dplyr)
#>
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#>
#>     filter, lag
#> The following objects are masked from 'package:base':
#>
#>     intersect, setdiff, setequal, union

# Observed scores
x_wisc <- c(
vci = 130,
vsi = 120,
fri = 123,
wmi = 116,
psi = 97)

# Reliability coefficients
rxx_wisc <- c(
vci = .92,
vsi = .92,
fri = .93,
wmi = .92,
psi = .88)

# Correlation matrix
R_wisc <- ("
index vci     vsi     fri     wmi     psi
vci   1.00    0.59    0.59    0.53    0.30
vsi   0.59    1.00    0.62    0.50    0.36
fri   0.59    0.62    1.00    0.53    0.31
wmi   0.53    0.50    0.53    1.00    0.36
psi   0.30    0.36    0.31    0.36    1.00") |>
.default = col_double(),
index = col_character())) |>
tibble::column_to_rownames("index") |>
as.matrix()
R_wisc
#>      vci  vsi  fri  wmi  psi
#> vci 1.00 0.59 0.59 0.53 0.30
#> vsi 0.59 1.00 0.62 0.50 0.36
#> fri 0.59 0.62 1.00 0.53 0.31
#> wmi 0.53 0.50 0.53 1.00 0.36
#> psi 0.30 0.36 0.31 0.36 1.00

d_ci <- multivariate_ci(
x = x_wisc,
rxx = rxx_wisc,
mu = rep(100, 5),
sigma = R_wisc * 225)

d_ci
#>     variable   x  rxx mu_univariate see_univariate mu_multivariate
#> vci      vci 130 0.92        127.60          4.069          127.88
#> vsi      vsi 120 0.92        118.40          4.069          119.70
#> fri      fri 123 0.93        121.39          3.827          122.33
#> wmi      wmi 116 0.92        114.72          4.069          115.95
#> psi      psi  97 0.88         97.36          4.874           98.86
#>     see_multivariate upper_univariate lower_univariate upper_multivariate
#> vci            3.912            135.6           119.62              135.5
#> vsi            3.897            126.4           110.42              127.3
#> fri            3.685            128.9           113.89              129.6
#> wmi            3.954            122.7           106.74              123.7
#> psi            4.803            106.9            87.81              108.3
#>     lower_multivariate
#> vci             120.21
#> vsi             112.07
#> fri             115.11
#> wmi             108.20
#> psi              89.44

# Relative Proficiency Index

When a typical same-age peer with an ability of W = 500 has a .90 probability of answering a question correctly, what is the probability that a person with ability of W = 520 will answer the question correctly?

rpi(x = 520, mu = 500)
#> 0.9878

Flipping the previous question, when a person with ability of W = 520 has a .90 probability of answering a question correctly, what is the probability that a typical same-age peer with an ability of W = 500 will answer the question correctly?

rpi(x = 520, mu = 500, reverse = TRUE)
#> 0.5

Criteria other than .9 proficiency are also possible. For example, when a typical same-age peer with an ability of W = 500 has a .25 probability of answering a question correctly, what is the probability that a person with ability of W = 520 will answer the question correctly?

rpi(x = 520, mu = 500, criterion = .25)
#> 0.75


text:
# Why is an advertisement for a closed beta site showing up?

I wasn't sure where this should be brought up, so here it is, probably something that a developer should look at.

I deleted the post from the community ads thread that I believe to be responsible for the ad appearing, so it should no longer appear.

• is there any way that Joe User can do that or is that strictly a moderator privilege? Nov 12 '13 at 23:19
• @Malachi You can vote down the ad. Only ads with (I think?) 6+ votes will be displayed on the sidebar.
– House Mod
Nov 12 '13 at 23:21
• I apologize, but where are these posted? link please. Nov 12 '13 at 23:23
• meta.gamedev.stackexchange.com/questions/983/…
– user1430
Nov 13 '13 at 0:35


text:
# Slope of secant line

• May 24th 2009, 06:19 PM
lisa1984wilson
Slope of secant line
Compute the slope of the secant line joining the points on the graph of f whose coordinates are x=1 and x=1.1
Let f(x)=x^3
• May 24th 2009, 06:35 PM
Shyam
Quote:

Originally Posted by lisa1984wilson
Compute the slope of the secant line joining the points on the graph of f whose coordinates are x=1 and x=1.1
Let f(x)=x^3

$f(1.1) = (1.1)^3=1.331$

$f(1)=1^3=1$

$slope= \frac{f(1.1)-f(1)}{1.1-1}=...........$

finish it.
• May 24th 2009, 06:39 PM
lisa1984wilson


text:
# How do I find the analytical solutions to Lotka Volterra equations?

The Lotka Volterra equations are used to calculate the rate of change of predator, $y$, and prey, $x$, populations over time ($t$),and are shown below:

$$\frac{dx}{dt}=ax-bxy$$

$$\frac{dy}{dt}=-cy+dxy$$

where $a$, $b$, $c$ and $d$ are parameters, and $x$ and $y$ are greater than zero.

I have been looking at online methods, used by mathematicians, however, researchers seem to be solving the parameters through computer softwares like Matlab and Maple. I could not find analytic i.e algebraic solutions to the parameters. My question is,can the parameters be determined algebraically? As a high school student, my school does not allow me to use high tech computer softwares like maple, and I have been to told find the parameters using algebra. However, online research and mathematical literature seems to suggest, that the only way to determine the parameters is through a computer analyses, and not with algebraic rigour.

It would be great, if I could get some sort of a direction, as to what I should do to figure out what the parameters are through algebra, if the population of the prey oscillates between 200 and 400 sheep and the predator population oscillates between 50 and 100.

• Are the equations written correctly? Did you mean $a x$?
– Moo
Jan 16 '18 at 17:35
• @Moo Hey, thanks! It was a mistake. I fixed it :) Jan 16 '18 at 17:41
• You can't find $x(t)$ and $y(t)$ explicitly, but you can find a constant of motion $V(x,y) = \text{constant}$. See here, for example: en.wikipedia.org/wiki/… Jan 16 '18 at 17:55
• @Hans Lundwark Thankyou! math.stackexchange.com/questions/1853612/… What do you think of the method presented here as an answer for the calculation of the parameters.? Jan 16 '18 at 18:03
• @HansLundmark I have already derived the linear function for the Lotka Volterra equations in terms of V(x,y)=constant; V (x,y)=constant. But the problem is still there, is there a method for calculating the parameters algebraically? Jan 16 '18 at 18:05


text:
The present investigation was aimed at establishing the distribution of${}^{237}{\rm Np}$ within the different structures of hepatocytes. Rats were contaminated experimentally by intravenous injection of${}^{237}{\rm Np}\ ({\rm V})$ and the subcellular structures of the liver were separated by ultracentrifugation. Twenty-four hours after contamination, the nuclear and cytosolic fractions bound 54 and 32%, respectively, of the total radionuclide. Purification of the nuclei followed by dissociation of the protein components in medium of increasing ionic strength showed a specific binding of neptunium to the structural proteins of the nuclear matrix.

This content is only available as a PDF.


text:
Home

#### The Fundamental Theorem of Calculus

Three Different Concepts
The Fundamental Theorem of Calculus (Part 2)
The Fundamental Theorem of Calculus (Part 1)
More FTC 1

#### The Indefinite Integral and the Net Change

Indefinite Integrals and Anti-derivatives
A Table of Common Anti-derivatives
The Net Change Theorem
The NCT and Public Policy

#### Substitution

Substitution for Indefinite Integrals
Examples to Try
Revised Table of Integrals
Substitution for Definite Integrals
Examples

#### Area Between Curves

Computation Using Integration
To Compute a Bulk Quantity
The Area Between Two Curves
Horizontal Slicing
Summary

#### Volumes

Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
More Practice

#### Integration by Parts

Integration by Parts
Examples
Integration by Parts with a definite integral
Going in Circles

#### Integrals of Trig Functions

Antiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (only even powers)
Product of Secants and Tangents
Other Cases

#### Trig Substitutions

How Trig Substitution Works
Summary of trig substitution options
Examples
Completing the Square

#### Partial Fractions

Introduction
Linear Factors
Improper Rational Functions and Long Division
Summary

#### Strategies of Integration

Substitution
Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

#### Improper Integrals

Type 1 - Improper Integrals with Infinite Intervals of Integration
Type 2 - Improper Integrals with Discontinuous Integrands
Comparison Tests for Convergence

#### Differential Equations

Introduction
Separable Equations
Mixing and Dilution

#### Models of Growth

Exponential Growth and Decay
Logistic Growth

#### Infinite Sequences

Examples of Infinite Sequences
Limit Laws for Sequences
Theorems for and Examples of Computing Limits of Sequences
Monotonic Covergence

#### Infinite Series

Introduction
Geometric Series
Limit Laws for Series
Test for Divergence and Other Theorems
Telescoping Sums and the FTC

#### Integral Test

The Integral Test
Estimates of Value of the Series

#### Comparison Tests

The Basic Comparison Test
The Limit Comparison Test

#### Convergence of Series with Negative Terms

Introduction, Alternating Series,and the AS Test
Absolute Convergence
Rearrangements

The Ratio Test
The Root Test
Examples

#### Strategies for testing Series

Strategy to Test Series and a Review of Tests
Examples, Part 1
Examples, Part 2

#### Power Series

Finding the Interval of Convergence
Power Series Centered at $x=a$

#### Representing Functions as Power Series

Functions as Power Series
Derivatives and Integrals of Power Series
Applications and Examples

#### Taylor and Maclaurin Series

The Formula for Taylor Series
Taylor Series for Common Functions
Adding, Multiplying, and Dividing Power Series
Miscellaneous Useful Facts

#### Applications of Taylor Polynomials

Taylor Polynomials
When Functions Are Equal to Their Taylor Series
When a Function Does Not Equal Its Taylor Series
Other Uses of Taylor Polynomials

#### Partial Derivatives

Visualizing Functions in 3 Dimensions
Definitions and Examples
An Example from DNA
Geometry of Partial Derivatives
Higher Order Derivatives
Differentials and Taylor Expansions

#### Multiple Integrals

Background
What is a Double Integral?
Volumes as Double Integrals

#### Iterated Integrals over Rectangles

How To Compute Iterated Integrals
Examples of Iterated Integrals
Cavalieri's Principle
Fubini's Theorem
Summary and an Important Example

#### Double Integrals over General Regions

Type I and Type II regions
Examples 1-4
Examples 5-7
Order of Integration

### How Trig Substitution Works

Regular $u$-substitution works by setting $u=g(x)$ for some function $g$.  We have to pick $u$ so that the integrand becomes a nice function of $u$ times $du$.  In the video, we explore what is called inverse substitution, which we use for trig substitution.  This is where we substitute something in for $x$, instead of creating a $u$ that is a function of $x$.

Trig substitution is appropriate when we have an integrand containing the sum or difference of the squares of a constant and a variable, i.e. one of the forms $x^2+a^2, a^2-x^2$, and $x^2-a^2$.  Examples integrals for which we would use trig substitution include those below.  Notice that regular substitution will not work with these integrals.  Also notice it is fine if the $x^2$ has a coefficient, and in the last example, we must complete the square in order to get one of our forms.  You should begin to recognize such integrals as trig substitution integrals.

$$\int\frac{\sqrt{9-x^2}}{x^2}\,dx,\qquad \int\frac{1}{x^2\sqrt{x^2+4}}\,dx\qquad\int\frac{dx}{(4x^2-25)^{3/2}},\quad\text{ and }\quad \int\frac{x^2}{(3+4x-4x^2)^{3/2}}\,dx$$In trig substitution, we let $x = g(\theta)$, where $g$ is a trig function, and then $dx = g'(\theta)\, d\theta$.  Since $x$ and $dx$ appear in the integrand, we can always rewrite the integrand in terms of $\theta$ and $d\theta$.  The question is whether the substitution helps us integrate.  Fortunately, we can teach you how to make good substitutions.  It is also non-trivial to convert everything back to $x$ at the end of the problem, or in the case of a definite integral, to change your limits of integration to be in terms of $\theta$.

The above three forms indicate the trig subsitutions we will use, and they are easy to remember since you know the derivatives of $\sin^{-1}x,\tan^{-1}x$, and (maybe) $\sec^{-1}x$.  These trig substitutions are derived in the following video, and summarized on the next page.


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# Pyrophosphoric Acid Formula

## Pyrophosphoric Acid

Last updated date: 21st Mar 2023
Total views: 191.4k
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Pyrophosphoric acid can be defined as a chemical compound as an anhydride of acrylic phosphorus acid. It is formed by the conjugation of the two molecules of phosphoric acid. The conjugate of this chemical compound is an acid. The chemical compound is naturally found as a colourless and odourless chemical.

The chemical nature of pyrophosphoric acid can be defined as a hygroscopic substance. The hygroscopic substance can be defined as a compound that has the ability to retain water through absorption or adsorption.

This article focuses on the chemical formula of pyrophosphoric acid, structural formula poof the compound, preparative methods, use, chemical and physical properties, and also some common hazards and risks associated with the handling of the compound.

### Chemical Formula of Pyrophosphoric Acid

Pyrophosphoric acid can be defined as a chemical compound with a conjugate base called pyrophosphoric acid and the conjugate acid of the compound is known as the diphosphate. The chemical compound is also known as diphosphoric acid. The chemical formula of pyrophosphoric acid can be represented as the following, H$_{4}$P$_{2}$O$_{7}$. The alternative method of representation of the pyrophosphoric acid formula is [(HO)$_{2}$P(O)]$_{2}$O. The chemical formula represents that the elements involved in the formation of the chemical compound include hydrogen, oxygen, phosphate. The compound is formed by condensation of the two molecules of phosphoric acid. The IUPAC name of the compound is phosphono dihydrogen phosphate

The structural formula of pyrophosphoric acid can be represented as mentioned below.

### Preparative Methods

There are two most widely used methods for commercial production of the chemical compound (pyrophosphoric acid). The first method of large scale commercial production of the compound includes the ion exchange method from sodium pyrophosphate. The second most commonly used method is the treatment of lead pyrophosphate with the hydrogen sulfide compound.

### Common Names of Pyrophosphoric Acid

The IUPAC name of the compound is phosphono dihydrogen phosphate. The other common name of the compound is as follows,

• Diphosphoric acid

• Pyrophosphoric acid

• Phosphonooxyphosphonic acid

• Diphosphoric acid.

### Physical Properties of the Chemical Compound

The physical property of any chemical compound can be defined as a property that can be quantified (measured or observed) without altering the chemical composition or chemical formula of the compound. The physical properties of a compound include conductivity, physical appearance, melting point, boiling point, density. Some of the physical properties of pyrophosphoric acid can be mentioned as follows,

1. The physical state of the chemical compound is defined as solid.

2. The melting point of pyrophosphoric acid is 71.5- degree Celcius

3. The pyrophosphoric acid is readily soluble in water.

4. Pyrophosphoric acid is also soluble in diethyl ether and ethyl alcohol.

5. They crystallize and form polymorphs.

### Chemical Properties of Pyrophosphoric Acid

The chemical properties of a compound can be defined as the poetry of the compound quantified or observed during a chemical reaction and the identity or chemical composition of the compound is changed during the chemical reaction. The chemical property of the compound includes reactivity with other chemicals, toxicity, coordination number, flammability, enthalpy of formation, the heat of combustion, oxidation states, and chemical stability. Some of the chemical properties of pyrophosphoric acid are mentioned below

1. The pyrophosphoric acid formula can be represented as  H$_{4}$P$_{2}$O$_{7}$

2. The alternative way of representation of the chemical formula of pyrophosphoric acid is [(HO)$_{2}$P(O)]$_{2}$O

3. The molecular weight of the compound is 177.973 g/mol

4. The conjugate base of the compound is phosphoric acid

5. The conjugate acid of the compound is diphosphate.

6. The hydrogen bond donor count is 4.

7. The hydrogen bond acceptor count is 7

8. The number of rotatable bond in the chemical compound is 2

9. There is 1  covalently bonded unit

10. The complexity of pyrophosphoric acid is 147.

11. The anions and esters of the compound are known as pyrophosphates.

### Risk and Hazards

Pyrophosphoric acid is a medium-strong acid, there is not much known about the industrial application of the compounds thus there is not a very reliable source of risk associated with it. Although the acid is corrosive in nature, they are not toxic in nature.

### Conclusion

Pyrophosphoric acid is a medium-strong acid that is formed by the condensation of two molecules of phosphoric acid. It polymorphs crystal, there is no vital industrial use of the compound. The pyrophosphoric acid formula is written as H$_{4}$P$_{2}$O$_{7}$. The chemical compound is also found as an intermediate metabolic product of E.coli.

## FAQs on Pyrophosphoric Acid Formula

Q.1) What are Pyrophosphates?

Ans: The salts, anion as well as an ester of the compound is known as pyrophosphate.

Q.2) What is the Melting Point of Pyrophosphoric Acid?

Ans: The melting point of the compound is 71.5 degrees Celcius.

Q.3) What Is the Conjugate Base of Pyrophosphoric Acid

Ans: The conjugate base of the compound is phosphoric acid.