instruction stringlengths 22 15.1k | completion stringlengths 2 49.3k | __original_row_idx int64 0 100k | conversations listlengths 2 2 |
|---|---|---|---|
If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
| If I were to choose a specific book for a mathematician at the beginning of their career, I would recommend "How to Prove It: A Structured Approach" by Daniel J. Velleman. This book provides a solid foundation in mathematical reasoning and proof techniques, which are essential skills in the field. It emphasizes the imp... | 57 | [
{
"from": "human",
"value": "If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?\n"
},
{
"from": "gpt",
"value": "If I were to choose a specific book for a mathematician at the beginning of their caree... |
I am solving the following inequality, please look at it and tell me whether am I correct or not. This is an example in Howard Anton's book and I solved it on my own as given below, but the book has solved it differently! I want to confirm that my solution is also valid.
| I'd be happy to help you with your inequality! Please provide the inequality that you solved, along with your solution and the method used in the book. Once I have that information, I can compare the two approaches and confirm whether your solution is valid. | 585 | [
{
"from": "human",
"value": "I am solving the following inequality, please look at it and tell me whether am I correct or not. This is an example in Howard Anton's book and I solved it on my own as given below, but the book has solved it differently! I want to confirm that my solution is also valid.\n\n"
... |
mathfactor is one I listen to. Does anyone else have a recommendation?
| If you enjoy MathFactor, you might also like the following math-related podcasts:
1. **Numberphile** - This podcast features discussions about interesting numbers and mathematical concepts with engaging guests.
2. **Math Encounters** - A podcast that showcases mathematical ideas in an accessible way, often featuring g... | 1 | [
{
"from": "human",
"value": "mathfactor is one I listen to. Does anyone else have a recommendation?\n"
},
{
"from": "gpt",
"value": "If you enjoy MathFactor, you might also like the following math-related podcasts:\n\n1. **Numberphile** - This podcast features discussions about interesting numb... |
What's the meaning of the double turnstile symbol in logic or mathematical notation? :
$\models$
| The double turnstile symbol, denoted as \(\models\), is used in logic and mathematical notation to signify semantic entailment or truth in a model. It indicates that a certain statement or set of statements (often referred to as a theory or a set of formulas) logically entails another statement.
In more formal terms, ... | 101 | [
{
"from": "human",
"value": "What's the meaning of the double turnstile symbol in logic or mathematical notation? :\n\n$\\models$\n\n"
},
{
"from": "gpt",
"value": "The double turnstile symbol, denoted as \\(\\models\\), is used in logic and mathematical notation to signify semantic entailment o... |
I'm looking for the name of a shape that is like a capsule, but where each circle can have different radii. The shape could be described using two circles (two centers and two radii). Something like this (this is a 2D shape, not a lame attempt at 3D):
| The shape you are describing sounds like a "lens" or more specifically, a "curvilinear quadrilateral" or "rounded rectangle" if you prefer, where two circles with different radii are connected by straight edges. In the context of a 2D shape formed by two circles with different radii, it could also be referred to as a "... | 297 | [
{
"from": "human",
"value": "I'm looking for the name of a shape that is like a capsule, but where each circle can have different radii. The shape could be described using two circles (two centers and two radii). Something like this (this is a 2D shape, not a lame attempt at 3D):\n\n"
},
{
"from": "... |
I'm looking for an online or software calculator that can show me the history of items I typed in, much like an expensive Ti calculator.
Can you recommend any?
| Certainly! If you're looking for a calculator that keeps a history of your calculations similar to a TI calculator, there are a few options you might consider:
1. **Desmos**: This is a powerful online graphing calculator that keeps a history of your calculations. You can create and save different projects, making it e... | 9 | [
{
"from": "human",
"value": "I'm looking for an online or software calculator that can show me the history of items I typed in, much like an expensive Ti calculator. \nCan you recommend any?\n"
},
{
"from": "gpt",
"value": "Certainly! If you're looking for a calculator that keeps a history of yo... |
This is problem taken from Problem 4.
I couldn't find the solution anywhere and I am curious to see a solution for this problem, as I can at least comprehend the question and it seems that the mechanism for the solution involved will be somewhat understandable.
| It seems like you are referencing a specific problem (Problem 4) but did not provide the details of that problem. If you could share the problem statement or describe the topic or concept involved, I would be more than happy to help you understand it and guide you through the solution! | 880 | [
{
"from": "human",
"value": "This is problem taken from Problem 4.\nI couldn't find the solution anywhere and I am curious to see a solution for this problem, as I can at least comprehend the question and it seems that the mechanism for the solution involved will be somewhat understandable.\n"
},
{
... |
Please suggest a suitable approach for this problem.
| Of course! However, I need more details about the specific problem you're facing in order to suggest a suitable approach. Could you please provide more context or describe the problem you're working on? | 1,332 | [
{
"from": "human",
"value": "Please suggest a suitable approach for this problem.\n"
},
{
"from": "gpt",
"value": "Of course! However, I need more details about the specific problem you're facing in order to suggest a suitable approach. Could you please provide more context or describe the probl... |
I can't seem to solve this one. I've tried Dirichlet's test but to no avail.
Any help is greatly appreciated.
| I'd be happy to help! However, I need more information about the specific problem you are trying to solve. Could you please provide the details of the series or integral you're working on? With that information, I can assist you better! | 1,623 | [
{
"from": "human",
"value": "I can't seem to solve this one. I've tried Dirichlet's test but to no avail.\nAny help is greatly appreciated.\n"
},
{
"from": "gpt",
"value": "I'd be happy to help! However, I need more information about the specific problem you are trying to solve. Could you please... |
I apologise for the really basic question. This didn't really fit on any other StackExchange website so the Maths one was the closest one where I could ask.
Really Basic Question- What are $4$ sided shapes called again?
Like how triangles are $3$ sided shapes, octagons are $8$ sided shapes, ... What are the $4$ sided ... | No need to apologize! A four-sided shape is called a **quadrilateral**. The term "quadrilateral" comes from the Latin words "quadri-" meaning four, and "latus" meaning side. There are many specific types of quadrilaterals, such as squares, rectangles, rhombuses, and trapezoids, but they all fall under the category of q... | 1,284 | [
{
"from": "human",
"value": "I apologise for the really basic question. This didn't really fit on any other StackExchange website so the Maths one was the closest one where I could ask.\nReally Basic Question- What are $4$ sided shapes called again? \nLike how triangles are $3$ sided shapes, octagons are $8... |
It seems like functions that are continuous always seem to be differentiable, to me. I can't imagine one that is not. Are there any examples of functions that are continuous, yet not differentiable?
The other way around seems a bit simpler -- a differentiable function is obviously always going to be continuous. But ... | It's a common intuition that continuous functions should be differentiable, but there are indeed continuous functions that are not differentiable. A classic example of this is the absolute value function:
\[ f(x) = |x| \]
This function is continuous everywhere, but it is not differentiable at \( x = 0 \). At that poi... | 36 | [
{
"from": "human",
"value": "It seems like functions that are continuous always seem to be differentiable, to me. I can't imagine one that is not. Are there any examples of functions that are continuous, yet not differentiable?\nThe other way around seems a bit simpler -- a differentiable function is obvi... |
Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?
It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely ... | The classification of zero as a natural number is a topic that has sparked discussion and variation in definitions within the mathematical community. There are primarily two conventions regarding the natural numbers:
1. **Including Zero**: In this convention, the set of natural numbers is defined to include zero. This... | 60 | [
{
"from": "human",
"value": "Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?\nIt seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions sayi... |
I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful.
Is there any good examples of their uses outside academia?
| Hyperbolic trigonometric functions indeed have several practical applications outside of academia. Here are a few examples that illustrate their usefulness in various fields:
1. **Physics**: Hyperbolic functions describe the shape of hanging cables or chains, known as catenaries. When a flexible chain or rope is suspe... | 31 | [
{
"from": "human",
"value": "I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful.\nIs there any good examples of their uses outside academia?\n"
},
{
"from": "gpt",
"value": "Hyperbolic trigo... |
The knight's tour is a sequence of 64 squares on a chess board, where each square is visted once, and each subsequent square can be reached from the previous by a knight's move. Tours can be cyclic, if the last square is a knight's move away from the first, and acyclic otherwise.
There are several symmetries among kni... | The enumeration of knight's tours on a chessboard is a well-studied problem in combinatorial game theory and graph theory. While the exact count of knight's tours that are distinct under all symmetries (including rotation and reflection) is complex, there have been notable findings.
As of the latest research, the numb... | 148 | [
{
"from": "human",
"value": "The knight's tour is a sequence of 64 squares on a chess board, where each square is visted once, and each subsequent square can be reached from the previous by a knight's move. Tours can be cyclic, if the last square is a knight's move away from the first, and acyclic otherwis... |
I just got my statistics test back and I am totally confused about one of the questions!
A study was done that took a simple random sample of 40 people and measured whether
the subjects were right-handed or
left-handed, as well as their ages.
The study showed that the proportion
of left-handed people and the... | Your understanding of correlation and causation is generally correct—correlation does not imply causation. However, the way you phrased your conclusion is problematic because it implies a specific directional relationship (i.e., that people become right-handed as they grow older), which is not supported by the informat... | 89 | [
{
"from": "human",
"value": "I just got my statistics test back and I am totally confused about one of the questions! \n\nA study was done that took a simple random sample of 40 people and measured whether\n the subjects were right-handed or\n left-handed, as well as their ages.\n The study showed that ... |
Is there a collective term for the $XY, YZ$ and $ZX$ planes in $3D$ co-ordinate geometry? I was thinking "principal planes" but I'm not sure where I heard that.
| Yes, the terms "principal planes" or "coordinate planes" can be used to refer to the \(XY\), \(YZ\), and \(ZX\) planes in 3D coordinate geometry. These planes are defined by the coordinate axes:
- The \(XY\) plane is defined by \(z = 0\).
- The \(YZ\) plane is defined by \(x = 0\).
- The \(ZX\) plane is defined by \(y... | 344 | [
{
"from": "human",
"value": "Is there a collective term for the $XY, YZ$ and $ZX$ planes in $3D$ co-ordinate geometry? I was thinking \"principal planes\" but I'm not sure where I heard that.\n"
},
{
"from": "gpt",
"value": "Yes, the terms \"principal planes\" or \"coordinate planes\" can be use... |
What is the proper definition of a Countable Set?
| A countable set is defined as a set that has the same cardinality as the set of natural numbers, which means it can either be finite or infinite but can be put into a one-to-one correspondence with the natural numbers \( \mathbb{N} = \{1, 2, 3, \ldots\} \).
In more formal terms:
1. **Finite Countable Set**: A finite ... | 704 | [
{
"from": "human",
"value": "What is the proper definition of a Countable Set?\n"
},
{
"from": "gpt",
"value": "A countable set is defined as a set that has the same cardinality as the set of natural numbers, which means it can either be finite or infinite but can be put into a one-to-one corres... |
In Intro Number Theory a key lemma is that if $a$ and $b$ are relatively prime integers, then there exist integers $x$ and $y$ such that $ax+by=1$. In a more advanced course instead you would use the theorem that the integers are a PID, i.e. that all ideals are principal. Then the old lemma can be used to prove that ... | Yes, it is indeed possible for a ring to have all finitely generated ideals as principal while also having some ideals that are not finitely generated.
To illustrate this, consider the ring of integers \(\mathbb{Z}\), which is a principal ideal domain (PID). In \(\mathbb{Z}\), every ideal is generated by a single ele... | 113 | [
{
"from": "human",
"value": "In Intro Number Theory a key lemma is that if $a$ and $b$ are relatively prime integers, then there exist integers $x$ and $y$ such that $ax+by=1$. In a more advanced course instead you would use the theorem that the integers are a PID, i.e. that all ideals are principal. Then... |
I need $n$ cakes for a party. I go to the cake shop and there are $k$ different kinds of cake. For variety, I'd like to get at least one of each cake.
How many ways can I do this?
| To determine how many ways you can select \( n \) cakes from \( k \) different kinds of cake with the requirement that you get at least one of each kind, we can utilize the "stars and bars" combinatorial method.
First, since we need at least one cake of each kind, we will initially allocate one cake of each kind. This... | 167 | [
{
"from": "human",
"value": "I need $n$ cakes for a party. I go to the cake shop and there are $k$ different kinds of cake. For variety, I'd like to get at least one of each cake. \nHow many ways can I do this?\n"
},
{
"from": "gpt",
"value": "To determine how many ways you can select \\( n \\) ... |
When given an algrebraic expression of the form
$$ \ 6A + 7{A^2}\ $$
and asked to simplify, would the correct answer be
a)
$$ \ A(6 + 7A)\ $$
or
b)
That this cannot be simplified further
| To simplify the expression \( 6A + 7A^2 \), we can factor out the common term, which is \( A \).
Starting with the expression:
\[
6A + 7A^2
\]
We can factor out \( A \):
\[
A(6 + 7A)
\]
So, the correct answer is:
a) \( A(6 + 7A) \)
Therefore, the expression can indeed be simplified further. | 1,119 | [
{
"from": "human",
"value": "When given an algrebraic expression of the form\n$$ \\ 6A + 7{A^2}\\ $$\nand asked to simplify, would the correct answer be\na)\n$$ \\ A(6 + 7A)\\ $$\nor\nb)\nThat this cannot be simplified further\n"
},
{
"from": "gpt",
"value": "To simplify the expression \\( 6A + ... |
Is a prime number in the decimal system still a prime when converted to a different base?
| Yes, a prime number remains prime when expressed in a different base, but the representation of the number changes, not its intrinsic properties.
A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. This definition is independent of the numeral system use... | 941 | [
{
"from": "human",
"value": "Is a prime number in the decimal system still a prime when converted to a different base?\n"
},
{
"from": "gpt",
"value": "Yes, a prime number remains prime when expressed in a different base, but the representation of the number changes, not its intrinsic properties... |
I have a square that's $10\mathrm{m} \times 10\mathrm{m}$. I want to cut it in half so that I have a square with half the area. But if I cut it from top to bottom or left to right, I don't get a square, I get a rectangle!
I know the area of the small square is supposed to be $50\mathrm{m}^{2}$, so I can use my calcul... | To find the side length of a square with an area of \(50 \, \text{m}^2\) without using a calculator, we can solve the problem step by step using some basic math principles.
1. **Understanding the Area of a Square**:
The area \(A\) of a square is given by the formula:
\[
A = s^2
\]
where \(s\) is the le... | 35 | [
{
"from": "human",
"value": "I have a square that's $10\\mathrm{m} \\times 10\\mathrm{m}$. I want to cut it in half so that I have a square with half the area. But if I cut it from top to bottom or left to right, I don't get a square, I get a rectangle! \nI know the area of the small square is supposed to ... |
The scatter graph at the Wikipedia article seems to suggest so. Has anyone attacked this before? Is there a known proof?
| It seems like you're referring to a specific concept or hypothesis related to a scatter graph that appears in a Wikipedia article. However, without additional context about which topic or concept you're discussing, it's difficult to provide a direct answer.
In general, many mathematical or statistical hypotheses repre... | 1,865 | [
{
"from": "human",
"value": "The scatter graph at the Wikipedia article seems to suggest so. Has anyone attacked this before? Is there a known proof?\n"
},
{
"from": "gpt",
"value": "It seems like you're referring to a specific concept or hypothesis related to a scatter graph that appears in a W... |
I have to divide $2860$ by $3186$. The question gives only $2$ minutes and that division is only half part of question. Now I can't possibly make that division in or less than $2$ minutes by applying traditional methods, which I can't apply on that division anyways.
So anyone can perform below division using faster tec... | To quickly estimate the division of \( 2860 \) by \( 3186 \), we can first simplify the fraction \( \frac{2860}{3186} \).
First, let's find approximate values for both numbers:
1. Calculate \( \frac{2860}{3186} \):
- Both numbers are fairly close to each other. We can round them to simpler numbers for estimation.
... | 351 | [
{
"from": "human",
"value": "I have to divide $2860$ by $3186$. The question gives only $2$ minutes and that division is only half part of question. Now I can't possibly make that division in or less than $2$ minutes by applying traditional methods, which I can't apply on that division anyways.\nSo anyone c... |
A quick question-Something that I learned that I forgot throughout the years.
If a question asked you for a multiple of 9 and 12, which would be less than 40, would "36" be an answer, as both 9 and 12 can multiply to get 36, yet it fits under the criteria for being under 40?
| Yes, you are correct!
The multiples of 9 are: 9, 18, 27, 36, 45, ...
The multiples of 12 are: 12, 24, 36, 48, ...
The least common multiple (LCM) of 9 and 12 is 36. Since 36 is indeed a multiple of both 9 and 12 and is less than 40, it fits the criteria perfectly.
So, 36 is a valid answer. | 1,546 | [
{
"from": "human",
"value": "A quick question-Something that I learned that I forgot throughout the years. \nIf a question asked you for a multiple of 9 and 12, which would be less than 40, would \"36\" be an answer, as both 9 and 12 can multiply to get 36, yet it fits under the criteria for being under 40?... |
This is such a basic question I'm sure, but I've been trying to find a robust solution to it for a while.
Lets say I have the numbers 0-100 in series.
As the number increases, I want another number say 1000, to decrease. so that when you got to 100, it was at 0
0 = 1000
50 = 500
100 = 0
What is this kind of series call... | What you're describing is a linear relationship between two variables, where one variable decreases as the other increases in a linear manner. Specifically, in your case, you want the values to be inversely proportional: when one variable increases from 0 to 100, the other variable decreases from 1000 to 0.
The mathem... | 931 | [
{
"from": "human",
"value": "This is such a basic question I'm sure, but I've been trying to find a robust solution to it for a while.\nLets say I have the numbers 0-100 in series.\nAs the number increases, I want another number say 1000, to decrease. so that when you got to 100, it was at 0\n0 = 1000\n50 =... |
Suppose $A$ is some algebraic structure, and $x$ and $y$ are two elements of the underlying set. Is there any more concise way of stating, "there exists an automorphism in $A$ which maps $x$ to $y$"?. It seems like the obvious thing to say is "$x$ and $y$ are symmetric in $A$", but I've never come across this phrasin... | The concept you're describing, where there exists an automorphism in the algebraic structure \( A \) that maps \( x \) to \( y \), can indeed be succinctly expressed. The term you might be looking for is "conjugate". Specifically, you can say that "elements \( x \) and \( y \) are conjugate in \( A \)" if such an autom... | 620 | [
{
"from": "human",
"value": "Suppose $A$ is some algebraic structure, and $x$ and $y$ are two elements of the underlying set. Is there any more concise way of stating, \"there exists an automorphism in $A$ which maps $x$ to $y$\"?. It seems like the obvious thing to say is \"$x$ and $y$ are symmetric in $... |
I forget for a while, we don't need the compactness condition here right?
| It seems like you're referring to a concept from topology or functional analysis, possibly in the context of a theorem or a property of certain spaces or sets. The compactness condition is often a crucial assumption in various mathematical results, such as the Arzelà-Ascoli theorem in analysis or certain theorems in to... | 1,180 | [
{
"from": "human",
"value": "I forget for a while, we don't need the compactness condition here right?\n"
},
{
"from": "gpt",
"value": "It seems like you're referring to a concept from topology or functional analysis, possibly in the context of a theorem or a property of certain spaces or sets. ... |
There is a theorem of Riemann to that effect. How to prove it?
Note: This was asked by Kenny in the beta for "calculus".
| It seems like you are referencing the Riemann Hypothesis or possibly theorems related to Riemann surfaces or Riemann sums in calculus, but it's not entirely clear from your question. Let's clarify a few possibilities:
1. **Riemann Hypothesis**: This is a conjecture in number theory about the distribution of the zeros ... | 204 | [
{
"from": "human",
"value": "There is a theorem of Riemann to that effect. How to prove it?\nNote: This was asked by Kenny in the beta for \"calculus\".\n"
},
{
"from": "gpt",
"value": "It seems like you are referencing the Riemann Hypothesis or possibly theorems related to Riemann surfaces or ... |
A map $s : \mathbb{N} \to X$ is a computable sequence in $(X,\nu_X)$ when there exists a computable map $f : \mathbb{N} \to \mathbb{N}$ such that $s(n) = \nu_X(f(n))$ for all $n \in \mathrm{dom}(\nu_X)$.
My best guess would be, "A map s taking N onto X is a computable sequence in the ??? (X, nu??) when there exists a c... | You can read the statement aloud in a way that maintains clarity while also conveying all the relevant elements. Here’s a suggested phrasing:
"A map \( s \) from the natural numbers to \( X \) is a computable sequence in the space \( (X, \nu_X) \) when there exists a computable function \( f \) from the natural number... | 1,625 | [
{
"from": "human",
"value": "A map $s : \\mathbb{N} \\to X$ is a computable sequence in $(X,\\nu_X)$ when there exists a computable map $f : \\mathbb{N} \\to \\mathbb{N}$ such that $s(n) = \\nu_X(f(n))$ for all $n \\in \\mathrm{dom}(\\nu_X)$.\nMy best guess would be, \"A map s taking N onto X is a computabl... |
You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise.
But recently a good friend named Euclid has raised doubts about the fifth postulate of Poincaré's Elements. This postula... | To model Euclidean geometry within the hyperbolic plane, you can use a creative approach. One intuitive way to accomplish this is by constructing what we'll call the "Conformal Projection Model" of Euclidean geometry in the hyperbolic plane.
### The Conformal Projection Model
1. **Choose a Hyperbolic Model:** Start ... | 314 | [
{
"from": "human",
"value": "You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise.\nBut recently a good friend named Euclid has raised doubts about the fifth post... |
I'm very interested in Computer Science (computational complexity, etc.). I've already finished a University course in the subject (using Sipser's "Introduction to the Theory of Computation").
I know the basics, i.e. Turing Machines, Computability (Halting problem and related reductions), Complexity classes (time and s... | It's great to hear that you've built a solid foundation in computability and complexity theory! Here are some recommendations for more advanced texts that delve deeper into various aspects of computational complexity:
1. **"Computational Complexity: A Modern Approach" by Sanjeev Arora and Boaz Barak** - This is an exc... | 74 | [
{
"from": "human",
"value": "I'm very interested in Computer Science (computational complexity, etc.). I've already finished a University course in the subject (using Sipser's \"Introduction to the Theory of Computation\").\nI know the basics, i.e. Turing Machines, Computability (Halting problem and related... |
Please suggest suitable approach for this problem
| I'd be happy to help! However, I need a bit more context about the specific problem you're referring to. Could you provide more details about the issue or situation you're facing? The more information you provide, the better I can tailor my suggestions to your needs. | 2,486 | [
{
"from": "human",
"value": "Please suggest suitable approach for this problem\n"
},
{
"from": "gpt",
"value": "I'd be happy to help! However, I need a bit more context about the specific problem you're referring to. Could you provide more details about the issue or situation you're facing? The ... |
Why are differential equations called differential equations?
| Differential equations are called so because they involve derivatives, which are a type of differential operation. In mathematics, derivatives measure how a function changes as its input changes. A differential equation is an equation that relates a function with its derivatives, thus capturing the relationship between... | 1,114 | [
{
"from": "human",
"value": "Why are differential equations called differential equations?\n"
},
{
"from": "gpt",
"value": "Differential equations are called so because they involve derivatives, which are a type of differential operation. In mathematics, derivatives measure how a function change... |
Lifted from Mathoverflow:
I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.
Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc.
One suggestion per answer please. Also, please include an explanation of why you like the book, or what makes... | One highly regarded second choice for a comprehensive resource in algebraic geometry is **"Algebraic Geometry" by Robin Hartshorne's own advisor, David Mumford**. This book is unique and valuable for several reasons:
1. **Geometric Intuition**: Mumford's approach emphasizes geometric intuition alongside formalism. His... | 243 | [
{
"from": "human",
"value": "Lifted from Mathoverflow:\nI think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.\nThen what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc.\nOne suggestion per answer please. Also, please include an explana... |
So you have $2$ dice and you want to get at least a $1$ or a $5$ (on the dice not added). How do you go about calculating the answer for this question.
This question comes from the game farkle.
| To solve the problem of finding the probability of rolling at least one \(1\) or one \(5\) with two dice, we can use the complementary counting method. Here are the steps:
1. **Calculate the Total Possible Outcomes**:
Each die has 6 faces, so with 2 dice, the total number of outcomes is:
\[
6 \times 6 = 36.
... | 70 | [
{
"from": "human",
"value": "So you have $2$ dice and you want to get at least a $1$ or a $5$ (on the dice not added). How do you go about calculating the answer for this question.\nThis question comes from the game farkle.\n"
},
{
"from": "gpt",
"value": "To solve the problem of finding the pr... |
Follow up to
Say I have triangles as shown:
Its easy to see that $ABC$ is adjacent to $ACJ: ABC$ and $ACJ$ share $2$ vertices ($A$ and $C$).
However $FEC$ and $ACJ$ are also adjacent, but only one vertex is shared.
How can I find if two triangles are adjacent when they differ in size?
| In the context of geometry, two triangles are considered adjacent if they share at least one vertex or one edge. The adjacency can be evaluated using the following criteria:
1. **Shared Vertices**: If two triangles share at least one vertex, they are adjacent. This is the case you mentioned with triangles \( ABC \) an... | 968 | [
{
"from": "human",
"value": "Follow up to\nSay I have triangles as shown:\n\nIts easy to see that $ABC$ is adjacent to $ACJ: ABC$ and $ACJ$ share $2$ vertices ($A$ and $C$).\nHowever $FEC$ and $ACJ$ are also adjacent, but only one vertex is shared.\nHow can I find if two triangles are adjacent when they di... |
This is a problem from "Introduction to Mathematics - Algebra and Number Systems" (specifically, exercise set 2 #9), which is one of my math texts. Please note that this isn't homework, but I would still appreciate hints rather than a complete answer.
The problem reads as follows:
If 3p2 = q2, where $p,q \in \mathbb{Z... | You're on the right track in showing that \( 3 \) divides \( q \). To show that \( 3 \) also divides \( p \), you can proceed with a proof by contradiction.
Here's a hint:
1. Assume for contradiction that \( 3 \) does not divide \( p \). This means that \( p \) can be expressed in the form \( p = 3k + r \) where \( ... | 671 | [
{
"from": "human",
"value": "This is a problem from \"Introduction to Mathematics - Algebra and Number Systems\" (specifically, exercise set 2 #9), which is one of my math texts. Please note that this isn't homework, but I would still appreciate hints rather than a complete answer.\nThe problem reads as fol... |
You know that a couple has two children. You go to the couple's house and one of their children, a young boy, opens the door. What is the probability that the couple's other child is a girl?
If you list all possibilities for the sexes of two children, $BB, BG, GB, GG$, you see that $2$ of the $3$ pairs that have B (for... | This question is a classic problem in probability that revolves around conditional probability. Let's analyze it step by step.
First, we list all possible combinations of two children, assuming they can either be a boy (B) or a girl (G). The possible combinations are:
1. BB (both are boys)
2. BG (one boy and one girl... | 641 | [
{
"from": "human",
"value": "You know that a couple has two children. You go to the couple's house and one of their children, a young boy, opens the door. What is the probability that the couple's other child is a girl?\nIf you list all possibilities for the sexes of two children, $BB, BG, GB, GG$, you see ... |
The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets:
If $C$ is the cantor set, then what is the measure of $C+C$?
| To find the measure of \( C + C \) where \( C \) is the Cantor set, we begin by examining the nature of the Cantor set and how its sum behaves.
1. **Understanding the Cantor Set:** The Cantor set \( C \) is a subset of the interval \([0, 1]\) that is constructed by iteratively removing the middle thirds of segments. I... | 568 | [
{
"from": "human",
"value": "The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets:\nIf $C$ is the cantor set, then what is the measure of $C+C$?\n"
},
{
"from": ... |
Context: Rings.
Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms?
Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck...
| In the context of rings in abstract algebra, the statements you've mentioned can be analyzed as follows:
1. **For the statement \( x \cdot 0 = 0 \)**: This is known as the *zero property of multiplication*. It states that multiplying any element \( x \) by the additive identity (zero) results in zero. This property is... | 79 | [
{
"from": "human",
"value": "Context: Rings.\n\nAre $x \\cdot 0 = 0$ and $x \\cdot 1 = x$ and $-(-x) = x$ axioms?\n\nArguably three questions in one, but since they all are properties of the multiplication, I'll try my luck...\n"
},
{
"from": "gpt",
"value": "In the context of rings in abstract ... |
I'm talking about the shape made up of a rectangle with a semi-circle at each end. Does it have a particular name? Does it begin with e?
| Yes, the shape you are describing is called an "ellipse." An ellipse can be thought of as a shape formed by a rectangle with semicircles at each end. However, it's worth noting that the term "ellipse" typically refers to the entire shape rather than just the construction you mentioned. If you're specifically referring ... | 1,356 | [
{
"from": "human",
"value": "I'm talking about the shape made up of a rectangle with a semi-circle at each end. Does it have a particular name? Does it begin with e?\n"
},
{
"from": "gpt",
"value": "Yes, the shape you are describing is called an \"ellipse.\" An ellipse can be thought of as a sh... |
Our math prof gave us this and we are not sure what to make of the notation, can someone give us a hand?
| Of course! I'd be happy to help. Please provide the math notation or problem you're working on, and I'll do my best to explain it to you. | 2,641 | [
{
"from": "human",
"value": "Our math prof gave us this and we are not sure what to make of the notation, can someone give us a hand? \n\n"
},
{
"from": "gpt",
"value": "Of course! I'd be happy to help. Please provide the math notation or problem you're working on, and I'll do my best to explain... |
I was musing about a particular limit,
$L = \prod\limits_{n > 0} \bigl(1 - 2^{-n}\bigr)$:
we may bound 0.288 < L < 0.308, which we may show by taking the logarithm:
$\ln(L) = \ln \bigl( \frac{315}{1024}\bigr) + \sum\limits_{n > 4} \ln\bigl(1 - 2^{-n}\bigr) > \ln\bigl(\frac{315}{1024}\bigr) - \sum\limits_{n > 4} 2^{-n... | The limit you're exploring, \( L = \prod_{n > 0} (1 - 2^{-n}) \), is a specific type of infinite product that can be related to various areas in number theory, analysis, and combinatorial mathematics. Products of this form are often studied in relation to generating functions and special series, and they can relate to ... | 889 | [
{
"from": "human",
"value": "I was musing about a particular limit,\n$L = \\prod\\limits_{n > 0} \\bigl(1 - 2^{-n}\\bigr)$:\nwe may bound 0.288 < L < 0.308, which we may show by taking the logarithm: \n$\\ln(L) = \\ln \\bigl( \\frac{315}{1024}\\bigr) + \\sum\\limits_{n > 4} \\ln\\bigl(1 - 2^{-n}\\bigr) > \... |
Sometimes when I am reading a mathematical text I will be unable to decipher the steps necessary to translate one mathematical statement into the next. Usually I have to ask for help, but I was wondering if there is a software package that would let me enter two mathematical statements and the it would find the steps n... | The desire to have a software package that can find the steps necessary to translate one mathematical statement to another is a common challenge in mathematical problem-solving. While there isn't a perfect solution that works universally for all mathematical statements, there are several tools and approaches that might... | 524 | [
{
"from": "human",
"value": "Sometimes when I am reading a mathematical text I will be unable to decipher the steps necessary to translate one mathematical statement into the next. Usually I have to ask for help, but I was wondering if there is a software package that would let me enter two mathematical sta... |
I'd like to write down proofs. I use first-order logic and natural deduction. Formulas tend to be long, too long for LaTeX. Writing formulas with the computer is also a slow process. Handwriting is a lot faster and more natural, but I can't trust paper. Time, water, dirt, fire are its enemies. I could write down someth... | It's great that you're considering both the clarity of your proofs and the environmental impact of your writing! Here are some strategies to help you balance both concerns:
1. **Use Reusable Materials**: Consider using a whiteboard or a reusable notebook (like a Rocketbook or similar products) that allows you to write... | 474 | [
{
"from": "human",
"value": "I'd like to write down proofs. I use first-order logic and natural deduction. Formulas tend to be long, too long for LaTeX. Writing formulas with the computer is also a slow process. Handwriting is a lot faster and more natural, but I can't trust paper. Time, water, dirt, fire a... |
What do you call this function?
$$f(x) = x(1 - x)$$
Does it have a name? It’s cropped up in at least two statistical formulas I've been asked to program so far. (In both cases, $x$ was something called a $p$-value (for what it's worth).)
| The function \( f(x) = x(1 - x) \) is commonly referred to as the **quadratic function** or **parabola** due to its polynomial form. However, it has specific applications and interpretations in various fields, particularly in statistics and probability.
In the context of statistics, this function is often associated w... | 1,171 | [
{
"from": "human",
"value": "What do you call this function?\n$$f(x) = x(1 - x)$$\nDoes it have a name? It’s cropped up in at least two statistical formulas I've been asked to program so far. (In both cases, $x$ was something called a $p$-value (for what it's worth).)\n"
},
{
"from": "gpt",
"v... |
I have just started auditing Linear Algebra, and the first thing we learned was how to solve a system of linear equations by reducing it to row echelon form (or reduced row echelon form). I know how to solve a system of linear equations using determinants, and so the using row echelon form seems to be very inefficient ... | Row echelon form (REF) and reduced row echelon form (RREF) are very important concepts in linear algebra, and they serve as foundational tools for various areas of mathematics and applied fields. While your friend may have a point that determinants can be used to solve linear systems, relying solely on determinants has... | 416 | [
{
"from": "human",
"value": "I have just started auditing Linear Algebra, and the first thing we learned was how to solve a system of linear equations by reducing it to row echelon form (or reduced row echelon form). I know how to solve a system of linear equations using determinants, and so the using row e... |
I am learning geometric algebra, and it is incredible how much it helps me understand other branches of mathematics. I wish I had been exposed to it earlier.
Additionally I feel the same way about enumerative combinatorics.
What are some less popular mathematical subjects that you think should be more popular?
| It's great to hear that you're finding geometric algebra and enumerative combinatorics so beneficial! There are several less popular mathematical subjects that also have the potential to enrich understanding in various fields. Here are a few:
1. **Category Theory**: Often regarded as "the mathematics of mathematics," ... | 49 | [
{
"from": "human",
"value": "I am learning geometric algebra, and it is incredible how much it helps me understand other branches of mathematics. I wish I had been exposed to it earlier. \nAdditionally I feel the same way about enumerative combinatorics. \nWhat are some less popular mathematical subjects th... |
In the comment thread of an answer, I said:
The computable numbers are based on the intuitionistic continuum, and are not finitary.
To which T.. replied:
Computable numbers are not based on the intuitionistic continuum.
This disagreement contains, I think, a good example of a philosophical question: are the comput... | The disagreement you mentioned touches on some foundational issues in mathematics and philosophy, particularly regarding the nature of computable numbers and the distinctions between various interpretations of real numbers, such as classical and intuitionistic viewpoints.
To unpack the statement, it's important to cla... | 446 | [
{
"from": "human",
"value": "In the comment thread of an answer, I said:\n\nThe computable numbers are based on the intuitionistic continuum, and are not finitary. \n\nTo which T.. replied:\n\nComputable numbers are not based on the intuitionistic continuum.\n\nThis disagreement contains, I think, a good ex... |
So first off all, this WAS homework. Submitted it about 3 hours ago.
Let $A$ be an $m \times m$ matrix. then show that if rank$ A = m$, then $Ax = 0$ has a unique solution.
My roommate said this is wrong but here is how I solved it.
very simply, i stated that if the rank of A is m that means the number of leading one... | Your solution is actually correct! You provided a valid argument to show that if the rank of matrix \( A \) is \( m \) (which implies that \( A \) is a full-rank square matrix), then the equation \( Ax = 0 \) indeed has a unique solution.
Here's a breakdown of your reasoning:
1. **Rank Condition**: Since the rank of ... | 1,271 | [
{
"from": "human",
"value": "So first off all, this WAS homework. Submitted it about 3 hours ago.\nLet $A$ be an $m \\times m$ matrix. then show that if rank$ A = m$, then $Ax = 0$ has a unique solution. \nMy roommate said this is wrong but here is how I solved it. \nvery simply, i stated that if the rank o... |
(Diagram and setup from UCSMP Precaluclus and Discrete Mathematics, 3rd ed.)
Above is a partial plot of the zeros of $p_c(x)=4x^4+8x^3-3x^2-9x+c$. The text stops at showing the diagram and does not discuss the shape of the locus of the zeros or describe the resulting curves. Are the curves in the locus some specific... | The polynomial \( p_c(x) = 4x^4 + 8x^3 - 3x^2 - 9x + c \) is a quartic polynomial in \( x \), and its roots can be expressed as functions of the constant \( c \). The behavior of the roots as \( c \) varies will typically generate a specific type of curve in the complex plane.
When you vary \( c \), the roots of the p... | 117 | [
{
"from": "human",
"value": "\n(Diagram and setup from UCSMP Precaluclus and Discrete Mathematics, 3rd ed.)\nAbove is a partial plot of the zeros of $p_c(x)=4x^4+8x^3-3x^2-9x+c$. The text stops at showing the diagram and does not discuss the shape of the locus of the zeros or describe the resulting curves.... |
Let U be a unitary transformation from Hilbert space H to Hilbert space K.
How do you call a *-homomorphism f from B(H) to B(K), defined by $f(a) = U a U^{-1}$?
I'm interested both in a symbol, which can be used in formulas, and a name for it, which can be used in texts or in speech.
| The *-homomorphism \( f: B(H) \to B(K) \) defined by \( f(a) = U a U^{-1} \) for a unitary transformation \( U \) is commonly referred to as a **unitary representation**.
In terms of symbols, you can denote this transformation simply as \( f_U \), where \( f_U(a) = U a U^{-1} \).
In texts or discussions, you might ... | 552 | [
{
"from": "human",
"value": "Let U be a unitary transformation from Hilbert space H to Hilbert space K. \nHow do you call a *-homomorphism f from B(H) to B(K), defined by $f(a) = U a U^{-1}$?\nI'm interested both in a symbol, which can be used in formulas, and a name for it, which can be used in texts or i... |
For a given $k>0$ constant, assuming that $x,y>0$. Also this equation has a particular name, or some mathematician associated with it?
| It seems you are referring to a certain type of inequality or mathematical expression, but your question is incomplete. There are many mathematical concepts and inequalities that involve a constant \( k \) and positive variables \( x \) and \( y \).
If you're referring to something like the **AM-GM inequality**, whic... | 410 | [
{
"from": "human",
"value": "For a given $k>0$ constant, assuming that $x,y>0$. Also this equation has a particular name, or some mathematician associated with it?\n"
},
{
"from": "gpt",
"value": "It seems you are referring to a certain type of inequality or mathematical expression, but your que... |
I know that the Fibonacci sequence can be described via the Binet's formula.
However, I was wondering if there was a similar formula for $n!$.
Is this possible? If not, why not?
| The Fibonacci sequence can indeed be expressed using Binet's formula, which gives an explicit formula for the \(n\)-th Fibonacci number \(F_n\) in terms of \(n\):
\[
F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}
\]
where \(\varphi = \frac{1 + \sqrt{5}}{2}\) is the golden ratio and \(\psi = \frac{1 - \sqrt{5}}{2}\).
Howe... | 77 | [
{
"from": "human",
"value": "I know that the Fibonacci sequence can be described via the Binet's formula.\nHowever, I was wondering if there was a similar formula for $n!$.\nIs this possible? If not, why not?\n"
},
{
"from": "gpt",
"value": "The Fibonacci sequence can indeed be expressed using ... |
A problem that is an example of this is $x^2+16$. I have to know how to factor this over the set of complex numbers. How do you do that? I used to know it's just been a long time.
| To factor the expression \( x^2 + 16 \) over the set of complex numbers, we start by rewriting it in a form that reveals its roots.
First, we can observe that \( 16 \) can be expressed as \( 4^2 \). This allows us to rewrite \( x^2 + 16 \) as:
\[
x^2 + 4^2
\]
The expression \( x^2 + a^2 \) (where \( a = 4 \)) can be... | 1,353 | [
{
"from": "human",
"value": "A problem that is an example of this is $x^2+16$. I have to know how to factor this over the set of complex numbers. How do you do that? I used to know it's just been a long time.\n"
},
{
"from": "gpt",
"value": "To factor the expression \\( x^2 + 16 \\) over the set... |
I have to do this for an upcoming exam, but cannot find anywhere (in the textbook or online) how to do this.
I only really need to know a couple points to plot it... when x = 0, and then the earliest crossing of the x-axis.
| To help you better, I need to know the specific function or equation you are working with. If you provide that information, I can guide you on how to find the points you need for your plot. | 2,954 | [
{
"from": "human",
"value": "I have to do this for an upcoming exam, but cannot find anywhere (in the textbook or online) how to do this.\nI only really need to know a couple points to plot it... when x = 0, and then the earliest crossing of the x-axis.\n"
},
{
"from": "gpt",
"value": "To help y... |
In measure theory, we have "lambda systems" and "pi systems". Pearl's message passing algorithm has "lambda messages" and "pi messages". Is there a reason that lambda and pi go together?
| Yes, the terms "lambda" and "pi" in both measure theory and Pearl's message passing algorithm draw from the same mathematical foundation, particularly in relation to system structures, functional mappings, and algebraic properties.
In measure theory, a **lambda system** (or **λ-system**) is a collection of sets that i... | 1,124 | [
{
"from": "human",
"value": "In measure theory, we have \"lambda systems\" and \"pi systems\". Pearl's message passing algorithm has \"lambda messages\" and \"pi messages\". Is there a reason that lambda and pi go together?\n"
},
{
"from": "gpt",
"value": "Yes, the terms \"lambda\" and \"pi\" ... |
Is there symbolism (or even a name) for groups of statements in which any fixed-number of them imply all the rest?
For example, in linear algebra, a basis is sometimes defined as a set of n-dimensional vectors which:
Is linearly independent
Spans the vector space
Contains exactly n elements
However, it is then shown ... | Yes, there is a concept in logic and mathematics known as a **dependency graph** or **implicational structure** that can capture the relationships among a set of statements in which any fixed number of them implies the others.
For your specific case involving the basis in linear algebra, we can express this relationsh... | 171 | [
{
"from": "human",
"value": "Is there symbolism (or even a name) for groups of statements in which any fixed-number of them imply all the rest?\nFor example, in linear algebra, a basis is sometimes defined as a set of n-dimensional vectors which:\n\nIs linearly independent\nSpans the vector space\nContains ... |
Basic definitions: a tiling of d-dimensional Euclidean space is a decomposition of that space into polyhedra such that there is no overlap between their interiors, and every point in the space is contained in some one of the polyhedra.
A vertex-uniform tiling is a tiling such that each vertex figure is the same: each v... | An example of a vertex-uniform tiling that is not vertex-transitive can be found in the context of three-dimensional space (d = 3). One of the classic examples is the **truncated octahedral tiling**, which consists of truncated octahedra and hexagonal faces.
In this tiling:
- Each vertex has the same vertex figure, w... | 43 | [
{
"from": "human",
"value": "Basic definitions: a tiling of d-dimensional Euclidean space is a decomposition of that space into polyhedra such that there is no overlap between their interiors, and every point in the space is contained in some one of the polyhedra.\nA vertex-uniform tiling is a tiling such t... |
Ok, that's a theorem. As it says in this question.
I just cannot understand how mathematics can prove such a thing; what does exactly this theorem say? How to prove it mathematically?
Thanks!
| It looks like your question references a specific theorem, but you didn't specify which theorem you're referring to. Mathematics encompasses a wide range of theorems, each with its own implications and proofs.
To help you better, could you please clarify which theorem you are talking about? Once I know the specific th... | 2,617 | [
{
"from": "human",
"value": "Ok, that's a theorem. As it says in this question.\nI just cannot understand how mathematics can prove such a thing; what does exactly this theorem say? How to prove it mathematically?\nThanks!\n"
},
{
"from": "gpt",
"value": "It looks like your question references a... |
I was reading this paper on position auctions for web ads. Basically, there are N slots each with an expected number of clicks (in a particular time period) $x$. Each agent makes a bid $B_i$ of how much they are willing to pay per click. The bids are put in decreasing order and agent who makes the $i^\text{th}$ highest... | The symmetric Nash equilibrium you described in the context of position auctions for web ads is indeed a generalization of the concept of Nash equilibrium in game theory, where the conditions apply uniformly across all players (or agents) involved. In this case, the equilibrium conditions ensure that no agent has an in... | 336 | [
{
"from": "human",
"value": "I was reading this paper on position auctions for web ads. Basically, there are N slots each with an expected number of clicks (in a particular time period) $x$. Each agent makes a bid $B_i$ of how much they are willing to pay per click. The bids are put in decreasing order and ... |
How to proceed in this problem ?
| Of course! Please provide the details of the problem you're facing, and I'll do my best to help you figure out how to proceed. | 3,167 | [
{
"from": "human",
"value": "How to proceed in this problem ?\n"
},
{
"from": "gpt",
"value": "Of course! Please provide the details of the problem you're facing, and I'll do my best to help you figure out how to proceed."
}
] |
The Wikipedia article on long division explains the different notations. I still use the European notation I learned in elementary school in Colombia. I had difficulty adapting to the US/UK notation when I moved to the US. However, I did enjoy seeing my classmates' puzzled faces in college whenever we had a professor t... | I don't have personal experiences or preferences, but I can tell you about the different notations for long division! The European notation typically uses a horizontal bar (or vinculum) to separate the dividend and divisor, whereas the US/UK notation often uses a "long division" symbol that resembles an elongated divis... | 1,845 | [
{
"from": "human",
"value": "The Wikipedia article on long division explains the different notations. I still use the European notation I learned in elementary school in Colombia. I had difficulty adapting to the US/UK notation when I moved to the US. However, I did enjoy seeing my classmates' puzzled faces... |
The word "quad" generally means 4. Quadratics don't have 4 of anything. Can anyone explain where the name comes from?
| The term "quadratic" comes from the Latin word "quadratus," which means "square." In the context of algebra, a quadratic equation is one that can be expressed in the standard form:
\[ ax^2 + bx + c = 0 \]
where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \). The key characteristic of a quadratic equa... | 1,106 | [
{
"from": "human",
"value": "The word \"quad\" generally means 4. Quadratics don't have 4 of anything. Can anyone explain where the name comes from?\n"
},
{
"from": "gpt",
"value": "The term \"quadratic\" comes from the Latin word \"quadratus,\" which means \"square.\" In the context of algebra,... |
Prime numbers are numbers with no factors other than one and itself.
Factors of a number are always lower or equal to than a given number; so, the larger the number is, the larger the pool of "possible factors" that number might have.
So the larger the number, it seems like the less likely the number is to be a prime.
... | The concept you are referring to touches on the distribution of prime numbers. Indeed, as numbers get larger, the density of primes decreases, meaning that there are more composite numbers (numbers that are not prime) among the larger integers. However, there is no finite "critical point" beyond which every integer mus... | 47 | [
{
"from": "human",
"value": "Prime numbers are numbers with no factors other than one and itself.\nFactors of a number are always lower or equal to than a given number; so, the larger the number is, the larger the pool of \"possible factors\" that number might have.\nSo the larger the number, it seems like ... |
In maths, you can use something as
simple as statistical analysis to
intuit the theory, and in comp-science
you can use simulators.
Is the above statement true for maths ?
| The statement you provided is somewhat true but requires clarification. In mathematics, while statistical analysis can be used to derive insights and intuit theories, the discipline is fundamentally different from empirical sciences like computer science.
Mathematics often relies on rigorous proofs, logical reasoning... | 2,480 | [
{
"from": "human",
"value": "\nIn maths, you can use something as\n simple as statistical analysis to\n intuit the theory, and in comp-science\n you can use simulators.\n\nIs the above statement true for maths ?\n"
},
{
"from": "gpt",
"value": "The statement you provided is somewhat true but ... |
As it is required for most students who wish to do a Ph.D in maths in the US to sit the GRE subject specific mathematics exam, I hope this question will be of interest to the mathematical community and will not be closed.
Essentially, the exam was "rescaled" (made more difficult) in 2001 and I have only been able to fi... | While the availability of past GRE Mathematics Subject Test papers can be quite limited, there are some resources you can explore to find practice materials:
1. **ETS Official Resources**: The Educational Testing Service (ETS), which administers the GRE, occasionally releases official practice materials and sample que... | 1,477 | [
{
"from": "human",
"value": "As it is required for most students who wish to do a Ph.D in maths in the US to sit the GRE subject specific mathematics exam, I hope this question will be of interest to the mathematical community and will not be closed.\nEssentially, the exam was \"rescaled\" (made more diffic... |
I keep looking at this picture and its driving me crazy. How can the smaller circle travel the same distance when its circumference is less than the entire wheel?
| This is a classic question that often comes up when discussing the motion of circles and wheels, particularly in relation to rolling without slipping. When a smaller circle rolls around a larger circle (like a wheel), both can have the same angular displacement despite their different circumferences.
Here’s how this w... | 45 | [
{
"from": "human",
"value": "I keep looking at this picture and its driving me crazy. How can the smaller circle travel the same distance when its circumference is less than the entire wheel? \n\n"
},
{
"from": "gpt",
"value": "This is a classic question that often comes up when discussing the m... |
Say I have $\sqrt{\frac{2}{9}}$. How would I convert this to a mixed radical where the radicand is a whole number?
I can convert the $\sqrt{8}$ to a mixed radical easily, $2\sqrt{2}$, but this fraction is tripping me up.
| To convert the expression \(\sqrt{\frac{2}{9}}\) to a mixed radical where the radicand is a whole number, we can start by simplifying the square root of a fraction.
Using the property of square roots that states \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\), we can rewrite the expression:
\[
\sqrt{\frac{2}{9}} ... | 1,506 | [
{
"from": "human",
"value": "Say I have $\\sqrt{\\frac{2}{9}}$. How would I convert this to a mixed radical where the radicand is a whole number? \nI can convert the $\\sqrt{8}$ to a mixed radical easily, $2\\sqrt{2}$, but this fraction is tripping me up.\n"
},
{
"from": "gpt",
"value": "To conv... |
I've been trying to find answer to this question for some time but in every document I've found so far it's taken for granted that reader know what $\mathbf ℝ^+$ is.
| The notation \( \mathbb{R}^+ \) typically refers to the set of all positive real numbers. In mathematical terms, this can be expressed as:
\[
\mathbb{R}^+ = \{ x \in \mathbb{R} \mid x > 0 \}
\]
This means that \( \mathbb{R}^+ \) includes all real numbers greater than zero, and it does not include zero itself or any n... | 391 | [
{
"from": "human",
"value": "I've been trying to find answer to this question for some time but in every document I've found so far it's taken for granted that reader know what $\\mathbf ℝ^+$ is.\n"
},
{
"from": "gpt",
"value": "The notation \\( \\mathbb{R}^+ \\) typically refers to the set of a... |
The arithmetic hierarchy defines the $\Pi_1$ formulae of arithmetic to be formulae that are provably equivalent to a formula in prenex normal form that only has universal quantifiers, and $\Sigma_1$ if it is provably equivalent to a prenex normal form with only existential quantifiers.
A formula is $\Delta_1$ if it is ... | The $\Delta_1$ formulae are often called recursive because they can be characterized in a way that relates them to computable functions. Specifically, $\Delta_1$ formulae are those that can be expressed in both $\Pi_1$ and $\Sigma_1$ forms, which means they can be reformulated in terms of quantifiers that exclusively i... | 63 | [
{
"from": "human",
"value": "The arithmetic hierarchy defines the $\\Pi_1$ formulae of arithmetic to be formulae that are provably equivalent to a formula in prenex normal form that only has universal quantifiers, and $\\Sigma_1$ if it is provably equivalent to a prenex normal form with only existential qua... |
I have a collection of 3D points in the standard $x$, $y$, $z$ vector space. Now I pick one of the points $p$ as a new origin and two other points $a$ and $b$ such that $a - p$ and $b - p$ form two vectors of a new vector space. The third vector of the space I will call $x$ and calculate that as the cross product of th... | You are on the right track with the idea of creating a new vector space based on a point \( p \) (which you choose as the origin) and two vectors \( a - p \) and \( b - p \). The third vector, \( x \), defined as the cross product of these two vectors, will serve as the third basis vector of your new vector space.
To ... | 42 | [
{
"from": "human",
"value": "I have a collection of 3D points in the standard $x$, $y$, $z$ vector space. Now I pick one of the points $p$ as a new origin and two other points $a$ and $b$ such that $a - p$ and $b - p$ form two vectors of a new vector space. The third vector of the space I will call $x$ and ... |
Suppose there are two correlated random variable and having very small correlation coefficient (order of 10-1). Is it valid to approximate it as independent random variables?
| In statistics, correlation measures the degree to which two random variables move in relation to one another. A correlation coefficient close to zero indicates a weak linear relationship between the variables, but it does not imply independence.
Even if two random variables have a very small correlation coefficient (e... | 623 | [
{
"from": "human",
"value": "Suppose there are two correlated random variable and having very small correlation coefficient (order of 10-1). Is it valid to approximate it as independent random variables? \n"
},
{
"from": "gpt",
"value": "In statistics, correlation measures the degree to which t... |
Is there a generally accepted formal notation for denoting the number of columns a matrix has (e.g to use in pseudocode/algorithm environment in LaTeX)? Something I could use in the description of an algorithm like:
if horizontaldim(V) > x then
end if
or
if size(V,2) > x then
end if
or should I just use a descript... | In pseudocode or algorithm descriptions, it's common to use notation that is both concise and clear. While there isn't a universally accepted formal notation like mathematical notation for the number of columns in a matrix, there are a few conventions that are widely recognized in the literature:
1. **Using `size(V, 2... | 571 | [
{
"from": "human",
"value": "Is there a generally accepted formal notation for denoting the number of columns a matrix has (e.g to use in pseudocode/algorithm environment in LaTeX)? Something I could use in the description of an algorithm like: \nif horizontaldim(V) > x then\nend if\n\nor \nif size(V,2) > x... |
This answer suggests that there are explicit polynomial equations for which the existence
(or nonexistence) of integer solutions is unprovable. How can this be?
| The question of whether certain polynomial equations have integer solutions (also known as Diophantine equations) is deeply connected to several important concepts in logic and mathematics, particularly in the context of Gödel's incompleteness theorems and the nature of undecidable propositions.
1. **Gödel's Incomplet... | 92 | [
{
"from": "human",
"value": "This answer suggests that there are explicit polynomial equations for which the existence \n(or nonexistence) of integer solutions is unprovable. How can this be?\n"
},
{
"from": "gpt",
"value": "The question of whether certain polynomial equations have integer solut... |
What is the name for I can only describe as the circumference of a sphere. I mean like the equator on a planet. What is the line called which goes around the entire sphere, it can be anywhere (so not just in the middle of the top and the bottom, if your talking about a planet or something) and the length of it equals 2... | The line you are describing is called a **great circle**. A great circle is the largest circle that can be drawn on a sphere, and it is formed by the intersection of the sphere with a plane that passes through the center of the sphere. All great circles have the same circumference, which is equal to \(2\pi r\), where \... | 2,246 | [
{
"from": "human",
"value": "What is the name for I can only describe as the circumference of a sphere. I mean like the equator on a planet. What is the line called which goes around the entire sphere, it can be anywhere (so not just in the middle of the top and the bottom, if your talking about a planet or... |
Page 13 of Tropical Algebraic Geometry by Itenberg, Mikhalkin, and Shustin mentions 1-valent vertices, but I haven't been able to find a source that actually defines this term or managed to guess the definition myself. Either a definition or reference would be greatly appreciated. Thanks!
| In the context of tropical geometry, a **1-valent vertex** refers to a vertex in a tropical curve (or a tropical graph) that has exactly one edge connected to it. In simpler terms, a 1-valent vertex is a vertex that is an endpoint of a single edge.
In graph theory, vertices (or nodes) can have varying degrees based o... | 1,943 | [
{
"from": "human",
"value": "Page 13 of Tropical Algebraic Geometry by Itenberg, Mikhalkin, and Shustin mentions 1-valent vertices, but I haven't been able to find a source that actually defines this term or managed to guess the definition myself. Either a definition or reference would be greatly appreciat... |
If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already:
Let $a = b$.
Then $a^2 = ab$
$a^2 - b^2 = ab - b^2$
Factor to $(a-b)(a+b) = b(a-b)$
Then divide out $(a-b)$ to get $a+b = b$
Since $a = b$, then $b+b = b$
The... | The proof you've provided is a classic example of a fallacious argument often referred to as "dividing by zero." Here's a step-by-step breakdown of the proof along with the precise reason why it is fallacious:
1. Start with the assumption \( a = b \).
2. From \( a = b \), deduce \( a^2 = ab \) (this step is valid).
3.... | 28 | [
{
"from": "human",
"value": "If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already:\n\n Let $a = b$.\n Then $a^2 = ab$\n $a^2 - b^2 = ab - b^2$\n Factor to $(a-b)(a+b) = b(a-b)$\n Then divide out $(a-b)$ to ge... |
Which is a better textbook - Dugundji or Munkres? I'm concerned with clarity of exposition and explanation of motivation, etc.
| Both Dugundji and Munkres are respected textbooks in the field of topology, but they serve slightly different purposes and audiences, which can affect their clarity and exposition depending on what you're looking for.
**Dugundji's "Topology":**
- Dugundji's book is often praised for its thoroughness and rigor. It is a... | 225 | [
{
"from": "human",
"value": "Which is a better textbook - Dugundji or Munkres? I'm concerned with clarity of exposition and explanation of motivation, etc.\n"
},
{
"from": "gpt",
"value": "Both Dugundji and Munkres are respected textbooks in the field of topology, but they serve slightly differe... |
I love R. Courant and F. John's Introduction to Calculus and Analysis because of its wide coverage, precise description and friendly written style.
Are there any classic textbooks like it on ODE/PDE?
thanks.
| If you appreciate the style and comprehensiveness of R. Courant and F. John's *Introduction to Calculus and Analysis*, you might enjoy the following classic textbooks on ordinary differential equations (ODEs) and partial differential equations (PDEs):
1. **Ordinary Differential Equations**:
- **"Ordinary Differenti... | 769 | [
{
"from": "human",
"value": "I love R. Courant and F. John's Introduction to Calculus and Analysis because of its wide coverage, precise description and friendly written style.\nAre there any classic textbooks like it on ODE/PDE?\nthanks.\n"
},
{
"from": "gpt",
"value": "If you appreciate the st... |
I was wondering if there was a resource that listed known algebraic groups and their corresponding coordinate rings.
Edit: The previous wording was terrible.
Given an algebraic group $G$, with Borel subgroup $B$ we can form the Flag Variety $G/B$ which is projective. I am hoping for a list of the graded ring $R$ such t... | Yes, there are several resources that discuss algebraic groups, their flag varieties, and the corresponding coordinate rings. In particular, for a given algebraic group \( G \) with a Borel subgroup \( B \), the flag variety \( G/B \) can be constructed, and its coordinate ring can be expressed in terms of certain grad... | 305 | [
{
"from": "human",
"value": "I was wondering if there was a resource that listed known algebraic groups and their corresponding coordinate rings.\nEdit: The previous wording was terrible.\nGiven an algebraic group $G$, with Borel subgroup $B$ we can form the Flag Variety $G/B$ which is projective. I am hopi... |
I have a two state dynamical system. The two state variables are $P$ and $Z$ and $a,b,c,d$ are parameters. The system equations are:
\begin{equation*}
\frac{dP}{dt}=a\cdot P-b\cdot PZ=P\left(a-bZ\right), \\
\frac{dZ}{dt}=c\cdot PZ-d\cdot Z=Z\left(cP-d\right).
\end{equation*}
Multiplying both sides by $\frac{1}{PZ}$ giv... | It looks like you've derived a conserved quantity \( E(P, Z) \) for your two-state dynamical system correctly, and it's good to see that you've recognized a calculus mistake along the way.
To clarify your final result, you found that the total time derivative of a quantity \( E(P, Z) \) is zero. This means \( E(P, Z) ... | 1,468 | [
{
"from": "human",
"value": "I have a two state dynamical system. The two state variables are $P$ and $Z$ and $a,b,c,d$ are parameters. The system equations are:\n\\begin{equation*}\n\\frac{dP}{dt}=a\\cdot P-b\\cdot PZ=P\\left(a-bZ\\right), \\\\\n\\frac{dZ}{dt}=c\\cdot PZ-d\\cdot Z=Z\\left(cP-d\\right).\n\\... |
I'm quite positive that I saw before an encyclopedia (more accurately, dictionary/glossary) on Elsevier website. I remember I looked up the definition of a hyperplane there, few months ago. But I can't find it through Google anymore.
Does anyone know the link?
| It sounds like you might be referring to the "Encyclopedia of Mathematics" or a specific dictionary available on the Elsevier website. However, since I cannot browse the internet to find current links, I recommend checking the Elsevier main site or searching for specific resources such as:
1. **Elsevier's ScienceDirec... | 2,163 | [
{
"from": "human",
"value": "I'm quite positive that I saw before an encyclopedia (more accurately, dictionary/glossary) on Elsevier website. I remember I looked up the definition of a hyperplane there, few months ago. But I can't find it through Google anymore.\nDoes anyone know the link?\n"
},
{
"... |
Does it exist a free interactive geometry software, like GeoGebra, which works for 3D geometry? I would be able to draw spheres, great circles, and so on.
| Yes, there are several free interactive geometry software options that support 3D geometry and allow you to draw shapes like spheres and great circles. Here are a few notable ones:
1. **GeoGebra 3D Graphing**: While you mentioned GeoGebra, it's worth noting that GeoGebra has a specific 3D Graphing tool that allows you... | 936 | [
{
"from": "human",
"value": "Does it exist a free interactive geometry software, like GeoGebra, which works for 3D geometry? I would be able to draw spheres, great circles, and so on.\n"
},
{
"from": "gpt",
"value": "Yes, there are several free interactive geometry software options that support ... |
How would I go about it?
| Could you provide more context or specify what you are referring to? Whether it's a task, project, or a specific question, I'd be glad to help you out! | 3,727 | [
{
"from": "human",
"value": "How would I go about it?\n"
},
{
"from": "gpt",
"value": "Could you provide more context or specify what you are referring to? Whether it's a task, project, or a specific question, I'd be glad to help you out!"
}
] |
Ok, so this might not be a permutation or combination but I'm curious as to what the answer is.
Question: There are 3 groups and 5 Objects in each group. How many combinations are there if your choosing one from each group?
Yes this is in reference to Call of Duty: Black Ops perks lol.
| To find the total number of combinations when choosing one object from each of the three groups, you can simply multiply the number of choices in each group together.
Since there are 3 groups and each group has 5 objects, the calculation would be:
\[
\text{Total combinations} = (\text{Choices from Group 1}) \times (... | 2,750 | [
{
"from": "human",
"value": "Ok, so this might not be a permutation or combination but I'm curious as to what the answer is.\nQuestion: There are 3 groups and 5 Objects in each group. How many combinations are there if your choosing one from each group?\nYes this is in reference to Call of Duty: Black Ops p... |
How can I calculate $27^{41}\ \mathrm{mod}\ 77$ as simple as possible?
I already know that $27^{60}\ \mathrm{mod}\ 77 = 1$ because of Euler’s theorem:
$$ a^{\phi(n)}\ \mathrm{mod}\ n = 1 $$
and
$$ \phi(77) = \phi(7 \cdot 11) = (7-1) \cdot (11-1) = 60 $$
I also know from using modular exponentiation that $27^{10} \mathr... | Yes, you can use the fact that \(27^{60} \equiv 1 \mod 77\) to simplify the calculation of \(27^{41} \mod 77\).
Since \(27^{60} \equiv 1 \mod 77\), we can reduce \(41\) modulo \(60\). In this case, \(41\) is already less than \(60\), so we have:
\[
27^{41} \equiv 27^{41} \mod 77.
\]
Now, since you have already disco... | 370 | [
{
"from": "human",
"value": "How can I calculate $27^{41}\\ \\mathrm{mod}\\ 77$ as simple as possible?\nI already know that $27^{60}\\ \\mathrm{mod}\\ 77 = 1$ because of Euler’s theorem:\n$$ a^{\\phi(n)}\\ \\mathrm{mod}\\ n = 1 $$\nand\n$$ \\phi(77) = \\phi(7 \\cdot 11) = (7-1) \\cdot (11-1) = 60 $$\nI also... |
Doing a little practice exercise, and came across the following question:
There are three functions: $f(x)$, $g(x)$, and $h(x)$.
Find $f(x)$ and $g(x)$, such that $h(x) = f \circ g(x)$ and $g(x) = \sqrt{4} + 8$.
The value of $h(x)$ is defined as $h(x) = (\sqrt{4} + 8) ^ 4$.
I'm not really sure how to approach the pro... | Let's break down the problem step by step.
You are given that \( h(x) = f \circ g(x) \) and that \( g(x) = \sqrt{4} + 8 \). First, let's simplify \( g(x) \).
We know that \( \sqrt{4} = 2 \), so:
\[
g(x) = 2 + 8 = 10
\]
Now that we have \( g(x) \), we can express \( h(x) \) in terms of \( g(x) \). The given expressi... | 1,061 | [
{
"from": "human",
"value": "Doing a little practice exercise, and came across the following question:\nThere are three functions: $f(x)$, $g(x)$, and $h(x)$. \nFind $f(x)$ and $g(x)$, such that $h(x) = f \\circ g(x)$ and $g(x) = \\sqrt{4} + 8$. \nThe value of $h(x)$ is defined as $h(x) = (\\sqrt{4} + 8) ^ ... |
I'd like to know the name of this kind of polynomial
$p(x)=x^n+a_{1}x^{n-1}+\ldots+a_{n-1}x+1$
where the $a_{i}\in\lbrace0,1\rbrace$.
Thanks.
| The polynomial you described,
\[
p(x) = x^n + a_1 x^{n-1} + a_2 x^{n-2} + \ldots + a_{n-1} x + 1
\]
where each \( a_i \) is either \( 0 \) or \( 1 \), is known as a **binary polynomial** or **polynomial over the binary field** \( \mathbb{F}_2 \).
In the context of coding theory and combinatorial design, these poly... | 1,270 | [
{
"from": "human",
"value": "I'd like to know the name of this kind of polynomial\n$p(x)=x^n+a_{1}x^{n-1}+\\ldots+a_{n-1}x+1$\nwhere the $a_{i}\\in\\lbrace0,1\\rbrace$.\nThanks.\n"
},
{
"from": "gpt",
"value": "The polynomial you described, \n\n\\[\np(x) = x^n + a_1 x^{n-1} + a_2 x^{n-2} + \\ldo... |
In a recent discussion, someone told me tuples in the modern meaning (in particular, tuples are heterogeneous: that is, different elements of a tuple can belong to different sets/have different types) first appeared in Codd's tuple calculus. I was surprised it would be so late, but searching Google Books before 1970, ... | Edgar F. Codd is indeed credited with introducing the concept of tuples in the context of relational databases and the relational model, specifically with his work on tuple calculus in the early 1970s. In his seminal paper "A Relational Model of Data for Large Shared Data Banks" (published in 1970), Codd introduced the... | 719 | [
{
"from": "human",
"value": "In a recent discussion, someone told me tuples in the modern meaning (in particular, tuples are heterogeneous: that is, different elements of a tuple can belong to different sets/have different types) first appeared in Codd's tuple calculus. I was surprised it would be so late,... |
When I used to compete in Olympiad Competitions back in high school, a decent number of the easier geometry questions were solvable by what we called a geometry bash. Basically, you'd label every angle in the diagram with the variable then use a limited set of basic geometry operations to find relations between the ele... | Yes, there are several software tools and systems designed to solve geometric problems, some of which can handle what you're describing—labeled angle relationships and basic geometric constructions. Here are a few notable ones:
1. **Geogebra**: This is an interactive geometry, algebra, statistics, and calculus applica... | 104 | [
{
"from": "human",
"value": "When I used to compete in Olympiad Competitions back in high school, a decent number of the easier geometry questions were solvable by what we called a geometry bash. Basically, you'd label every angle in the diagram with the variable then use a limited set of basic geometry ope... |
Each person from a group of 3 people can choose his dish from a menu of 5 options. Knowing that each person eats only 1 dish what are the number of different orders the waiter can ask the chef?
| To determine the number of different orders the waiter can ask the chef based on the choices made by the 3 people from a menu of 5 options, we can break down the problem as follows:
1. **Choices for Each Person**: Each of the 3 people has 5 options to choose from. This means for each person, there are 5 possible choic... | 1,930 | [
{
"from": "human",
"value": "Each person from a group of 3 people can choose his dish from a menu of 5 options. Knowing that each person eats only 1 dish what are the number of different orders the waiter can ask the chef?\n"
},
{
"from": "gpt",
"value": "To determine the number of different ord... |
Let us suppose to have a set of $N$ elements and to choose $m$ elements out of the set. I know I can do it in $ N \choose m $ ways; but I want to know how many of these combinations contain $k$ predefined elements (say $1,2,...,k$).
| To determine the number of combinations of \(m\) elements from a set of \(N\) elements that include \(k\) predefined elements, we can use combinatorial reasoning.
1. **Choose the predefined elements**: Since we are required to include the \(k\) predefined elements (let's denote them as \(1, 2, \ldots, k\)), we will f... | 910 | [
{
"from": "human",
"value": "Let us suppose to have a set of $N$ elements and to choose $m$ elements out of the set. I know I can do it in $ N \\choose m $ ways; but I want to know how many of these combinations contain $k$ predefined elements (say $1,2,...,k$). \n"
},
{
"from": "gpt",
"value": ... |
Note: This question has been posted on StackOverflow. I have moved it here because:
I am curious about the answer
The OP has not shown any interest in moving it himself
In the Communications of the ACM, August 2008 "Puzzled" column, Peter Winkler asked the following question:
On the table before us are 10 dots,
a... | You're correct in observing that the probabilistic nature of the proof implies that it is possible for all 10 dots to be covered in many configurations, but it doesn't guarantee that this will happen for every specific arrangement of the dots.
The key point of the proof, as presented by Peter Winkler, is that when yo... | 1,750 | [
{
"from": "human",
"value": "Note: This question has been posted on StackOverflow. I have moved it here because:\n\nI am curious about the answer\nThe OP has not shown any interest in moving it himself\n\n\nIn the Communications of the ACM, August 2008 \"Puzzled\" column, Peter Winkler asked the following q... |
An off-topic question asked at Mathoverflow by Andrew Stacey; but one which fits here:
I'm interested in hearing of examples of mathematical (or, at a pinch, scientific) websites with serious content where the design of the website actually makes it easy to read and absorb the material. To be absolutely clear, the math... | Here are several examples of websites and platforms where the design aids in the readability and absorption of mathematical or scientific content:
1. **Khan Academy**:
- The design of Khan Academy provides a clean interface that is easy to navigate. Lessons are broken down into manageable segments, often accompanie... | 247 | [
{
"from": "human",
"value": "An off-topic question asked at Mathoverflow by Andrew Stacey; but one which fits here:\nI'm interested in hearing of examples of mathematical (or, at a pinch, scientific) websites with serious content where the design of the website actually makes it easy to read and absorb the ... |
Consider a situation where a person has partial knowledge, but we have a more complete picture. For example, suppose that we want to know the probability that a fish is red. Suppose that the person with partial information knows 1/3 of all fish are red, but we know that the particular species is actually red 2/3 of the... | In the scenario you describe, the two probabilities can be considered as different types of probabilities based on the information available. The first probability, which the person with partial knowledge uses (1/3), can be referred to as the **prior probability** or **marginal probability**. This represents the person... | 1,020 | [
{
"from": "human",
"value": "Consider a situation where a person has partial knowledge, but we have a more complete picture. For example, suppose that we want to know the probability that a fish is red. Suppose that the person with partial information knows 1/3 of all fish are red, but we know that the part... |
The classic text of Deligne and Rapoport giving a model for the mod p reduction of the modular curve is found in a "Lecture Notes in Mathematics" volume.
Has it been translated from the original french and/or LaTeX-ed so that it can be searched more easily?
| Yes, the classic text by Pierre Deligne and Michael Rapoport, which provides a model for the mod \( p \) reduction of the modular curve, is available in the "Lecture Notes in Mathematics" series, specifically in volume 349. The text was originally written in French and has been translated into English.
The English ver... | 2,392 | [
{
"from": "human",
"value": "The classic text of Deligne and Rapoport giving a model for the mod p reduction of the modular curve is found in a \"Lecture Notes in Mathematics\" volume.\nHas it been translated from the original french and/or LaTeX-ed so that it can be searched more easily?\n"
},
{
"f... |
What I currently meet are all fiber bundles.
| It sounds like you might be referencing the concept of fiber bundles in mathematics, particularly in the context of topology or differential geometry. A fiber bundle is a structure that consists of a "total space," a "base space," and "fibers" that are associated with each point in the base space.
In simpler terms, y... | 1,644 | [
{
"from": "human",
"value": "What I currently meet are all fiber bundles.\n"
},
{
"from": "gpt",
"value": "It sounds like you might be referencing the concept of fiber bundles in mathematics, particularly in the context of topology or differential geometry. A fiber bundle is a structure that con... |
I want to refer to functions of the form $f(x) = \sum_{i=1}^n a_i x^{\alpha_i}$ where $\alpha_i < 1$.
This is not a polynomial, because $\alpha_i$ could be just real arbitrary numbers (though positive ones).
Is there a name for this class of functions that I can refer to? Wikipedia does claim these are not polynomials.... | The functions of the form \( f(x) = \sum_{i=1}^n a_i x^{\alpha_i} \) where \( \alpha_i < 1 \) are commonly referred to as "fractional power functions" or "power series of fractional order." Since the exponents \( \alpha_i \) are not restricted to being non-negative integers, these functions do not fall under the tradit... | 2,050 | [
{
"from": "human",
"value": "I want to refer to functions of the form $f(x) = \\sum_{i=1}^n a_i x^{\\alpha_i}$ where $\\alpha_i < 1$.\nThis is not a polynomial, because $\\alpha_i$ could be just real arbitrary numbers (though positive ones).\nIs there a name for this class of functions that I can refer to? ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.