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science1
You use a proof by contradiction. Basically, you suppose that $\sqrt{2}$ can be written as $p/q$. Then you know that $2q^2 = p^2$. As squares of integers, both $q^2$ and $p^2$ have an even number of factors of two. $2q^2$ has an odd number of factors of 2, which means it can't be equal to $p^2$.
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science2
Suppose no one ever taught you the names for ordinary numbers. Then suppose that you and I agreed that we would trade one bushel of corn for each of my sheep. But there's a problem, we don't know how to count the bushels or the sheep! So what do we do? We form a "bijection" between the two sets. That's just fancy langu...
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science3
The basic concept is thus: A 'countable' infinity is one where you can give each item in the set an integer and 'count' them (even though there are an infinite number of them) An 'uncountable' infinity defies this. You cannot assign an integer to each item in the set because you will miss items. The key to seeing this ...
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science5
Well, I am not sure where you want to embed the graphs, but Wolfram Alpha is pretty handy for graphing. It has most of the features of Mathematica, can handle 3D functions, and fancy scaling and such. I highly recommend it.
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science8
The closed form calculation for Fibonacci sequences is known as Binet's Formula.
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science9
You can use Binet's formula, described at http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html (see also Wikipedia for a proof: http://en.wikipedia.org/wiki/Binet_formula#Closed_form_expression )
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science10
Given (by long division): $\frac{1}{3} = 0.\bar{3}$ Multiply by 3: $3\times \left( \frac{1}{3} \right) = \left( 0.\bar{3} \right) \times 3$ Therefore: $\frac{3}{3} = 0.\bar{9}$ QED.
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science12
Hilbert's Hotel is a classic demonstration.
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science13
Lots of people like to use Instacalc which lets you do unit conversions and store intermediate calculations in variables.
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science14
Natural numbers The "counting" numbers. (That is, all integers, that are one or greater). Whole numbers The Natural numbers, and zero. Integers The Whole numbers, and the negatives of the Natural numbers. Rational numbers Any number that may be expressed by any integer A divided by any integer B, where B is not zero. I...
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science16
You can visualise it by thinking about it in infinitesimals. The more $9's$ you have on the end of $0.999$, the closer you get to $1$. When you add an infinite number of $9's$ to the decimal expansion, you are infinitely close to $1$ (or an infinitesimal distance away). And this isn't a rigorous proof, just an aid to v...
0.182492
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science17
What I really don't like about all the above answers, is the underlying assumption that $1/3=0.3333\ldots$ How do you know that? It seems to me like assuming the something which is already known. A proof I really like is: $$\begin{align} 0.9999\ldots × 10 &= 9.9999\ldots\\ 0.9999\ldots × (9+1) &= 9.9999\ldots\\ \text{b...
0.175452
0
science18
Google's calculator is very powerful: http://www.googleguide.com/help/calculator.html and your use history will be stored in your browser history.
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science19
You might as well start at the 'lowest': Integers are all the whole numbers, Natural numbers (N) are the set of positive integers, {1, 2, ...} (0 optional), Rational numbers (Q) are any number that can be represented a/b with a and b being Integers (|b| < 0. Real numbers are all the Rational numbers and all the others....
0.531828
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science20
What does it mean when you refer to $.99999\ldots$? Symbols don't mean anything in particular until you've defined what you mean by them. In this case the definition is that you are taking the limit of $.9$, $.99$, $.999$, $.9999$, etc. What does it mean to say that limit is $1$? Well, it means that no matter how small...
0.634401
0
science25
Suppose this was not the case, i.e. $0.9999... \neq 1$. Then $0.9999... < 1$ (I hope we agree on that). But between two distinct real numbers, there's always another one in between, say $x=\frac{0.9999... +1}{2}$, hence $0.9999... < x < 1$. The decimal representation of $x$ must have a digit somewhere that is not $9$ (...
0.322959
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science26
I've heard of it being as a rough check to see if accounting numbers were being made up
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science27
Assuming this is relative to the origin (as John pointed out): Given two position vectors $\vec p_1$ and $\vec p_2$, their dot product is: $$\vec p_1\cdot \vec p_2 = |\vec p_1| \cdot |\vec p_2| \cdot \cos \theta$$ Solving for $\theta$, we get: $$\theta = \arccos\left(\frac{\vec p_1 \cdot \vec p_2}{|\vec p_1| \cdot |\ve...
0.228263
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science28
There are several important kinds of relations, each of which satisfy a different collection of properties: Equivalence relations: These are reflexive, symmetric, and transitive. Essentially they're relations that "behave like equality." The most important elementary one is "equivalence modulo m," where say 1 = 6 = 11 ...
0.293714
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science30
You can see that there are infinitely many natural numbers 1, 2, 3, ..., and infinitely many real numbers, such as 0, 1, pi, etc. But are these two infinities the same? Well, suppose you have two sets of objects, e.g. people and horses, and you want to know if the number of objects in one set is the same as in the othe...
0.092105
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science31
Forensic accountancy is a popular use, and is actually admissible as evidence in the USA.
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science32
Here is a super nice powerpoint on the subject! http://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf
0.430863
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science33
An inner product space is a vector space for which the inner product is defined. The inner product is also known as the 'dot product' for 2D or 3D Euclidean space. An arbitrary number of inner products can be defined according to three rules, though most are a lot less intuitive/practical than the Euclidean (dot) produ...
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science34
Asymmetric means simply "not symmetric". So in the binary case, it is NOT the case that if a is related to b, b is related to a. Antisymmetric means that if a is related to b, and b is related to a, a = b. To explain your third example: "is older than" is asymmetric because if Alice is older than Bob, Bob is NOT older ...
0.42583
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science35
I believe they are used in quantum physics as well, because rotation with quaternions models Spinors extremely well (due to the lovely property that you need to rotate a point in quaternionic space around 2 full revolutions to get back to your 'original', which is exactly what happens with spin-1/2 particles). They are...
0.312261
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science36
The natural numbers can be defined by Peano's Axioms (sometimes called the Peano Postulates): Zero is a number. If n is a number, the successor of n is a number. zero is not the successor of a number. Two numbers of which the successors are equal are themselves equal. (induction axiom.) If a set S of numbers contains z...
0.426351
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science39
The dot product can be used to find the length of a vector or the angle between two vectors. The cross product is used to find a vector which is perpendicular to the plane spanned by two vectors.
0.501837
1
science41
As for the utility of inner product spaces: They're vector spaces where notions like the length of a vector and the angle between two vectors are available. In this way, they generalize $\mathbb R^n$ but preserve some of its additional structure that comes on top of it being a vector space. Familiar friends like Cauchy...
0.115618
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science42
\begin{align} x &= 0.999... \\ 10x &= 9.999... \\ &= 9 + 0.999... \\ &= 9 + x \\ 10x - x &= (9 + x) - x \\ (10 - 1)x &= 9 + (x - x) \\ 9x &= 9 \\ x &= 1 \end{align}
0.317285
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science43
Graph theory! It's essentially connecting the dots, but with theorems working wonders behind the scenes for when they're old enough. Simple exercises like asking how many colors you need to color the faces or vertices of a graph are often fun (so I hear). (Also, most people won't believe the 4-color theorem.)
0.414826
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science45
You're right to think that the definitions are very similar. The main difference between groups and rings is that rings have two binary operations (usually called addition and multiplication) instead of just one binary operation. If you forget about multiplication, then a ring becomes a group with respect to addition (...
0.250455
1
science46
Two good general references: Wikipedia MathWorld
0.483034
99
science48
Khan Academy, http://www.khanacademy.org/ You'll find tons of explanatory videos from various branches of mathematics; plus, each subject is explained pretty good, and the videos are easy to follow
0.519485
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science50
Not always pure math, but I think John Baez' This Week in Mathematical Physics contains a lot of really interesting math reads. I should add Terry Tao's What's new. It's a very active math blog (both in posts and comments) and definitely covers some cutting edge math, even if it can be way over my head.
0.120629
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science53
This really depend on how smart the kid is. I lean toward discrete math, elementary number theory related topics when talking to non-math people about math. They requires little background knowledge. There are some fun problems in discrete math, especially combinatorics. Simple probability is also nice. So are logic pr...
0.545068
3
science54
One place they are frequently used is in computer games when you want to smoothly transition from one rotation to another. An artist might have said "at this time I want the head oriented like this and at that time I want it like this". The computer needs to work out what happens in-between these poses. It's quite easy...
0.342764
99
science55
First you are doing it in the wrong end, second, the statement in general is still not correct. for example: 9999999999 1000000001 Say if you want the first 2 digits, you will get 10 if you discard the last 2 digit and do the sum. The right answer is 11
0.304121
99
science56
If you were supposed to find the last ten digits, you could just ignore the first 40 digits of each number. However you're supposed to find the first ten digits, so that doesn't work. And you can't just ignore the last digits of each number either because those can carry over.
0.417022
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science59
The field is the smaller/left circle, centered at A. The cow is tied to the post at E. The larger/right circle is the grazing radius. Let the radius of the field be R and the length of the rope be L. The grazable area is the union of a segment of the circular field and a segment of the circle defined by the rope length...
0.510422
99
science61
During one math class I had the same problem. the professor told us the answer can be a arbitrary integer, depend on how you remove the balls. Since you said every ball you put in was eventually removed, I will make it simple and assume you first number the balls to natural numbers, and then add and remove the balls wi...
0.585937
0
science65
The following reddit post has a decent list of math resources: http://www.reddit.com/r/math/comments/bqbex/lets_list_all_the_useful_free_online_math/ One site I did not see it their list that I've found very helpful: http://betterexplained.com/
0.083195
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science67
If you take a rope, fix the two ends, and let it hang under the force of gravity, it will naturally form a hyperbolic cosine curve.
0.243666
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science68
There came a time in mathematics when people encountered situations where they had to deal with really, really, really small things. Not just small like 0.01; but small as in infinitesimally small. Think of "the smallest positive number that is still greater than zero" and you'll realize what sort of problems mathemati...
0.194223
99
science70
Calculus is a field which deals with two seemingly unrelated things. (1) the area beneath a graph and the x-axis. (2) the slope (or gradient) of a curve at different points. Part (1) is also called 'integration' and 'anti-differentiation', and part (2) is called 'differentiation'.
0.095713
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science74
I listen to Math Mutation Podcast. The topics are interesting and understandable by a layman.
0.016129
2
science77
For non-zero bases and exponents, the relation $ x^a x^b = x^{a+b} $ holds. For this to make sense with an exponent of $ 0 $, $ x^0 $ needs to equal one. This gives you: $\displaystyle x^a \cdot 1 = x^a\cdot x^0 = x^{a+0} = x^a $ When the base is also zero, it's not possible to define a value for $0^0$ because there is...
0.15896
99
science78
$$0^x = 0, \quad x^0=1$$ both are true when $x>0$. What happens when $x=0$? It is undefined because there is no way to chose one definition over the other. Some people define $0^0 = 1$ in their books, like Knuth, because $0^x$ is less 'useful' than $x^0$.
0.153071
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science80
A nice thing to notice. It's basically because $(n + 1) ^ 2 - n ^ 2 = 2n + 1$
0.318766
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science82
Take a pair of compasses and draw an arc between two opposite corners, centred at another corner; then draw a diagonal that bisects the arc. If you now draw two lines from the point of intersection, parallel to the sides of the square, the biggest of the resulting squares will have half the area of the original square....
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science83
Andrea Feretti's MathOnline page.
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science86
It's easy to find a function which is continuous but not differentiable at a single point, e.g. $f(x) = |x|$ is continuous but not differentiable at $0$. Moreover, there are functions which are continuous but nowhere differentiable, such as the Weierstrass function. On the other hand, continuity follows from differenti...
0.357398
99
science87
To understand the benefits of using quaternions you have to consider different ways to represent rotations. Here are few ways with a summary of the pros and cons: Euler angles Rotation matrices Axis angle Quaternions Rotors (normalized Spinors) Euler angles are the best choice if you want a user to specify an orientati...
0.043591
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science88
Quantum mechanics, and hence physics and everything around us, fundamentally involves complex numbers.
0.304768
1
science89
The argument isn't worth having, as you disagree about what it means for something to 'exist'. There are many interesting mathematical objects which don't have an obvious physical counterpart. What does it mean for the Monster group to exist?
0.398186
1
science92
It is a theorem of Liouville, reproven later with purely algebraic methods, that for rational functions $f$ and $g$, $g$ non-constant, the antiderivative of $$f(x)\exp(g(x)) \, \mathrm dx$$ can be expressed in terms of elementary functions if and only if there exists some rational function $h$ such that it is a solutio...
0.355915
3
science96
This was first "guessed at" by Archimedes, by what fraction of the volume of a cylinder a sphere is. That is, a sphere that is contained within that cylinder. From that, he figured out $4/3$. You can do it these days with the tools of calculus. One way would to use the [Disk Method], over the graph of a semicircle. Ano...
0.151127
1
science97
It's important to understand that this is not something that can be proved: it's a definition. We choose not to regard 1 as a prime number, simply because it makes writing lots of theorems much easier. Noah gives the best example in his answer: Euclid's theorem that every positive integer can be written uniquely as a p...
0.398876
0
science98
Let $|S|$ be the cardinality of $S$. We know that $|S| < |2^S|$, which can be proven with generalized Cantor's diagonal argument. Theorem The set of all sets does not exist. Proof Let $S$ be the set of all sets, then $|S| < |2^S|$. But $2^S$ is a subset of $S$. Therefore $|2^S| \leq |S|$. A contradiction. Therefore the...
0.240856
3
science99
You may be interested to read the MathOverflow question "Demystifying Complex Numbers," here. A teacher is asking how to motivate complex numbers to students taking complex analysis.
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science100
Pappus's centroid theorem (second theorem) says that the volume of a solid formed by revolving a region about an axis is the product of the area of the region and the distance traveled by the centroid of the region when it is revolved. A sphere can be formed by revolving a semicircle about is diameter edge. The area of...
0.513128
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science102
Does this give you any ideas?
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science103
We will will first consider the most common definition of $i$, as the square root of $-1$. When you first hear this, it sounds crazy. $0$ squared is $0$; a positive times a positive is positive and a negative times a negative is positive too. So there doesn't actually appear to be any number that we can square to get $...
0.130895
1
science104
An informal explanation is Russel's Paradox. The wiki page is informative, here's the relevant quote: Let us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all squares. That set is not itself a square, and therefore is not a member of the set of all squares. So i...
0.321981
3
science107
Dan's algebraic justification is correct, but you may get more intuition about why this is happening from the above picture. Each time you want to enlarge the square by one unit, you have to add an extra row, an extra column, and one more square to fill in the corner. These correspond directly to the $n+n+1=2n+1$ that ...
0.553257
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science109
No number does "really exist" the way trees or atoms exist. In physics people however have found use for complex numbers just as they have found use for real numbers.
0.384838
1
science110
If you take two real numbers x and y then there per definition of the real number z for which x < z < y or x > z > y is true. For x = 0.99999... and y = 1 you can't find a z and therefore 0.99999... = 1.
0.316788
0
science111
You are changing your basis vectors, call your new ones $i$, $j$, and $k$ where $i$ is defined from $a-p$, $j$ from $b-p$, and $k$ the cross product. Now recall that your basis vectors should be unit, so take the length of your three vectors and divide the vectors by their length, making $i$, $j$, and $k$ unit. Now $a ...
0.354265
99
science112
What you are describing is an Affine Transformation, which is a linear transformation followed by a translation. We know this because any straight lines in your original vector space is also going to be a straight line in your transformed vector space.
0.171082
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science114
From the Wikipedia article about the prime number theorem: Roughly speaking, the prime number theorem states that if a random number nearby some large number N is selected, the chance of it being prime is about 1 / ln(N), where ln(N) denotes the natural logarithm of N. For example, near N = 10,000, about one in nine nu...
0.338671
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science115
The "set of all sets" is not so much a paradox in itself as something that inevitably leads to a contradiction, namely the well-known (and referenced in the question) Russell's paradox. Given any set and a predicate applying to sets, the set of all things satisfying the predicate should be a subset of the original set....
0.55237
3
science117
The distinction between constructive mathematics and traditional mathematics has nothing to do with Russell's Paradox. Constructive mathematics simply requires working with one less basic postulate that many mathematicians have believed to be sensible and on which some proofs are based, namely the Axiom of Choice
0.521533
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science118
Personally, I always use Python's console. It has history and allows all kinds of math operations. It is available for Linux, Windows, Mac, ChromeOS, Android, and others.
0.002688
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science121
Euclid's famous proof is as follows: Suppose there is a finite number of primes. Let $x$ be the product of all of these primes. Then look at $x+1$. It is clear that $x$ is coprime to $x+1$. Therefore, no nontrivial factor of $x$ is a factor of $x+1$, but every prime is a factor of $x$. By the fundamental theorem of ari...
0.207636
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science122
According to XKCD, we have the following Haiku: Top Prime's Divisors' Product (Plus one)'s factors are...? Q.E.D B@%&$ I wonder if we can edit it to make it correct
0.292489
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science123
The existence of universal set is incompatible with the Zermelo–Fraenkel axioms of set theory. However, there are alternative set theories which admit a universal set. One such theory is Quine's New Foundations.
0.52001
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science126
Brian Conrad explains this in the following: Impossibility theorems on integration in elementary terms (archived PDF)
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science129
Though I'm sure it's not unpopular, I don't think many people learn it early: Group Theory. It's a real nice area with a lot of cool math and some neat applications (like cryptography).
0.39437
2
science131
Theory of computation, information theory and logic/foundation of mathematics are very interesting topics. I wish I knew them earlier. They are not unpopular(almost every university have a bunch of ToC people in CS depatment...) , but many math major I know have never touched them. They show you the limits of mathemati...
0.161069
2
science135
That picture confuses things by making it look as though the red line is being "unwound" from the circle like paper towel being unwound from a roll. Our brains pick up on that, since it is a real-world example. Both circles complete a single revolution, and both travel the same distance from left to right. If these rea...
0.079366
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science136
I don't know how I missed that one, indeed: http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables Thanks Kaestur Hakarl!
0.428347
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science137
On a map using the Mercator projection, the relationship between the latitude L of a point and its y coordinate on the map is given by $y = \operatorname{arctanh}(\sin(L))$, where $\operatorname{arctanh}$ is the inverse of the hyperbolic tangent function.
0.204543
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science138
To the degree that anything actually "exists" in math, yes complex number exist. Once you accept that groups, rings and fields exist, and that isomorphism of rings makes sense, complex numbers can be recognized as (isomorphic to) the subring (which happens to be a field) of the ring of $2 \times 2$ real matrices. Gener...
0.450636
1
science139
I will assume the polygon has no intersections between the edges except at corners. Call the point $(x_0, y_0)$. First we determine whether we are on a line - this is simple using substitution and range checking. For the range checking, suppose we have a segment $(x_1, y_1)$, $(x_2, y_2)$. We check that $x_1\leq x_0\le...
0.547764
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science140
When I was that age, I discovered Raymond Smullyan's classic logic puzzle books in the library (such as What is the name of this book?), and really got into it. I remember my amazement when I first understood how a complicated logic puzzle could become trivial, just symbolic manipulation really, with the right notation...
0.093327
3
science141
Nearly always the direct proof is easier to understand, shorter, and more helpful!
0.296861
3
science144
Speaking from my own experience with elementary mathematics, yours is a very hard question to answer because there is little in the elementary math curriculum worth getting excited about. Before students are going to get excited about math, the math curriculum has to be changed so that the process of doing mathematics ...
0.457412
3
science147
In the sciences (as opposed to in mathematics) people are often a bit vague about exactly what assumptions they are making about how "well-behaved" things are. The reason for this is that ultimately these theories are made to be put to the test, so why bother worrying about exactly which properties you're assuming when...
0.048579
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science150
All of three dimensional space can be filled up with an infinite curve.
0.165938
0
science152
The two envelopes problem is a good one. See also: Card doubling paradox and: https://mathoverflow.net/questions/9037
0.286537
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science153
The Monty Hall problem fits the bill pretty well. Almost everyone, including most mathematicians, answered it wrong on their first try, and some took a lot of convincing before they agreed with the correct answer. It's also very easy to explain it to people.
0.30647
0
science155
There are true statements in arithmetic which are unprovable. Even more remarkably there are explicit polynomial equations where it's unprovable whether or not they have integer solutions with ZFC! (We need ZFC + consistency of ZFC)
0.111392
0
science159
Polya's "How To Solve It"
0.440328
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science160
Every simple closed curve that you can draw by hand will pass through the corners of some square. The question was asked by Toeplitz in 1911, and has only been partially answered in 1989 by Stromquist. As of now, the answer is only known to be positive, for the curves that can be drawn by hand. (i.e. the curves that ar...
0.438214
0
science162
This is an extremely broad question, especially given the wide variety of mathy people here, but I'll bite. HSM Coxeter's Introduction to Geometry is a book that was very important to the development of my interest in mathematics and inclination towards its geometric aspects.
0.565642
0
science163
Nicolas Bourbaki's Éléments de mathématique (specifically Topologie Générale and Algèbre).
0.084904
0
science166
Journey Through Genius A brilliant combination of interesting storytelling and large amounts of actual Mathematics. It took my love of Maths to a whole other level.
0.337066
0
science169
Euclid's Elements Newton's Principia Mathematica Ideally in the original languages of Ancient Greek and Latin respectively! No, just kidding. But they are true classics that any accomplished mathematician should read at some point during their career. Not because they'll teach you something you don't already know, but ...
0.574064
0
science171
Purplemath has a list for math lessons and tutoring. It's a list with various links and short reviews referring to tutoring and instructional resources.
0.079149
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science175
John D Cook writes The Endeavor One of the MathWorks blogs: Loren on the Art of Matlab ... a few more: eon Peter Cameron's Blog Walking Randomly Todd and Vishal's Blog (Check their blogrolls for more)
0.128631
0
science176
No, this is not a proof of your statement. If you look very closely, you will see that the hypotenuse of the triangles aren't straight. You can also verify this algebraically by calculating the internal angles of the triangle using trigonometry. The other way to demonstrate the non-straightness of the hypotenuse is in ...
0.08178
0
End of preview. Expand in Data Studio

Simulated Query and Document Streams of LoTTE Forum Dataset

This dataset contains the simulated streams for the paper "MURR: Model Updating with Regularized Replay for Searching a Document Stream". Please refer to the paper for the detailed sampling process.

The code for sampling the queries, qrels, and documents are documents in sampling.py. It is not intended to be ran but for recording purposes.

The random numbers for sampling are also recorded in the files for futrue references.

Dataset Structure

root
├───sampling.py
│
├───docs
│   └───test_D{1,2,3}_{0,1,2,3,4}.parquet
│   
├───queries
│   └───test_D{1,2,3}_{0,1,2,3,4}.jsonl
│   
└───qrels
    └───test_D{1,2,3}_{0,1,2,3,4}.qrels

D{1,2,3} indicates different streams and the subsequent {0,1,2,3,4} are the sessions ids.

Citation

@inproceedings{ecir2025murr,
    author = {Eugene Yang, Nicola Tonellotto, Dawn Lawrie, Sean MacAvaney, James Mayfield, Douglas Oard and Scott Miller},
    title = {MURR: Model Updating with Regularized Replay for Searching a Document Stream},
    booktitle = {Proceedings of the 47th European Conference on Information Retrieval (ECIR)},
    year = {2025},
}
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