query stringlengths 42 731 | positive_passages listlengths 1 1 | negative_passages listlengths 12 12 | score float64 100 100 |
|---|---|---|---|
Query: When designing a simple physics experiment to calculate the kinetic energy of an object moving along a straight path, if I need to determine the relationship between the distance traveled and the area under the velocity-time graph, which fundamental physical quantity should I consider as a measure? | [
{
"doc_id": null,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C$... | [
{
"doc_id": 37813,
"text": "['', \"Consider a simple undirected graph with 10 vertices and 15 edges. Determine the diameter of the graph.\\nTo determine the diameter of the graph, we need to find the longest shortest path between any two vertices in the graph. The diameter is the maximum distance between an... | 100 |
Query: When analyzing the performance of a video game, I need to understand how the graphical rendering process and user interaction can impact the game's frame rate, which is essentially the number of frames displayed per second. What broader concept explains this relationship in terms of a device that converts visual... | [
{
"doc_id": null,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C$... | [
{
"doc_id": 21158,
"text": "['', \"The frame rate of a video refers to how many times an image appears onscreen\\nper second. It's measured in frames per second (fps). A higher frame rate\\nprovides a smoother, more detailed picture. A lower frame rate will create a\\nshabbier appearance.\"]",
"title": ... | 100 |
Query: When measuring the duration of repeated electrical pulses and I want a single representative value without applying different importance to each reading, which basic averaging method should I use? | [
{
"doc_id": null,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C$... | [
{
"doc_id": 4655,
"text": "['', 'Suppose that a car travels along a straight road at a speed of 60 miles per hour for a duration of 3 hours. If the car started at a mile marker reading 30, what is the estimated mile marker reading where the car was halfway through the journey? Use the mean value theorem to ... | 100 |
Query: I'm trying to grasp the technical specifications of an analog to digital converter in the context of converting audio signals into digital format. Could you explain what 'frequency' means in relation to the speed at which this conversion happens? | [
{
"doc_id": null,
"text": "\\section{Area of Triangle in Terms of Two Sides and Angle} Tags: Triangles, Area of Triangle in Terms of Two Sides and Angle, Euclidean Geometry, Area Formulas, Areas of Triangles \\begin{theorem} The area of a triangle $ABC$ is given by: :$\\displaystyle \\frac 1 2 a b \\sin C$... | [
{
"doc_id": 8096,
"text": "['', '4x analog or digital servo control, with precision 16-bit timers.\\n2 x bi-directional brushed DC motor control, 1 Amp current-limited each, with 8-bit PWM speed control (or one stepper).\\n4 x high-current “Darlington” 500mA drive outputs with kick-back diode protection. Fo... | 100 |
Query: If I have a set of numbers and I want to determine a single value that represents the central tendency of this set, what term encompasses the concept of a mean and applies to any set of numbers, regardless of its size or composition? | [
{
"doc_id": null,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a s... | [
{
"doc_id": 3479,
"text": "['', 'Suppose the height of a certain population follows a normal distribution with a mean of 68 inches and a standard deviation of 3 inches. If a sample of 36 individuals is taken, what is the probability that the sample mean height is greater than 69 inches?\\nTo solve this prob... | 100 |
Query: When analyzing the timing of repeated electrical pulses in a communication system and wanting a single representative value without considering the variations between each pulse, which type of statistical measure should I use that is a specialization of the Pythagorean mean and directly related to determining pu... | [
{
"doc_id": null,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a s... | [
{
"doc_id": 76213,
"text": "['', 'To measure spike-evoked calcium influx, each protocol consisted of five sweeps\\neach with 1, 3, 5, 7 or 9 current injection pulses. Current was injected with\\na 4-ms square-wave pulse every 20 ms to evoke 50-Hz spike trains. The\\namplitude of the current injection was se... | 100 |
Query: Given a generalized mean and an uncountable set, what is the broader mathematical concept that they both fall under, especially in the context of understanding the nature of averages and sets in advanced mathematics? | [
{
"doc_id": null,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a s... | [
{
"doc_id": 12829,
"text": "['', '## tl;dr\\n\\n 1. Replication Requirements : What you’ll need to replicate the analysis in this tutorial \\n 2. Centered Moving Averages : Technique for calculating and displaying a simple moving average \\n 3. Moving Averages of Moving Averages : Using the concept o... | 100 |
Query: In the context of signal processing, if I'm looking for a type of statistic that generalizes the concept of a mean and specifically includes the generalized mean, what term would best describe this family of measures used to summarize data? | [
{
"doc_id": null,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a s... | [
{
"doc_id": 35428,
"text": "['', \"\\\\begin{definition}[Definition:Countable Set/Also defined as]\\nSome sources define a countable set to be what is defined on {{ProofWiki}} as a countably infinite set.\\nThat is, they use '''countable''' to describe a set which has ''exactly the same'' cardinality as $\\... | 100 |
Query: If I’m measuring the center point of a range of pulse widths in a signal processing application and I don’t care about giving more weight to certain readings, what general concept should I refer to, keeping in mind that the mean might not always be the right choice? | [
{
"doc_id": null,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a s... | [
{
"doc_id": 22270,
"text": "['', 'There are a couple of tricks you can do that will help birds put weight on\\nfaster in any scenario or any breed. First, leave the lights on longer. Longer\\ndaylight hours means your birds will be eating more. Even just having a light\\non a timer that turns on for an hour... | 100 |
Query: When measuring the duration of a series of electrical pulses in a communication system and you need a generalized method that goes beyond just the weighted mean of pulse widths, which concept should I apply to represent the central tendency of these pulse widths? | [
{
"doc_id": null,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a s... | [
{
"doc_id": 75146,
"text": "['', \"2.2. Painful stimulation\\nAn infrared neodymium yttrium aluminium perovskite laser with a wavelength of 1340 nm (El.En. S.p.A., Calenzano, Italy) was used to apply a total of 250 laser stimuli to the dorsum of the subjects' left hand. Laser stimuli were transmitted throug... | 100 |
Query: When analyzing a series of electrical pulses in a communication system and needing to determine a single representative duration without considering any particular pulse more important than others, what general type of average should I use? | [
{
"doc_id": null,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a s... | [
{
"doc_id": 74689,
"text": "['', '3.4.5. Lurching pulses\\nCan lurching pulses with more complicated spatiotemporal form be obtained in our model, without synaptic delays? We find such lurching pulse, with spatiotemporal periodicity in the firing pattern, if the E-to-E excitation is made slow, whereas all t... | 100 |
Query: Given a set of complex numbers, if I want to find a central value that represents the typical or representative number in this set, what fundamental concept am I aiming to apply? | [
{
"doc_id": null,
"text": "\\begin{definition}[Definition:Arithmetic Mean] Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \\ldots, x_n$ is defined as: :$\\ds A_n := \\dfrac 1 n \\sum_{k \\mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a s... | [
{
"doc_id": 37338,
"text": "['', 'Find the maximum and minimum values of $|z - 1|$ in the region defined by $|z| \\\\leq 2$.\\nTo find the maximum and minimum values of $|z - 1|$ in the region defined by $|z| \\\\leq 2$, we can use geometric interpretation. Here, $z$ is a complex number, and we can represen... | 100 |
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