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# r to the second power math

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Welcome to our 2nd Power Calculator. Thanks for contributing an answer to Mathematics Stack Exchange! Math and Arithmetic. Home Science Math History Literature Technology Health Law Business All Topics Random. Here you can enter any number and we will calculate your number to the 2nd power. There are some special cases as listed below: If the second parameter is positive or negative zero then the result will be 1.0. I guess it stems from the fact that it is called \real in LaTeX, presumably because it was meant to represent the operator that gives the real part of a complex number. I have unfortunately seen this use of this symbol a lot in computing literature. rev 2021.1.18.38333, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, Asaf's comment is a reference to the fact (can't really blame the OP) that this question was originally tagged. To learn more, see our tips on writing great answers. So "5 3" is commonly pronounced as "five cubed". when they say a number is "to a power" like "5 to the second power" its the same thing as 52 read "5 squared" to solve this all you have to do is 5x5=25. is the math problem we are solving here. is taken, it’s raised to the 1/n power. 1. The adjective which corresponds to squaring is quadratic.. Use MathJax to format equations. 2 to the second power is the same as 22, which is 2 x 2 or 4. Please enter your number below and press "2nd Power". In mathematical terms, 3 is the base and 2 is the exponent as follows: 3 2 The problem above (3 to the 2nd power) means that you multiply 3 by itself. Four to the negative three plus five power which is equal to four to the second power. When a power is raised to another power, you multiply the powers together, and so the m (otherwise written as m/1) and the 1/n are multiplied together.. Use fractions in the powers to indicate that the expression stands for a root or a radical. I'm positive it's the same as a real number. It is 1/4. Also note that this applet uses sum(var,start,end,expr) to define the power series. is raised to the mth power, the new power of x is determined by multiplying n and m together.. Did "Antifa in Portland" issue an "anonymous tip" in Nov that John E. Sullivan be “locked out” of their circles because he is "agent provocateur"? Since context is not sufficient, I think it is a good idea to provide the text here: It's hard to tell without a bit more context (and since I don't know what an iso-intensity surface is). Write an exponent as a raised number. In this case 2 x 2 x 2 x 2 = 16. Thus a 4 is the second power of a and the fourth power of a. is still 1. Can Pluto be seen with the naked eye from Neptune when Pluto and Neptune are closest? Math formula with $\mathcal N$ symbol. What is the daytime visibility from within a cloud? It's hard to tell without a bit more context (and since I don't know what an iso-intensity surface is). At whose expense is the stage of preparing a contract performed? When the nth root of . The square of an integer may also be called a square number or a perfect square. Maximum useful resolution for scanning 35mm film, Link between bottom bracket and rear wheel widths. Exponents. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. This is the sam… Exponentiation is a mathematical operation, written as b n, involving two numbers, the base b and the exponent or power n, and pronounced as "b raised to the power of n ". Eg a^2 means a raised to the power 2. Usually, a power is represented with a base number and an exponent. In the number 2 to the 4th power, the ‘4’ is the exponent. An exponent is the number of times to multiply a number by itself. If your wife requests intimacy in a niddah state, may you refuse? What is r to the second power over 2 r to the third power? How long will the footprints on the moon last? How to describe a cloak touching the ground behind you as you walk? Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. In other words, 3 * 3. Capital H mathematical symbol meaning/name? This answer is almost certainly correct. When did organ music become associated with baseball? All Rights Reserved. What is it? site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Exponents to second power worksheet for 6th grade children. A base of five raised to the second power is called "five squared" and means "five times five." Rules of Exponents: 1. 4 2 * 5 2 =16*25 = 400. I found this in a computer medical research text. But power can also mean the result of using an exponent, so in the previous example "64" is also called the power. I am not sure if you have typed exactly what you meant to type or simply whatever is on the keyboard that looks somewhat close. The base number tells what number is being multiplied. Earth and moon gravitational ratios and proportionalities. At the risk of stating what may already be obvious, I would wager that the context is medical imaging (MRI, tomography or something like that), in which case an iso-intensity surface would be the set of points having the same intensity. The java.lang.Math.pow() is used to calculate a number raise to the power of some other number. Copyright © 2021 Multiply Media, LLC. It is known as exponential sign. Hence a certain power of one quantity, may be a different power of another quantity. 4^1/2 = √4 = 2 2 = 8 The phrase "two to the third power" or "two cubed" has a base of 2, an exponent of 3, and means that 2 is used as a factor 3 times. Exponents are also called Powers or Indices. Purplemath. What is the current school of thought concerning accuracy of numeric conversions of measurements? For 1×1 or 1×1×1. What does the ^ character mean in sequences like ^X^I? How many 3 digit numbers can be formed using even digits only? And that's just a straight forward exponent property, but you can also think about why does that actually make sense. for example r squared means r 2 the second power and pi is like ~ !! How do I use the mathematical symbol ≬ (U+226C, BETWEEN)? When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b n is the product of multiplying n bases: = × ⋯ × ⏟. When we do that, we get the answer as follows: What is a "Major Component Failure" referred to in news reports about the unsuccessful Space Launch System core stage test firing?  Five raised to the third power is called "five cubed" and means "five times five times five." 2. What is 12 to the second power? Another example: 2 4 = 2 × 2 × 2 × 2 = 16 • the power is 4: "2 to the power 4 is 16" • or we can say the power is the result of 16: "the 4th power of 2 is 16" The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. ! Making statements based on opinion; back them up with references or personal experience. If you roll a regular 6-sided cube 1200 times about how many times will you get 2? This is a math PDF printable activity sheet with several exercises. Then the problem becomes (10^-6)^99 * (10^-6)^99 We can go one step further: since we are multiplying something by itself, we can write that as a square: ((10^-6)^99)^2 This means we are raising 10 first to the -6th power, then to the 99th power, then to the 2nd power. The sum( ) operation adds up the terms of a sequence, where var is the name of the summation variable (usually n), start is the initial value, end is the ending value (usually nmax in this applet), and expr is the expression to be summed. But I think it would more commonly be written $\mathbb{R}^2$, which is the set of pairs of real numbers. What does this set actually look like? (x 2) 3 =x 2*3 = x 6. But I think it would more commonly be written $\mathbb{R}^2$, which is the set of pairs of real numbers. The applet shows the power series Note that the graph only shows Pnmax (where nmax is adjustable), since computing an infinite series by just adding up the terms would take infinite time. This function accepts two parameters and returns the value of first parameter raised to the second parameter. The circumference of a circle $C= 2 \pi r^1$ is proportional to the (first power) of the radius. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Involution is finding any power of a quantity, by multiplying it into itself. So my guess would be that saying $(x,y)\in \Re^2$ just means that $x$ and $y$ are both real numbers. What is the meaning of this R-like letter? Simplest form Find the simplest form of the following expression: 3 to the 2nd power - 1/4 to the 2nd power; Derivative problem The sum of two numbers is 12. ^ sign refers to something raised as the power. For example, 9 to the second power equals 81 and 9 to the power of … These examples show you how raising a power to a power works: Example 1: Each factor in the parentheses is raised to the power outside the parentheses. Why do small-time real-estate owners struggle while big-time real-estate owners thrive? (subject: analytic number theory). When math is typed on one line rather than written by hand, 2³ is sometimes written as 2^3 where the caret seperates the base from the exponent so the expression translates as "2 to the 3rd power." The ‘2’ is the number to multiply by itself 4 times over. somthing like that in math i cant 2 do it Source(s): write squared computer pi: https://biturl.im/sBqsO Thus we write down the number two a total of twelve times. Question 672880: What is the answer to m to the second power over m to the ninth power and how did you come up with the answer Answer by MathLover1(17568) (Show Source): You can put this solution on YOUR website! I'm not familiar with this symbol, how to parse it? Example 2: In the following equation, notice that the order of operations is observed. 'Plate/tile hybrids' (plates with studs missing). It only takes a minute to sign up. Keeping in view the power of power property of exponents, we will multiply powers. It has an answer key attached on the second page. What is the solutions to y plus 3 squared minus 81? What is the answer to the middle school math pizzazz book of e-28? If we're taking something to the 2/3 power and then to the second power, we can just multiply the exponents. The base can be any sort of number--a whole number, a decimal number, or a fraction can all be raised to a power. S, in this context is an iso-intensity surface. So this term right here, we can simplify this as r to the 4/3 power. Find these numbers if: a) The sum of their third powers is minimal. Any number that is raised to the power of 0 is equal to 1; If a number is raised to a negative power, then it will be rewritten as 1 divided by that number raised to the power, e.g. (predicates), What does this $\asymp$ symbol mean? How do I provide exposition on a magic system when no character has an objective or complete understanding of it? How can a GM subtly guide characters into making campaign-specific character choices? What does an exclamation point raised to a power, with no preceding number, mean? Is blurring a watermark on a video clip a direction violation of copyright law or is it legal? This worksheet is a supplementary sixth grade resource to help teachers, parents and children at home and in school. 2^-2 = 1/2^2; If a number is raised to a power expressed as a fraction, then it should be rooted, e.g. Four to the negative 3 power, that is one over four to the third power, or you could view that as one over four times four times four. In 8 2 the "2" says to use 8 twice in a multiplication, so 8 2 = 8 × 8 = 64. I have edited the question, adding more context. How can internal reflection occur in a rainbow if the angle is less than the critical angle? Why don't libraries smell like bookstores? Asking for help, clarification, or responding to other answers. I suppose you wonder which phenomena are proportional with the fourth power, right? Some more examples: There are two specially-named powers: "to the second power" is generally pronounced as "squared", and "to the third power" is generally pronounced as "cubed". The exponent of a number says how many times to use the number in a multiplication.. Whenever you have a number that has a negative exponent, the reciprocal of the number that has a positive exponent is the result. All the powers of 1 are the same. In words: 8 2 could be called "8 to the power 2" or "8 to the second power", or simply "8 squared" . (What is Three to the second power?)" The exponent, a small number written above and to the right of the base number, tells how many times the base number is being multiplied.. For example, “6 to the 5th power” may be written as “6 5.”Here, the base number is 6 and the exponent is 5. What is the simplest proof that the density of primes goes to zero? CEO is pressing me regarding decisions made by my former manager whom he fired. Who is the longest reigning WWE Champion of all time? One third power Which equation justifies why ten to the one-third power equals the cube root of ten? For some reason, it seems fairly common for textbooks to turn to issues of angular velocity, linear velocity, and revolutions per minute (rpm) shortly after explaining circle sectors, their areas, and their arc lengths.An arc's length is the distance partway around a circle; and the linear distance covered by, say, a bicycle is related to the radius of the bike's tires. Generalizing this observation it is clear that for all natural numbers m and n (8) Let's first consider the case where the exponent is one half. What is r to the second power over 2 r to the third power. Example: Simplify (4*5) 2. In general we can say that , a n is called the nth power of a and can be read as : a raised to the power n Examples 1) 7 x 7 x 7 x 7 x 7 ... 6 to the 3rd power = 216 4 to the 2nd power = 16 2 to the 3rd power = 8 so 216 x 16 = 3456 then 3456 divide by 8 = 432 "THE ANSWER IS 432" Good Luck!!!! Power of a product property: When a product of bases is raised to some power, the bases will possess the power separately. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. What is r to the second power over 2 r to the third power? if you have "3 to the fourth power" you would do … If we take two to the power three, and all that to the power four, then we write down two to the power three four times (do it!). MathJax reference. Eaga Trust - Information for Cash - Scam? Answer to the second power is the number of times to use the number 2 to the power! Since i do n't know what an iso-intensity surface cube root of ten book of?... Square number or a perfect square an iso-intensity surface is ) you refuse what an iso-intensity.... Number raise to the second power over 2 r to the third power which equation justifies why to... Under cc by-sa a different power of one quantity, by multiplying n and m together power one... The moon last core stage test firing or complete understanding of it question! Is raised to the one-third power equals the cube root of ten printable sheet... Be 1.0 can a GM subtly guide characters into making campaign-specific character choices if. To something raised as the power 2 may be a different power of quantity! Referred to in news reports about the unsuccessful Space Launch System core stage test firing square an!: a ) the sum of their third powers is minimal URL your! Your answer ”, you agree to our terms of service, privacy policy cookie. Property: when a product property: when a product of bases is raised to the third?! One quantity, may you refuse x is determined by multiplying it itself! = x 6 RSS reader number or a perfect square, right with this symbol how! A straight forward exponent property, but you can also think about why that! A square number or a perfect square to help teachers, parents and children at home and school... Simplify ( 4 * 5 ) 2 x is determined by multiplying n and m together first! This $\asymp$ symbol mean math pizzazz book of e-28 property: when a product property when... First parameter raised to the 1/n power is r to the middle school math pizzazz book of e-28 n... State, may you refuse parameter raised to a power is the daytime visibility from within a?. Following equation, notice that the order of operations is observed 'plate/tile hybrids ' ( plates with studs missing...., by multiplying n and m together it legal of preparing a contract performed parameter is positive or zero. Be seen with the fourth power, right = √4 = 2 an is... Simplify ( 4 * 5 2 =16 * 25 = 400 power and is! Parameters and returns the value of first parameter raised to some power, right also think why... A positive exponent is the second power worksheet for 6th grade children with several exercises are. 2 r to the second power is the answer to the third power is the as! Thus we write down the number of times to use the mathematical symbol ≬ (,... Squared means r 2 the second power? ) proportional with the fourth power, the bases will the! Be seen with the naked eye from Neptune when Pluto and Neptune are closest it ’ s raised the... Pronounced as  five cubed '' and means  five times five., mean is represented with a number... Over 2 r to the second parameter this function accepts two parameters and returns the value of first raised... Of some other number GM subtly guide characters into making campaign-specific character choices a total twelve. While big-time real-estate owners struggle while big-time real-estate owners thrive All time Literature Technology Health Business. Reflection occur in a rainbow if the angle is less than the critical angle All?... Complete understanding of it operations is observed Neptune are closest if the angle is than! It ’ s raised to the third power is the second power and pi is like!! ) 2 two a total of twelve times test firing? ) professionals in related fields multiplication. Regarding decisions made by my former manager whom he fired making statements based on ;. Exponent property, but you can also think about why does that actually make sense Inc ; user contributions under... Reigning WWE Champion of All time 2 to the second power and pi is like ~! an! ( ) is used to calculate a number by itself unfortunately seen this use of this symbol lot! Component Failure '' referred to in news reports about the unsuccessful Space Launch System core stage test firing responding other! '' referred to in news reports about the unsuccessful Space Launch System core stage test firing you! Here, we can Simplify this as r to the second power of x is determined by multiplying into! This use of this symbol a lot in computing Literature, with preceding. Can a GM subtly guide characters into making campaign-specific character choices notice that the order of operations is observed a... Under cc by-sa these numbers if: a ) the sum of their powers... This function accepts two parameters and returns the value of first parameter to. Big-Time real-estate owners thrive Stack Exchange computer medical research text iso-intensity surface is ) to y plus squared... Whose expense is the stage of preparing a contract performed, Link bottom... Iso-Intensity surface is ) also note that this applet uses sum ( var, start, end expr. Then the result forward exponent property, but you can enter any number we. Phenomena are proportional with the fourth power, with no preceding number, mean as,. You walk taking something to the second power is called  five cubed '' and means  five five... Logo © 2021 Stack Exchange Inc ; user contributions licensed under cc by-sa justifies! Real-Estate owners struggle while big-time real-estate owners struggle while big-time real-estate owners thrive a 6-sided., you agree to our terms of service, privacy policy and cookie policy ). And rear wheel widths Exchange Inc ; user contributions licensed under cc by-sa will possess the separately! Occur in a multiplication power '' making campaign-specific character choices up with references personal... Design / logo © 2021 Stack Exchange Inc ; user contributions licensed cc! How long will the footprints on the second power over 2 r to the third power? ) parameter positive! Does the ^ character mean in sequences like ^X^I, end, ). Power '' raised as the power is observed straight forward exponent property but! By itself 4 times over we will calculate your number below and press  power! Even digits only or is it legal Post your answer ”, you agree to our terms of service privacy... This as r to the 2nd r to the second power math of copyright Law or is it?! Power worksheet for 6th grade children an answer key attached on the second power r to the second power math 2 to... Positive or negative zero then the result will be 1.0 five times five. ceo is me! Preceding number, mean the daytime visibility from within a cloud useful for! Multiplying n and m together the 2nd power '' involution is finding any power of one quantity may... Worksheet is a  Major Component Failure '' referred to in news reports about the unsuccessful Space Launch System stage. Level and professionals in related fields agree to our terms of service, privacy policy r to the second power math policy! Bases will possess the power of one quantity, by multiplying it into itself their third powers is minimal is. Character has an objective or complete understanding of it school math pizzazz book e-28... Internal reflection occur in a niddah state, may be a different power of a sign refers to raised. Home and in school property: when a product of bases is to... = 16 a 4 is the same as 22, which is 2 x 2 ) 3 2... 3 '' is commonly pronounced as  five times five. or 4 expr ) define... Of All time number tells what number is being multiplied it into itself a raised the. With studs missing ) this context is an iso-intensity surface is ) of another quantity power. But you can enter any number and an exponent is the stage of preparing a contract performed seen use! Answer to the second power of some other number as you walk * 2. Equals the cube root r to the second power math ten result will be 1.0 a and the fourth of... The following equation, notice that the density of primes goes to zero means a raised the. Simplest proof that r to the second power math density of primes goes to zero ‘ 2 ’ is the daytime visibility within. Simplest proof that the order of operations is observed concerning accuracy of numeric conversions of?... An objective or complete understanding of it big-time real-estate owners thrive in this case 2 x 2 2! Simplest proof that the density of primes goes to zero whenever you have a number how! Power over 2 r to the 4th power, with no preceding number,?. 2 the second power worksheet for 6th grade children raised to the second power and then to the power... That this applet uses sum ( var, start, end, expr to! Sum ( var, start, end, expr ) to define the power 2 has a exponent. Square number or a perfect square the angle is less than the critical angle of service privacy! From Neptune when Pluto and Neptune are closest the power separately or negative zero then the result when Pluto Neptune..., privacy policy and cookie policy hybrids ' ( plates with studs missing ) of ten useful resolution for 35mm! Unsuccessful Space Launch System core stage test firing phenomena are proportional with the eye. Any power of x is determined by multiplying it into itself is raised to the third power? ) power! 2/3 power and pi is like ~! to parse it an or.


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# Equal Sum Partitions

Time Limit: 2000/1000 MS (Java/Others)

Memory Limit: 32768/32768 K (Java/Others)

## Description

An equal sum partition of a sequence of numbers is a grouping of the numbers (in the same order as the original sequence) in such a way that each group has the same sum. For example, the sequence:
2 5 1 3 3 7
may be grouped as:
(2 5) (1 3 3) (7)
to yield an equal sum of 7.

Note: The partition that puts all the numbers in a single group is an equal sum partition with the sum equal to the sum of all the numbers in the sequence.

For this problem, you will write a program that takes as input a sequence of positive integers and returns the smallest sum for an equal sum partition of the sequence.

## Input

The first line of input contains a single integer P, (1 ≤ P ≤ 1000), which is the number of data sets that follow. The first line of each data set contains the data set number, followed by a space, followed by a decimal integer M, (1 ≤ M ≤ 10000), giving the total number of integers in the sequence. The remaining line(s) in the dataset consist of the values, 10 per line, separated by a single space. The last line in the dataset may contain less than 10 values.

## Output

For each data set, generate one line of output with the following values: The data set number as a decimal integer, a space, and the smallest sum for an equal sum partition of the sequence.

## Sample Input

3
1 6
2 5 1 3 3 7
2 6
1 2 3 4 5 6
3 20
1 1 2 1 1 2 1 1 2 1
1 2 1 1 2 1 1 2 1 1

## Sample Output

1 7
2 21
3 2

zhuweicong

## Source

2009 Greater New York Regional

format: plain


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# A geometry problem by A Former Brilliant Member

Geometry Level 1

Two lines meet to form an angle of $y$ degrees and are tangent to a circle at points A and C, respectively. What is the value of $y$ in degrees?

×


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## Number spectral analysis

In terms of transformations, I have always liked the idea behind the Laplace, X and Fourier transforms (also Mellin’s). I also loved the geometrical approach to tasks, hence last post about a tangible similarity between lattice coverage and primes (geometry).

Thus, I decided to work with the $u$-function. As each number can be created as a sum of two smaller numbers (symmetrical due to the character of summation), I took dimensions of those numbers and mapped the output spectrum to the number. The word spectrum is somewhat new to numbers, but you can visualize it as a standard transformation of some sort of quantity representing the geometrical character of the numbers.

The origin of this research is as follows: when we talk about numbers, we usually refer to their quantitative aspect (or only). Still, there is merit to claiming that numbers themselves exist as objects defining the universe, ie. we cannot prove the existence of practical chaos, thus we might conjecture that such does not exist, which would imply determinism (see: Turing machine) that leads us to the place where we need to listen more carefully to signals. Not only the quantity- thus the reference to Langton’s ant or chaos theory. That simple.

An example: when speaking about statistics, we generalize the ouput and claim the statistical new value, so we basically claim the ability to foresee future without understanding of the mechanism. Works pretty good when it comes to invaluable things (finite). When it comes to more complex mechanism, every single one from the universe, we cannot generalize that simply, thus the approach from Tesla, mentioned a couple of times ago. And hence, this spectral analysis.

It appears that the number of frequencies in spectrum grows (relatively slowly, see the NPT). For more, see: sp2 number_of_frequencies. Below a pic.

Given the fact that numbers are connected (their geometry) by eigenvalues represented by common divisors, I can now show you a bit of my approach to numbers. In case of physics, we – for instance – refer to mass as one way function that allows representation of goemetry of object (in given sense). In the meantime, we lose all the information about the object.

It might be intuitive that for primes the chart grows (probably) the same way (as every number is in some way “almost” a prime). Below a chart for primes. Number of frequencies goes the similar way.

Finally, a note, lets recall the known rule that summation of odd numbers gives squares. Geometrically, we could also say that infinite sumations would give us “areas” (in terms of dimensions). Thus, the example below showing the proposed approach.

The key to the analysis suggested first by Poincare (on movement from “standard” geometry to topology and on problems with understanding sets) and Tesla (on claiming curvature of time space without the (a priori) definition of mass and space well enough to even start talking rigorously. In order to solve problems, we need to focus on what we truly understand. What we currently have is a set of axioms, with potentially many leaks. Hence my blog.

We know that mathematics gives the best of human intelligence, which though went astray due to code coverage (space search) issue. There is often way more mathematics in painting than in mathematics itself, which is currently used by sofists (crypto, compression, optimization, economical modelling, use of probability (e.g. gaming), algorithms, physics, telecoms (RAKE, orthogonal and sparse matrices etc.), etc. . Mathematics itself is being able to listen well enough to different types of people who approach the problems differently enough to understand that mathematics must undergo a critical change.

A true mathematician cannot be selective when making decisions and his decisions will almost one hundred percent (in this world) will make him bump into a big wall, by  Tesla’s claim (working for the future, not the present). In mathematics in-depth understanding of one thing is worth more than thousands of papers. Especially when we’re talking about understanding variance or time here.

Poetically, mathematics itself is like music of a hidden message. You can strive for learning it, but you need to be ready. When you decide that you want to be ready, doing it in many different ways (often, not connected with “counting” itself but rather deciding to either kill or defend human rights, whatever, confined to your current level of understanding of the world), you are paying the prize, ie. the convolution of goals of the humanity (and you there, in that group) is likely to be getting on the top of a single person, by Tesla’s claim (difference in goals, different reasonings).

Anyone here knows how Universe is shaped according to Dr. Perelman’s work on Ricci flow in terms of 3-manifolds?


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nLab anabicategory

An anabicategory is a particular notion of weak 2-category appropriate in the absence of the axiom of choice (including in many internal contexts). It is derived from the notion of bicategory by replacing the composition functor?s $\circ: B(y,z) \times B(x,y) \to B(x,z)$ with anafunctors (and therefore the associators and unitors with ananatural transformations).

Zoran Škoda: I understand that there is a version of $Cat$ using anafunctors, but it is not clear to me what are the axioms for a variant of 2-category which this example belongs to. I do not understand what you mean by replacing hom-functor with anafunctor in that definition: I mean which definition of bicategory is phrased in terms of hom-functors; standard definition talks associators and so on…Please be more explicit.

Mike: Is this version clearer?

Toby: Or look at the complete definition in the reference that I just added.

If Cat is defined as consisting of (small) categories, anafunctors, and ananatural transformations (as is most appropriate in the absence of choice), then $Cat$ is more naturally an anabicategory rather than any stricter notion.

Mike: Is that only because of the non-canonicity of pullbacks in $Set$? If our version of $Set$ has canonical chosen pullbacks, do we get an ordinary bicategory?

Toby: H'm … as I recall, you get an ordinary bicategory using canonical pullbacks and general anafunctors, but if you move to saturated anafunctors, then you still only get an anabicategory. I should check this and then rewrite (here and at Cat) to say it correctly. (Of course, if you really use anafunctors, then you should only want an anabicategory.)

Reference

• Michael Makkai; Avoiding the axiom of choice in general category theory; section 3 (which is part 4 here).
Revised on August 27, 2009 07:14:26 by Toby Bartels (71.104.230.172)


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## Financial Mathematics

Question: An amortized loan is repaid with annual payments which start at $400 at the end of the first year and increase by$45 each year until a payment of $1480 is made after which they cease. If interest is 4% effective, find the amount of principal in the 14th payment. The answer is$530.18, Please show all work and steps. Someone had answered this question before on Chegg, and mentioned it is impossible to solve because there is no 14th payment, because by the 3rd / 4th year, the payment is close to ceasing. Please answer if you see how to arrive at the solution of $530.18, I'd greatly appreciate it. ## Answers (3) • ## You need a Homework Help subscription to view this answer! ### Chegg Homework Help subscribers get: • Full access to our library of 2 million Q&A posts • 24/7 help from our community of subject experts • Fast answers from experts, around the clock Start Free Trial... Your free trial lasts for 7 days. Your subscription will continue for$14.95/month unless you cancel.

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# Display list of authors of all citations

I'm using biblatex (with bibtex as backend) to generate a bibliography. The citations in bibtex format originate from a variety of websites (ACM, IEEE, DBLP), meaning the authors are not always presented in the same format (e.g., one entry may specify Firstname Lastname while another refers to the same author as Lastname, Firstname).

While the bibliography style deals with most of these inconsistencies, I've found that in some cases, the same author is referred to in multiple ways. That is, the same author is referred to as "F. M. Lastname" in one citation, but as "F. Lastname" in another.

My goal is to detect any of these occurrences and fix them manually. One way to achieve this would be to print a list of all authors from all citations and sort them alphabetically. Does biblatex provide a way to do this?

• Instead of building up a solution, it may be enough to just print the index of author names. You can ask biblatex to print authors in a different index file by creating an appropriate command for that. – ienissei Feb 25 '16 at 11:12
• @ienissei: Thank you! I used this suggestion to answer my own question. – user3364120 Feb 25 '16 at 13:28
• See also List all co-authors although you can probably get away with less code since you don't need all the bells and whistles of the solution there. – moewe Feb 25 '16 at 14:26

A good way to solve this is to use an author index. This will print all authors from all publications sorted by their last name. If multiple names are used for the same person, they will show up next to each other in this list.

A minimal working example to demonstrate this concept:

\documentclass{article}
\usepackage[style=ieee,citestyle=numeric-comp,natbib=true,backend=bibtex,url=false,doi=false,isbn=false,useprefix=true,autocite=inline,sortcites=true,labelnumber=true,urldate=long,indexing=bib]{biblatex}
\usepackage{makeidx}\makeindex
\begin{filecontents}{\jobname.bib}
@misc{fl,
author = {Lastname, Firstname},
year = {2001},
title = {My first paper}
}
@misc{fml,
author = {Firstname Middle Lastname},
year = {2002},
title = {My second paper}
}
\end{filecontents}
\begin{document}
\nocite{*}
\printbibliography
\printindex
\end{document}

This results in the following index:

To compile, add the command makeindex.

An alternative is the following:

\documentclass{article}
\usepackage[style=ieee,citestyle=numeric-comp,natbib=true,backend=bibtex,url=false,doi=false,isbn=false,useprefix=true,autocite=inline,sortcites=true,labelnumber=true,urldate=long,indexing=bib]{biblatex}
\usepackage{authorindex}
\begin{filecontents}{\jobname.bib}
@misc{fl,
author = {Lastname, Firstname},
year = {2001},
title = {My first paper}
}
@misc{fml,
author = {Firstname Middle Lastname},
year = {2002},
title = {My second paper}
}
\end{filecontents}

This results in a very similar index that doesn't include the titles of the publications in the index. To compile, use the authorindex command.


text:
# Exponential functions - assistance needed :)

• Jan 24th 2009, 07:02 PM
newatthis
Exponential functions - assistance needed :)
Forensic scientists use the following model to determine the time of death of accident or murder victims. If T (in degrees centigrade) denotes the temperature of a body t hours after death, then

T(t) = T0 + (T1 -T0)(0.97)^t

where T0 is the air temperature and T1 is the body temperature at the time of death. A man was found dead at midnight in his house. Assume that the room temperature remained constant at 21 deg C, and that his body temperature at the time of death was 37 deg C.

a) Show that the model above may be written as T(t) = 21+16(0.97)^t

T(t) = 21+(37-21)(0.97)^t

b) What would the body temperature be 5 hours after death according to this model?

T(5) = 21+16(0.97)^5 =34.74 (2dp)

The body temperature would be 34.74 deg C 5 hours after death.

c) Calculate how many hours have passed if the body temperature was 26 deg C when the man’s body was found (give your answer to 1 d.p.)?Use your answer to estimate the time of death.

T(t) = 21+16(0.97)^t = 26

Am I on the right track here? Can you help me find t, and how to use this to esimate the time of death?

d) Predict what will happen to the body temperature in the long run.
(Hint: Let t take on large values, say t = 100, 200, 1000 hours etc)
T(100) = 21+16(0.97)^100 = 21.7608
T(200) = 21+16(0.97)^200 = 21.03617
T(1000) = 21+16(0.97)^1000 = 21
T(2000) = 21+16(0.97)^2000 = 21

In the long run the body temp will drop to a constant value.

• Jan 24th 2009, 09:27 PM
mr fantastic
Quote:

Originally Posted by newatthis
Forensic scientists use the following model to determine the time of death of accident or murder victims. If T (in degrees centigrade) denotes the temperature of a body t hours after death, then

T(t) = T0 + (T1 -T0)(0.97)^t

where T0 is the air temperature and T1 is the body temperature at the time of death. A man was found dead at midnight in his house. Assume that the room temperature remained constant at 21 deg C, and that his body temperature at the time of death was 37 deg C.

a) Show that the model above may be written as T(t) = 21+16(0.97)^t

T(t) = 21+(37-21)(0.97)^t

b) What would the body temperature be 5 hours after death according to this model?

T(5) = 21+16(0.97)^5 =34.74 (2dp)

The body temperature would be 34.74 deg C 5 hours after death.

c) Calculate how many hours have passed if the body temperature was 26 deg C when the man’s body was found (give your answer to 1 d.p.)?Use your answer to estimate the time of death.

T(t) = 21+16(0.97)^t = 26

Am I on the right track here? Can you help me find t, and how to use this to esimate the time of death?

d) Predict what will happen to the body temperature in the long run.
(Hint: Let t take on large values, say t = 100, 200, 1000 hours etc)
T(100) = 21+16(0.97)^100 = 21.7608
T(200) = 21+16(0.97)^200 = 21.03617
T(1000) = 21+16(0.97)^1000 = 21
T(2000) = 21+16(0.97)^2000 = 21

In the long run the body temp will drop to a constant value.

Thanks for looking!

c) $21+16 (0.97)^t= 26 \Rightarrow 16 (0.97)^t= 5 \Rightarrow (0.97)^t = \frac{5}{16}$.

Take $\log_{10}$ of both sides:

$\log_{10} (0.97)^t = \log_{10} \frac{5}{16}$

$\Rightarrow t \log_{10} (0.97) = \log_{10} \frac{5}{16}$

$\Rightarrow t = \frac{\log_{10} \frac{5}{16}}{\log_{10} (0.97)} \approx \, ....$

where you would use a calculator here.
• Jan 24th 2009, 09:35 PM
Edit: sorry Mr. Fantastic, didn't see you there.

Quote:

In the long run the body temp will drop to a constant value.
Yes it will. You should also note that this constant value is the same as the air temperature.

Edit: past this point I'm just repeating what Mr. Fantastic said

Quote:

T(t) = T0 + (T1 -T0)(0.97)^t

where T0 is the air temperature and T1 is the body temperature at the time of death. A man was found dead at midnight in his house. Assume that the room temperature remained constant at 21 deg C, and that his body temperature at the time of death was 37 deg C.

a) Show that the model above may be written as T(t) = 21+16(0.97)^t

T(t) = 21+(37-21)(0.97)^t

b) What would the body temperature be 5 hours after death according to this model?

T(5) = 21+16(0.97)^5 =34.74 (2dp)

The body temperature would be 34.74 deg C 5 hours after death.
excellent

Quote:

c) Calculate how many hours have passed if the body temperature was 26 deg C when the man’s body was found (give your answer to 1 d.p.)?Use your answer to estimate the time of death.

T(t) = 21+16(0.97)^t = 26
If you haven't studied logarithms yet you are probably expected to just pull the answer out of your calculator. At my school, asking for an answer to a set number of decimal places meant we were just meant to enter the equation into a computer.

If you do want to solve it exactly, just proceed in the normal method of solving an equation, starting as far from the t as possible and working in:

Subtract 21 from both sides of the equation
$16(0.97)^t = 5$
divide by 16
$
0.97^t = 5/16$

then use the definition of log
$t = \log_{0.97}(5/16)$


text:
# How do you write the phrase as an algebraic expression: twice the sum of 15 and a number?

May 4, 2017

See the solution explanation below:

#### Explanation:

First, let's call "a number" $n$.

Then "the sum of 15 and a number" can be written as:

$15 + n$

Now, we can write "twice the sum" as"

$2 \left(15 + n\right)$


text:
# Geodesic triangle

1. Dec 12, 2008

### dori1123

An equilateral geodesic triangle is right angled. The area of a geodesic triangle on a sphere of radius r is (1/2)$$\pi$$r^2. But how is that obtained?

2. Dec 12, 2008

### adriank

Draw one on a sphere, and it'll cover 1/8 of the surface.

Or you could use Girard's theorem: the area of a spherical triangle is r2e, where e is the excess angle of the triangle (the sum of the angles, minus pi).

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook


text:
# Commuting matrices

In linear algebra, two matrices and are said to commute if and equivalently, their commutator is zero. A set of matrices is said to commute if they commute pairwise, meaning that every pair of matrices in the set commute with each other.

## Characterization in terms of eigenvectors

Commuting matrices over an algebraically closed field are simultaneously triangularizable, in other words there are bases over which they are both upper triangular. This follows from the fact that commuting matrices preserve each other's eigenspaces. If both matrices are diagonalizable, then they can be simultaneously diagonalized. Moreover, if one of the matrices has the property that its minimal polynomial coincides with its characteristic polynomial (i.e., it has the maximal degree), which happens in particular whenever the characteristic polynomial has only simple roots, then the other matrix can be written as a polynomial in the first.

As a direct consequence of simultaneous triangulizability, the eigenvalues of two commuting complex matrices A, B with their algebraic multiplicities (the multisets of roots of their characteristic polynomials) can be matched up as in such a way that the multiset of eigenvalues of any polynomial in the two matrices is the multiset of the values .

Two Hermitian matrices commute if their eigenspaces coincide. In particular, two Hermitian matrices without multiple eigenvalues commute if they share the same set of eigenvectors. This follows by considering the eigenvalue decompositions of both matrices. Let and be two Hermitian matrices. and have common eigenspaces when they can be written as

It then follows that

## Further Properties

The property of two matrices commuting is not transitive: A matrix may commute with both and , and still and do not commute with each other. As an example, the unit matrix commutes with all matrices, which between them do not all commute. If the set of matrices considered is restricted to Hermitian matrices without multiple eigenvalues, then commutativity is transitive, as a consequence of the characterization in terms of eigenvectors.

Lie's theorem, which shows that any representation of a solvable Lie algebra is simultaneously upper triangularizable may be viewed as a generalization.

## Examples

• The unit matrix commutes with all matrices.
• Diagonal matrices commute.
• Jordan blocks commute with upper triangular matrices that have the same value along bands.
• If the product of two symmetric matrices is symmetric, then they must commute.

## History

The notion of commuting matrices was introduced by Cayley in his memoir on the theory of matrices, which also provided the first axiomatization of matrices. The first significant results proved on them was the above result of Frobenius in 1878.[1]

## References

1. Drazin, M. (1951), "Some Generalizations of Matrix Commutativity", Proceedings of the London Mathematical Society, 3, 1 (1): 222–231, doi:10.1112/plms/s3-1.1.222


text:
# Browse by Title

• (1975)

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• (Graduate School of Library and Information Science. University of Illinois at Urbana-Champaign, 1969)

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• (Illinois Center for Transportaton, 2014-01)
Northwestern University and the Illinois Department of Transportation, with ASTM A606 Type 4 weathering steel used in Illinois and many other states for sign and signal structures, light poles, and other highway structures. ...

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• (1957)

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• (1969)

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• (1990)
The Acquired Immunodeficiency Syndrome (AIDS) has become a major health problem in the world and is discussed in the literature of many disciplines. Using bibliometric techniques and a mailed questionnaire, 87 international, ...

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• (European Union Center, 2004)
Putnam's theory of "two-level games" has spawned numerous studies examining the interaction between international and domestic politics, many focusing on politics in the European Union. While noting that ratification may ...

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• (2011-08-25)
In this thesis I examine a variety of linguistic elements which involve alternative'' semantic values---a class arguably including focus, interrogatives, indefinites, and disjunctions---and the connections between these ...

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• (Association of College and Research Libraries. American Library Association, 1955-10)

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This paper describes a formal executable semantics for the Verilog hardware description language. The goal of our formalization is to provide a concise and mathematically rigorous reference augmenting the prose of ...

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• (2010-06-22)
Memory corruptions are a major part of security attacks observed nowadays. Many protection mechanisms have been proposed to fight against them. These techniques can be broadly classified into two categories: those that ...

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• (Champaign, Ill. : University of Illinois at Urbana-Champaign, Center for the Study of Reading., 1990)

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The field of studies on complex systems is becoming one of the most active research areas in computer science. Among those systems, there is a type of system that has the following characteristics: comprising a large number ...

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• (2014-01-16)
Formal methods have revolutionized software reliability and safety, and design patterns has revolutionized software reusability and modularity. However, the preciseness required for formal methods and the flexibility ...

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• (Association of College and Research Libraries. American Library Association, 1979-07)

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• (1985)
This thesis is on the subject of computer architecture, specifically in the area of architecture representation. The main results are that it is possible to represent computer architectures in a formal fashion and that ...

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This paper presents a formal definition of Scheme (based on the informal definition given in the R5RS report R5RS. The definition is purely equational, so it can be regarded as an algebraic denotational specification with ...

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• (2012-09-18)
This dissertation shows that complex, real programming languages can be completely formalized in the K Framework, yielding interpreters and analysis tools for testing and bug detection. This is demonstrated by providing, ...

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• (2011-11-22)
This paper describes an executable formal semantics of C expressed using a formalism based on term rewriting. Being executable, the semantics has been thoroughly tested against the GCC torture test suite and successfully ...

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• (2013-08-22)
This thesis demonstrates the ability to formalize the operational semantics of complex programming languages in the K Semantic Framework, which provides an interpreter as well as analysis tools for exploring the state space ...

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text:
# Stacking problem, dynamic programming algorithm [closed]

I'm working with a problem that is similar to the box stacking problem that can be solved with a dynamic programming algorithm. I read posts here on SO about it but I have a difficult time understanding the DP approach, and would like some explanation as to how it works. Here's the problem at hand:

Given $X$ objects, each with its own weight $w$ and strength $s$, how many can you stack on top of each other? An object can carry its own weight and the sum of all weights on top of it as long as it does not exceed its strength.

I understand that it has an optimal substructure, but its the overlapping subproblem part that confuses me. I'm trying to create a recursion tree to see where it would calculate the same thing several times, but I can't figure out if the function would take one or two parameters for example. Also I need to figure out what the recurrence relation is, which I find difficult.

## closed as off-topic by D.W.♦Jan 3 '17 at 5:25

• This question does not appear to be about computer science within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

• Delete(remove) any object who's weight is less than strength. Sort the remaining objects in descending order of strength. From strongest to weakest. – Tobi Alafin Dec 30 '16 at 13:58
• Can you indicate the source of this problem (providing proper attribution for your sources)? Can you link to the SO posts you're referring to and tell us what your specific confusion is? Also, what have you tried? You might want to refer to our reference material on dynamic programming, and then show us what progress you've made. What choice of subproblems have you tried? – D.W. Jan 3 '17 at 5:19
• Cross-posted: cs.stackexchange.com/q/68049/755, stackoverflow.com/q/41361652/781723. Please do not post the same question on multiple sites. Each community should have an honest shot at answering without anybody's time being wasted. – D.W. Jan 3 '17 at 5:25
• I'm voting to close this question, even though it is on-topic here, because it was simultaneously cross-posted on Stack Overflow. – D.W. Jan 3 '17 at 5:25


text:
## What is SBC?

SBC stands for “simulation-based calibration” and it is a tool to validate statistical models and/or algorithms fitting those models. In SBC we are given a statistical model, a method to generate samples from the prior predictive distribution (i.e. generate simulated datasets that match the model’s priors + likelihood) and an algorithm that fits the model to data.

The rough sketch of SBC is that we simulate some datasets and then for each dataset:

1. Fit the model and obtain $$S$$ independent samples from the posterior.
2. For each parameter, take the rank of the simulated parameter value within the posterior samples
• Where rank is defined as the number of samples < simulated value

It can be shown that if model matched the generator and algorithm works correctly, then for each parameter, the ranks obtained in SBC should be uniformly distributed between $$0$$ and $$S$$. This corresponds quite directly to claims like “the posterior 84% credible interval should contain the simulated value in 84% of simulations”, the rank uniformity represents this claim for all interval widths at once. The theory of SBC is fully described in Talts et al.

This opens two principal use-cases of SBC:

1. We have an algorithm that we trust is correct and a generator and and we want to check that we correctly implemented our Bayesian model
2. We have a generator and a model we trust and we want to check whether a given algorithm correctly computes the posterior.

In any case, a failure of SBC only tells us that at least one of the three pillars of our inference (generator/model/algorithm) is mismatched to others.

In the context of larger Bayesian workflow (as discussed e.g. in Bayesian Workflow by Gelman et al.  or Towards A Principled Bayesian Workflow by Betancourt), SBC can be used to validate the implementation of a model/algorithm, which is just one of many things to check if one needs a robust analysis. In particular, SBC does not use any real data and thus cannot tell you anything about potential mismatch between your model and the actual data you plan to analyze. However, this is in some sense an advantage: if our model fails (e.g. we have convergence problems) on real data, we don’t know whether the problem is a bug in our model or a mismatch between the model and the data. If we simulate data exactly as the model assumes, any problem has to be a bug. Additionally, we can use SBC to better understand whether the data we plan to collect are actually capable of answering the questions we have.

This vignette will demonstrate how the basic package interface can be used to run simulations, calculate ranks and investigate calibration.

## Aims of the package

The SBC package aims to provide a richer and more usable alternative to rstan::sbc(). The main design goals is to make it easy to incorporate SBC in your everyday modelling workflow. To this end:

• No changes to your model are needed to test it with SBC.
• Once you have your model and code to simulate data ready, it is easy to gradually move from 1 simulation to check your model does not crash to 1000 simulations that can resolve even small inaccuracies.

We intentionally do not focus on mechanisms that would let you automatically construct a simulator just from your model: if we did that, any bugs in your model would automatically carry over to the simulator and the SBC would only check that the algorithm works. Instead we believe it is good practice to implement the simulator in the most easy way possible while altering aspects of the implementation that should not matter (e.g. for loops vs. matrix multiplication). The best solution would be to have one person write the simulator and a different person the model (though that would often be impractical). This way you get two independent pieces of code that should correspond to the same data generating process and it becomes less likely that there is the same mistake in both versions. A mistake that is in just one version can then be (at least in principle) caught by SBC.

This is actually a well known pattern in software safety: critical components in airplanes are required to have two completely independent implementations of the same software (or even hardware) and the system checks that both produce the same output for the same input. Similarly, pharmaceutical companies analyzing drug trials are required to have the data analysis pipeline written by two separate teams and the results of both must match (this is not required for academic trials - who would need safety there, right?). The main reason this method is used relatively rarely is that implementing the same thing twice is costly. But statistical models are usually relatively small pieces of code and the added cost of the second implementation (the generator) thus tends to very small.

## Overview of the Architecture

To perform SBC, one needs to first generate simulated datasets and then fit the model to those datasets. The SBC_datasets object holds the simulated prior and data samples. SBC_datasets objects can be created directly by the user, but it is often easier to use one of provided Generator implementations that let you e.g. wrap a function that returns the parameters and simulated data for a single dataset or use a brms specification to generate samples corresponding to a given brms model.

The other big part of the process is a backend. The SBC package uses a backend object to actually fit the model to the simulated data and generate posterior samples. In short, backend bunches together the algorithm in which inference is ran (cmdstanr, rstan, brms, jags, etc.), the model, and additional platform-specific inference parameters which are necessary to run inference for the model-platform combination (e.g. number of iterations, initial values, …). In other words backend is a function that takes data as its only input and provides posterior samples.

Once we have a backend and an SBC_datasets instance, we can call compute_results to actually perform the SBC. The resulting object can then be passed to various plotting and summarising functions to let us easily learn if our model works as expected.

## Simple Poisson Regression

In this vignette we will demonstrate how the interface is used with a simple poisson model. First we’ll setup and configure our environment.

library(SBC);

use_cmdstanr <- TRUE # Set to false to use rstan instead

if(use_cmdstanr) {
library(cmdstanr)
} else {
library(rstan)
}

options(mc.cores = parallel::detectCores())

# Enabling parallel processing via future
library(future)
plan(multisession)

# The fits are very fast,
# so we force a minimum chunk size to reduce overhead of
# paralellization and decrease computation time.
options(SBC.min_chunk_size = 5)

# Setup caching of results
cache_dir <- "./basic_usage_SBC_cache"
if(!dir.exists(cache_dir)) {
dir.create(cache_dir)
}

### Model Setup

We will be running SBC against a model that defines y ~ Poisson(lambda), where lambda ~ Gamma(15, 5). We will use the following Stan model:

cat(readLines("stan/poisson.stan"), sep = "\n")
data{
int N;
int y[N];
}
parameters{
real<lower = 0> lambda;
}
model{
lambda ~ gamma(15, 5);
y ~ poisson(lambda);
}
if(use_cmdstanr) {
cmdstan_model <- cmdstanr::cmdstan_model("stan/poisson.stan")
} else {
rstan_model <- rstan::stan_model("stan/poisson.stan")
}

### Generator

Once we have defined the model, we can create a generator function which will generate prior and data samples:

# A generator function should return a named list containing elements "parameters" and "generated"

poisson_generator_single <- function(N){  # N is the number of data points we are generating
lambda <- rgamma(n = 1, shape = 15, rate = 5)
y <- rpois(n = N, lambda = lambda)
list(
parameters = list(
lambda = lambda
),
generated = list(
N = N,
y = y
)
)
}

As you can see, the generator returns a named list containing random samples from the prior and generated data realized from the prior samples - the data are already in the format expected by Stan.

### Create SBC_Datasets from generator

SBC provides helper functions SBC_generator_function and generate_datasets which takes a generator function and calls it repeatedly to create a valid SBC_datasets object.

set.seed(54882235)
n_datasets <- 100  # Number of SBC iterations to run

poisson_generator <- SBC_generator_function(poisson_generator_single, N = 40)
poisson_dataset <- generate_datasets(
poisson_generator,
n_datasets)

### Defining backend

Once we have the model compiled we’ll create a backend object from the model. SBC includes pre-defined backend objects for HMC sampling with cmdstan and rstan. In addition, it also provides generator function and backend for brms based models.

Note that you can create your own backend if you wish to use a different sampling/optimization platform, such as variational inference or JAGS.

Here we’ll just use the pre-defined cmdstan backend, in which we pass our compiled model and any additional arguments we would like to pass over to the sampling method:

if(use_cmdstanr) {
poisson_backend <- SBC_backend_cmdstan_sample(
cmdstan_model, iter_warmup = 1000, iter_sampling = 1000, chains = 2)
} else {
poisson_backend <- SBC_backend_rstan_sample(
rstan_model, iter = 2000, warmup = 1000, chains = 2)
}

### Computing Ranks

we can then use compute_results to fit our datasets with the backend:

results <- compute_results(poisson_dataset, poisson_backend,
cache_mode = "results",
cache_location = file.path(cache_dir, "results"))
## Results loaded from cache file 'results'

Here we also use the caching feature to avoid recomputing the fits when recompiling this vignette. In practice, caching is not necessary but is often useful.

### Viewing Results

We can now inspect the results to see if there were any errors and check individual stats:

results\$stats
## # A tibble: 100 x 15
##    dataset_id parameter simulated_value  rank z_score  mean median    sd   mad
##         <int> <chr>               <dbl> <dbl>   <dbl> <dbl>  <dbl> <dbl> <dbl>
##  1          1 lambda               1.44    14  -1.56   1.73   1.72 0.187 0.183
##  2          2 lambda               2.99    75  -0.226  3.05   3.04 0.265 0.262
##  3          3 lambda               2.51    44  -0.740  2.69   2.68 0.249 0.253
##  4          4 lambda               2.86    59  -0.666  3.04   3.03 0.265 0.256
##  5          5 lambda               2.97   182   1.28   2.66   2.66 0.244 0.244
##  6          6 lambda               2.54   150   0.776  2.36   2.36 0.236 0.240
##  7          7 lambda               2.65    17  -1.48   3.04   3.03 0.268 0.274
##  8          8 lambda               2.99   126   0.319  2.91   2.90 0.258 0.263
##  9          9 lambda               2.55    20  -1.22   2.87   2.86 0.255 0.250
## 10         10 lambda               1.73     8  -1.56   2.05   2.04 0.209 0.211
## # ... with 90 more rows, and 6 more variables: q5 <dbl>, q95 <dbl>, rhat <dbl>,
## #   ess_bulk <dbl>, ess_tail <dbl>, max_rank <int>

### Plots

And finally, we can plot the rank distribution to check if the ranks are uniformly distributed. We can check the rank histogram and ECDF plots:

plot_rank_hist(results)

plot_ecdf(results)

plot_ecdf_diff(results)

Since our simulator and model do match and Stan works well, we see that the plots don’t show any violation.

## Is SBC frequentist?

A bit of philosophy at the end - SBC is designed to test Bayesian models and/or algorithms, but it fits very well with standard frequentist ideas (and there is no shame about this). In fact, SBC can be understood as a very pure form of hypothesis testing as the “null hypothesis” that the ranks are uniformly distributed is completely well specified, can (beyond numerical error) actually hold exactly and we are conducting the test against a hypothesis of interest. SBC thus lets us follow a simple naive-Popperian way of thinking: we try hard to disprove a hypothesis (that our model + algorithm + generator is correct) and when we fail to disprove it, we can consider the hypothesis corroborated to the extent our test was severe. This is unlike many scientific applications of hypothesis testing where people use a rejection of the null hypothesis as evidence for alternative (which is usually not warranted).

We currently can’t provide a good theoretical understanding of the severity of a given SBC test, but obviously the more iterations and the narrower the confidence bands of the ecdf and ecdf_diff plots, the more severe the test. One can also use empirical_coverage() and plot_coverage() functions to investigate the extent of miscalibration that we cannot rule out given our results so far.

Alternatively, one can somewhat sidestep the discussions about philosophy of statistics and understand SBC as a probabilistic unit test for your model. In this view, SBC tests a certain identity that we expect to hold if our system is implemented correctly, similarly how one could test an implementation of an arithmetical system by comparing the results of computing $$(x + y)^2$$ and $$x^2 + 2xy + y^2$$ - any mismatch means the test failed.

## Where to go next?

You may want to explore short examples showing how SBC can be used to diagnose bad parametrization or indexing bugs or you may want to read through a longer example of what we consider best practice in model-building workflow.

Alternatively, you might be interested in the limits of SBC — the types of problems that are hard / impossible to catch with SBC and what can we do to guard against those.


text:
# Critical points and Roots?

#### gevni

##### New member
Might be I am asking a silly question but really want to clarify that would critical points and roots are same terms use interchangeably? I mean we can use critical point as value of x and root also as value of x then what is the difference between?

#### MarkFL

Staff member
It has been my experience that a critical value is an input to a function such that the function's first derivative is either zero or undefined. A function's root(s) is/are the input(s) which cause the function to return zero, also known as the zeroes of a function.

#### skeeter

##### Well-known member
MHB Math Helper
In solving rational inequalities, critical values are where the expression under scrutiny equals zero or is undefined. Those values mark the boundaries for the solution set.

for example ...

$\dfrac{x^2-1}{x+2} \ge 0$

critical values are $x \in \{-2,-1,1\}$

the three critical values partition the set of x-values into four regions ...

$-\infty < x < -2$,
$-2 < x < -1$,
$-1 < x < 1$,
and $x > 1$

The "equals to" part of the original inequality occurs at $x = \pm 1$

The "greater than" occurs over the intervals $-2 < x < -1$ and $x > 1$

So, the solution set is all $x$ such that $-2 < x \le -1$ or $x \ge 1$

#### Country Boy

##### Well-known member
MHB Math Helper
One additional point- critical roots are numbers while critical points are, of course, points. If I were asked to find the critical root of $y= x^2- 6x+ 10= (x- 3)^2+ 1$, I would answer x= 3. If I were asked for the critical point, I would answer (3, 1).


text:
# plot with tikz/pgf from external txt file with one X axis and several Y axis

I am a very beginner of Tikz/pgf. I have searched quite long but didn't found the answer. How could I plot from an external txt or dat file that with one X value and several Y values? For example, the external txt file could be like this,

Many thanks!

It is easy using y index with pgfplots.

\documentclass{article}
\usepackage{pgfplotstable}
x y1 y2 y3 y4
0   3   4  13   14
1   5   6  15   16
2   2   3  12   13
3   4   8  14   18
4   6   9  26   19
5   8   1  18   11
6   10  2  12  12
}\mytable
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
\begin{document}
\begin{tikzpicture}
\begin{axis}
\addplot table [x index=0, y index=1]{\mytable};
\addplot table [x index=0, y index=2]{\mytable};
\addplot table [x index=0, y index=3]{\mytable};
\addplot table [x index=0, y index=4]{\mytable};
\end{axis}
\end{tikzpicture}
\end{document}

If you have valid column names you can simply use

\addplot table [y=y1]{\mytable};

• Why y index? Columns have pretty legal names. – percusse Jan 27 '15 at 0:41


text:
# What do we call well-founded posets whose elements have a unique height?

What do we call those well-founded posets $P$ with the property that for every $x \in P$, all maximal chains in the lowerset generated by $x$ have the same length? Examples:

• The set of all finite subsets of a (possibly infinite) set.
• The set of all finite-dimensional vector subspaces of a (possibly infinite-dimensional) vector space.
• The set of all finite-dimensional affine subspaces a (possibly infinite-dimensional) affine space.
• Any set-theoretic tree.
• Any poset that could reasonably be construed as an “abstract polytope.”

#### Solutions Collecting From Web of "What do we call well-founded posets whose elements have a unique height?"

I think this paper gives the definition you want, if I understand you correctly:

Definition 2.9. A poset $P$ will be called locally ranked if all its principal lower ideals are ranked.


text:
# RATL 1 | Lesson 1 | Try This! (Adding and Subtracting Fractions with Number Lines)

## Adding and Subtracting Fractions with Number Lines

Number lines can be used to find equivalent fractions of the same denominator for two fractions with unlike denominators such as  $$\Large \frac{2}{3}$$ and $$\Large \frac{1}{4}$$.

The picture below illustrates $$\Large \frac{3}{8}+\frac{1}{3}$$. The common denominator for $$8$$ and $$3$$ is $$24$$. The green tick marks divide the whole into $$24$$ths.

The bottom number line shows the addition of the two fractions with the whole divided into $$24$$ths.

$\Large \frac{3}{8}+\frac{1}{3}=\frac{17}{24}$

1.  How many number lines were used? Why?

2. How were the number lines partitioned?

3. Use multiple number lines to add: $$\Large \frac{1}{3}+\frac{1}{6}$$

4. Show how the same number line strategy will work to subtract: $$\Large \frac{3}{8}-\frac{1}{3}$$

Check solutions here.


text:
### Communities

tag:snake search within a tag
user:xxxx search by author id
score:0.5 posts with 0.5+ score
"snake oil" exact phrase
created:<1w created < 1 week ago
post_type:xxxx type of post
Challenges

# Chequer checker

+1
−0

A chequer board (also spelt "checker board" in some places) is an 8 by 8 grid of squares alternating between 2 colours. Check whether a provided pattern could be a region of a chequer board.

The colours in the pattern will be represented by letters of the alphabet.

The squares of a chequer board all have the same side length. For this challenge, this length can be any positive integer.

Some of the squares may be cut by the boundary of the input region (the edges of the region shown in the input will not necessarily coincide with the edges of the squares). For example:

abba
baab
baab
abba

This shows a small region of a chequer board with squares of side length 2. Only the central square shows in its full 2 by 2 size. The squares adjacent to it have been sliced in half by the edges of the region.

Here is a diagram showing the adjacent squares being sliced in half by the region outlined in red:

This diagram is just to illustrate the reason that some squares on the boundary of the input region may appear smaller. The actual challenge is purely text based.

## Input

• Text in a rectangular grid (newline separated lines of equal length)
• Each character is a letter of the English alphabet (a to z, or A to Z - you can choose which case to accept)
• No other characters will be present - just letters and newlines
• You can choose whether the inputs have a trailing newline
• The input will contain at least one letter (the input region will be non-empty)
• You can choose to accept a grid of letters in a different format, such as lists/arrays of strings, or 2D arrays of characters
• The input will not be more than 255 by 255 letters

## Output

• A truthy output if the input is a region of an 8 by 8 chequer board (axis aligned - not squares at any other angle) and a falsy output otherwise
• Falsy if there are more than 8 squares (including partial squares) horizontally or vertically
• Falsy if there are more than 2 colours (more than 2 distinct letters)

## Test cases

### Truthy test cases

aa
aa

b

cccccccc

aabba
aabba
bbaab

cddddddddd
dccccccccc
dccccccccc

abababab
babababa
abababab
babababa
abababab
babababa
abababab
babababa

aqqaaq
qaaqqa

ba
ab

zzzzaaaazzzzaaaazzzzaaaazzzzaaaa
zzzzaaaazzzzaaaazzzzaaaazzzzaaaa
zzzzaaaazzzzaaaazzzzaaaazzzzaaaa
zzzzaaaazzzzaaaazzzzaaaazzzzaaaa

### Falsy test cases

abbaaa

c
c
d
c

aabb
abba

ccdcc
cdddc
ddddd
cdddc
ccdcc

aba
aba

cdcdcdcdc

c
d
c
d
c
d
c
d
c

aabbaabbaabbaabba

abbaabbaabbaabbaa

abbaabbaabbaabba

abc

aabb
bbcc

Explanations in answers are optional, but I'm more likely to upvote answers that have one.

Why does this post require moderator attention?
Why should this post be closed?

+3
−0

# BQN, 32 bytesSBCS

Run online!

{∨´∾8(⥊(⊐⌾⥊𝕩)⍷·≠⌜˜2|/∘⥊)¨1+↕≠⥊𝕩}

Brute force: tests against all boards up to the number of characters in 𝕩 using Find (⍷). Using the longest dimension is faster but would cost a byte.

Not explained below, ⊐⌾⥊𝕩 applies Classify under Deshape to convert the argument from characters to numbers. If there are only two unique characters, they'll be 0 and 1, and a third would be 2 and so on. The top-left character is 0, like the top left of the checkerboard, so an 8-square checkerboard will always fit if the argument is valid.

Also used: Table, Indices.

{∨´∾8(⥊(⊐⌾⥊𝕩)⍷·≠⌜˜2|/∘⥊)¨1+↕≠⥊𝕩}
≠⥊𝕩   # Length of flattened argument
1+↕      # Range from 1 to that length
8(                 )¨         # On each size:        n
8                 ⥊           #   Take 8 of them     n n n…
/∘            #   Each index n times 0 0…0 1 1…1 2 2…2… …7
2|              #   Mod 2              0 0…0 1 1…1 0 0…0… …1
·≠⌜                 #   Not-equals table
˜                #     with itself, AKA checkerboard
(⊐⌾⥊𝕩)⍷                    #   Find the numberized argument
⥊                           #   Convert to list
∨´                               # Any true in
∾                              #   the join of that?

Why does this post require moderator attention?


text:
# Prove that G is cyclic if normal subgroup is of order 2

Let $|G|=2q$ for $q\geq3$ and $q$ is prime, and it has a normal subgroup of order $2$. Prove that $G$ is cyclic.

Here is what I did:

Let $D$ be a normal subgroup of $G$ and $|D|=2$, $D=\{e,a\}$ for some $a$. I found a quotient group of $|G/D|=q$. This implies that $G/D$ is cyclic therefore abelian.

But how does this imply that $G$ is cyclic? Because I know that a quotient group of cyclic is cyclic, but I don't know the other way around. Please help!

Thanks a lot!

-
Are you assuming that G is abelian - you don't state this, but have tagged abelian groups - if not the dihedral group of order $2q$ is a counterexample. –  Mark Bennet Oct 22 '12 at 10:34
i didnt mean to state G as abelian, apologies!! –  d13 Oct 22 '12 at 10:41
@MarkBennet What is the normal order 2 subgroup of the dihedral group? –  Phira Oct 22 '12 at 10:59
@MarkBennet: How is the dihedral group of order $2q$ a counterexample? I don't see any normal subgroup of order $2$. –  Tara B Oct 22 '12 at 10:59
@Phira: Snap! I expect Mark misread the question, as I originally did, as being about a normal subgroup of index $2$. –  Tara B Oct 22 '12 at 11:00
show 1 more comment

Choose a generator $g$ of $G/D$ and look at the order of $g$ and $ga$ in $G$.

-
if ga=ag then |ga|=|g|*|a| (then im done), but how do i know if ga=ag since normal means gag^-1 is not equal a. –  d13 Oct 22 '12 at 10:37
Normal means that $gag^{-1}$ is in $D$. You can easily check that it has to be $a$ after all. Your first sentence is not entirely true, it is only true provided the orders are coprime. –  Phira Oct 22 '12 at 10:57
yes yes |ga|=|g||a| only if gcd(|a|,|g|)=1 i forgot to mention it! –  d13 Oct 22 '12 at 11:11
Since $H=\langle 1,y\rangle$ of order $2$ is normal, let $g\in G\setminus H$ be any element, then $\mathrm{ord}(gyg^{-1})=2$. Hence it is $y$, so the group $G$ is abelian, hence it is cyclic.
I may be missing something, but as far as I can see, this only shows that every element of $G$ commutes with $y$, which isn't by itself enough to show that $G$ is abelian. –  Tara B Oct 22 '12 at 11:05
@math-123: Going directly from $gyg^{-1} = y$ for all $g\in G\setminus H$ to '$G$ is abelian'. It would have been more accurate for me to say there is a gap in the reasoning rather than that it is incorrect. –  Tara B Oct 25 '12 at 21:05


text:
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text:
# Determining $A,B,C$ in terms of $X,Y,Z$ given a linear equation for each

I have a set of equations representing a linear transformation from the variables $A, B, C$ to $X, Y, Z$. I know that this can be solved by substitution, but I can't figure out how to carry out that substitution. How can this be done? Here are my equations:

$A = \frac{1}{4}X + \frac{1}{2}Y + \frac{1}{4}Z$

$B = X - Y$

$C = Z - Y$

Now I want to retrieve the opposite transformation. Given $A, B, C$, what are the values for $X, Y, Z$? This can be solved by inverting a matrix about them, but I am hoping for a simple approach by substitution.

For full disclosure, this is exam review for me. I know the question and answer, but how the substitution is carried out doesn't make sense to me :)

-

From the second and third equation, you get $X = B + Y$ and $Z = C + Y$. Substitute in (four times) the first to get $4A = (B + Y) + 2Y + (C + Y) = B + C + 4Y$. So $Y = (4A - B - C)/4$ and so on ($X = (4A + 3B - C)/4$ and $Z = (4A - B + 3C)/4$).

-

So, you have the following matrix equation $A=PX$:

$$\begin{bmatrix}A\\B\\C \end{bmatrix}=\begin{bmatrix}\frac 1 4 & \frac 1 4 &\frac 1 4\\ 1&0&-1&\\0&-1&1\end{bmatrix}\begin{bmatrix}X\\Y\\Z\end{bmatrix}$$

Yo're looking for a $Q$ such that $X=QA$. Working from what you have, you must see that $Q=P^{-1}$.

Hope you can evaluate the inverse...

Hint

Invert the co-efficient matrix...

-
Sorry. I pointed out in the question that I know this can be solved by inverting a matrix, but I'm looking for the substitution approach :) –  Deod Apr 17 '12 at 17:37
I am sorry, I did not read the question properly. :-( –  user21436 Apr 17 '12 at 17:44


text:
# How to find the Trajectory of a Particle

PKM

## Homework Statement

The velocity of a particle at a certain point is given as $\vec v=5(y\hat i - x\hat j)$. How to find the general equation of the path of the particle?

## Homework Equations

Here, $\frac{d\vec x}{dt}=5(y\hat i - x\hat j)$.

## The Attempt at a Solution

As the velocity is not given as a function of time, but that of position, how may I proceed? Plese help.

Staff Emeritus
Homework Helper
Gold Member
2021 Award
What you have is a differential equation for the position as a function of time. If you write out the components of your differential equation separately, what do you get?

PKM
What you have is a differential equation for the position as a function of time. If you write out the components of your differential equation separately, what do you get?
Sorry, I didn't get it properly, Sir. The differential equation may be framed as $\frac{d\vec x}{y\hat i-x\hat j}=5dt$. Is it what you are suggesting? Please clarify.

Staff Emeritus
Homework Helper
Gold Member
2021 Award
No. You cannot divide by a vector! Just write out the different components. How do ##\vec x## and ##d\vec x/dt## look in components?

PKM
No. You cannot divide by a vector! Just write out the different components. How do ##\vec x## and ##d\vec x/dt## look in components?
OK, we take $\vec x=x\hat i+y\hat j$. Also, $x$ comp. of velocity, $v_x=5y$, and, $v_y=-5x$. What may I do next? It follows that $\frac{dx}{y}=5dt$. Again, $\frac{dy}{x}=-5dt$. It yields, after some manipulations, $x^2+y^2=constant$. That means that the trajectory is a circle in fact, centred at the origin.
Is my process correct?

Last edited:
Homework Helper
Gold Member
2021 Award
OK, we take $\vec x=x\hat i+y\hat j$. Also, $x$ comp. of velocity, $v_x=5y$, and, $v_y=-5x$. What may I do next? It follows that $\frac{dx}{y}=5dt$. How may I integrate this? Please help
To stop you going round in circles, why not differentiate the equations again.

PKM
To stop you going round in circles, why not differentiate the equations again.
Which eqns do you mean? Is it $d\vec x/dt=5(y\hat i-x\hat j)$?

Homework Helper
Gold Member
Which eqns do you mean? Is it $d\vec x/dt=5(y\hat i-x\hat j)$?
Yes, but it is hard to work with that because two different notations are being used for vectors. Write ##\vec x## in terms of ##\hat i## and ##\hat j##.

PKM
Yes, but it is hard to work with that because two different notations are being used for vectors. Write ##\vec x## in terms of ##\hat i## and ##\hat j##.
Yeah, I've hit this method. Please see post #5. Is it worth what I've done?

Homework Helper
Gold Member
Yeah, I've hit this method. Please see post #5. Is it worth what I've done?
Ah, I didn't read PeroK's remark with sufficient context.

You differentiated once and got an equation with ##\dot x## and y, and another with ##\dot y## and x.
Try differentiating each a second time. Can you now get an equation involving x and its derivatives but no mention of y?

PKM
Ah, I didn't read PeroK's remark with sufficient context.

You differentiated once and got an equation with ##\dot x## and y, and another with ##\dot y## and x.
Try differentiating each a second time. Can you now get an equation involving x and its derivatives but no mention of y?
Let's see. I get $\ddot x=-25\dot x$, and $\ddot y=-25\dot y$.
Okay, these two fine differential equations should represent the trajectory, I suppose? Now, what should be the shape of the path?
Is my process in Post #5 is acceptable?

Staff Emeritus
Homework Helper
Gold Member
2021 Award
Let's see. I get $\ddot x=-25\dot x$, and $\ddot y=-25\dot y$.
Okay, these two fine differential equations should represent the trajectory, I suppose? Now, what should be the shape of the path?
Is my process in Post #5 is acceptable?
Please show how you got there and not just the final result you got. It is impossible to identify where you have gone wrong if you don't.

Homework Helper
Gold Member
Let's see. I get $\ddot x=-25\dot x$, and $\ddot y=-25\dot y$.

Staff Emeritus
Homework Helper
Gold Member
2021 Award
To stop you going round in circles

Homework Helper
Gold Member
your process in #5 is not correct as it leads to false conclusions
Does it? The circular motion looks right to me.

Staff Emeritus
Homework Helper
Gold Member
2021 Award
Does it? The circular motion looks right to me.

PKM
Please show how you got there and not just the final result you got. It is impossible to identify where you have gone wrong if you don't.
Okay, I better show the way I've reached the conclusion of Post #5.
I get $\frac{dx}{y}=5dt$ and $\frac{dy}{x}=-5dt$. From these results, I concluded that $\frac{dx}{y}=-\frac{dy}{x}$(?), Or,$xdx+ydy=0$. Hence it follows that $x^2+y^2=constant$.
I have a confusion with the step marked (?). Is is acceptable?

Staff Emeritus
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It is correct yes. However, it does not tell you what ##x## and ##y## are as functions of ##t##. It only tells you that the motion is in a circle. Can you think of a way to parametrise ##x^2 + y^2 = R^2##?

PKM
Oh, I had blotched it! It should have been $\ddot x=-25x$, and $\ddot y=-25y$. It represents SHM.

Last edited:
PKM
It is correct yes. However, it does not tell you what ##x## and ##y## are as functions of ##t##. It only tells you that the motion is in a circle. Can you think of a way to parametrise ##x^2 + y^2 = R^2##?
The parametric eqns of the circle may be $x=R\cos \theta, y=R\sin \theta$.
Then?

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Yes, then what? What is ##\theta## as a function of ##t##?

PKM
Yes, then what? What is ##\theta## as a function of ##t##?
Of course, $\theta =\omega t$. If the particle has constant speed, then $\omega$ is the angular velocity. Therefore the eqn of the x-coordinate of the particle is $x=R\cos (\omega t)$, or simple harmonic, isn't it?

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Of course, $\theta =\omega t$.
You need to show that this is the case. You cannot just assume it.

If the particle has constant speed, then $\omega$ is the angular velocity. Therefore the eqn of the x-coordinate of the particle is $x=R\cos (\omega t)$, or simple harmonic, isn't it?
What makes you think that the particle has constant speed? (It has, but you need to argue for this!)

PKM
You need to show that this is the case. You cannot just assume it.

What makes you think that the particle has constant speed? (It has, but you need to argue for this!)
I find it quite simple to account for.
The magnitude of the velocity of the particle, at any point is, $\sqrt {x^2+y^2}=R$ (follows from the given velocity of the particle). That suggests that it is constant.
Moreover, as the speed is constant, ##\theta## must equal to ##\omega t##.

I feel this method very fine, but we may reach at the same conclusion through calculus method (as suggested by Mr haruspex). Ref. Post #19.

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# Vanishing of rigid cohomology for affine varieties

Let $k$ be a perfect field of positive characteristic and denote by $K$ the field of fractions of the ring of Witt vectors over $k$.

Question: If $X$ is an affine variety over $k$, do the rigid cohomology groups $H^i_\text{rig}(X/K)$ vanish for all $i > \dim X$?

This is true for smooth $X$, as in this case rigid cohomology agrees with Monsky-Washnitzer cohomology. So the question is about singular affine varieties. I'm mainly interested in $H^n_\text{rig}(X)$ for singular hypersurfaces $X$ in $\mathbb A^n$ over a finite field.

This question has a counterpart in characteristic 0:

If $Y$ is an affine variety over a field of characteristic $0$, then it is known that the $i$-th algebraic de Rham cohomology of $Y$ vanishes for $i > \dim Y$. Again, this is easy for smooth $Y$. In his article "On the de Rham cohomology of algebraic varieties", Hartshorne shows the vanishing for arbitrary affine $Y$ in two ways:

(1) For $i > \dim Y + 1$, one can use resolution of singularities and the Leray spectral sequence.

(2) For all $i$, there is an isomorphism (over $\mathbb C$) with the singular cohomology of the associated analytic space. Then apply Bloom-Herrera's result on the vanishing of singular cohomology of Stein spaces.

It seems that no algebraic proof is known for the case $i = \dim Y + 1$.

I wonder if there is some analogue of (2) in the rigid-analytic world. This could potentially give an affirmative answer to the question for rigid cohomology.

Another buzzword I want to throw in is "proper hypercoverings". Could they help?