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mlfoundations-dev
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math_375000_samples

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I solved the following problem from by book, but the answer of this problem at the end of book is $x \leq 3$. Please tell me how I can get this answer.	To help you better, could you please provide the specific problem that you solved? Once I have the details, I'll be able to guide you through the solution and how to arrive at the answer \( x \leq 3 \).	672	[ { "from": "human", "value": "I solved the following problem from by book, but the answer of this problem at the end of book is $x \\leq 3$. Please tell me how I can get this answer.\n\n\n" }, { "from": "gpt", "value": "To help you better, could you please provide the specific problem that you so...
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 certain types of vertex-uniform tilings of three-dimensional space, particularly those based on the Cartesian product of polygons. One classical example is the truncated octahedral tiling in three-dimensional space, which is known ...	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...
I know that the product of two Gaussian functions is also Gaussian function. This is stated in Wikipedia, but I might need to cite a classical (text)book/paper stating this result. A book containing the proof is welcome. Could you name such a book?	One classical reference that discusses the product of Gaussian functions and provides a proof is "The Gaussian Integral" by G. B. Folland in his book "Real Analysis: Modern Techniques and Their Applications". While this book primarily focuses on real analysis, it does touch on properties of Gaussian functions and integ...	830	[ { "from": "human", "value": "I know that the product of two Gaussian functions is also Gaussian function. This is stated in Wikipedia, but I might need to cite a classical (text)book/paper stating this result. A book containing the proof is welcome. Could you name such a book? \n" }, { "from": "gpt"...
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. This includes all real numbers that are greater than zero. In mathematical terms, you can express it as: \[ \mathbb{R}^+ = \{ x \in \mathbb{R} \mid x > 0 \} \] This set includes numbers like \(1, 2.5, \pi, \sqrt{2}\), and so forth...	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 all...
Here is a classic theorem of Herstein: $G$ is a finite group, $M$ a maximal subgroup, which is abelian. Then $G$ is solvable. The proof is pretty easy, but it uses character theory (specifically, Frobenius' theorem on Frobenius groups). Is there a character-theory-free proof? To get things going, note that we can red...	To prove that if \( G \) is a finite group and \( M \) is a maximal abelian subgroup of \( G \), then \( G \) is solvable without using character theory, we can follow these steps: 1. Use the Hall subgroup condition: Since we can assume \( M \) is a Hall subgroup of \( G \) (meaning that \( M \) is a maximal subgr...	355	[ { "from": "human", "value": "Here is a classic theorem of Herstein:\n$G$ is a finite group, $M$ a maximal subgroup, which is abelian. Then $G$ is solvable.\nThe proof is pretty easy, but it uses character theory (specifically, Frobenius' theorem on Frobenius groups). Is there a character-theory-free proof...
For two matrices $\textbf{S}$ and $\textbf{T}$, a proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ is given below in the diagrammatic tensor notation. Here $\det$ denotes the determinant. Why can the antisymmetrizing bar be inserted in the middle because "there is already antisymmetry in the index lines"?...	In the context of the diagrammatic tensor notation you mentioned, we can better understand the manipulation of the antisymmetrizing bar by considering the nature of tensors and the properties of determinants. The determinant of a matrix can be expressed as a sum over permutations of its rows (or columns), weighted by ...	677	[ { "from": "human", "value": "For two matrices $\\textbf{S}$ and $\\textbf{T}$, a proof of $\\det(\\textbf{ST})=\\det(\\textbf{S})\\det(\\textbf{T})$ is given below in the diagrammatic tensor notation.\n\nHere $\\det$ denotes the determinant.\nWhy can the antisymmetrizing bar be inserted in the middle becaus...
Back in the day, before approximation methods like splines became vogue in my line of work, one way of computing the area under an empirically drawn curve was to painstakingly sketch it on a piece of graphing paper (usually with the assistance of a French curve) along with the axes, painstakingly cut along the curve, w...	In the context of calculating the area under a curve that crosses the horizontal axis multiple times, you were correct in arguing that the pieces should have been weighed separately. The reason for this is that in mathematical terms, the area above the horizontal axis is considered positive, while the area below the ho...	918	[ { "from": "human", "value": "Back in the day, before approximation methods like splines became vogue in my line of work, one way of computing the area under an empirically drawn curve was to painstakingly sketch it on a piece of graphing paper (usually with the assistance of a French curve) along with the a...
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 find the number of ways to choose \( m \) elements from a set of \( N \) elements that includes \( k \) predefined elements (let's denote these elements as \( 1, 2, \ldots, k \)), we can break the problem into two parts: 1. Choosing the predefined elements: Since we want to include all \( k \) predefined elemen...	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": ...
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; however, there are specific aspects of the question and the phrasing of your response that may have led to your teacher marking you wrong. 1. Misinterpretation of Correlation: The question mentions a strong negative correlation between the propo...	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 ...
In $b^a = x$, $b$ is the base, a is the exponent and $x$ is the result of the operation. But in its logarithm counterpart, $\log_{b}(x) = a$, $b$ is still the base, and $a$ is now the result. What is $x$ called here? The exponent?	In the logarithmic equation \(\log_{b}(x) = a\), \(x\) is referred to as the argument of the logarithm. In this context, \(b\) is the base, \(a\) is the logarithm (which represents the exponent), and \(x\) is the value for which we are finding the logarithm. So to summarize: - In \(b^a = x\), \(x\) is the result of r...	1,002	[ { "from": "human", "value": "In $b^a = x$, $b$ is the base, a is the exponent and $x$ is the result of the operation. But in its logarithm counterpart, $\\log_{b}(x) = a$, $b$ is still the base, and $a$ is now the result. What is $x$ called here? The exponent?\n" }, { "from": "gpt", "value": "In...
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 discussion regarding computable numbers and their relation to finitistic mathematics indeed raises important philosophical questions. Let's break down the key concepts involved. 1. Computable Numbers: These are real numbers that can be computed to any desired precision by a finite, deterministic algorithm. In ...	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...
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 varies among different mathematical communities and contexts. There are two main conventions regarding the definition of natural numbers: 1. Including Zero: In this convention, natural numbers are defined as the set \(\{0, 1, 2, 3, \ldots\}\). This definition is often...	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...
What software do you use to accurately draw geometry diagrams?	There are several software options available for accurately drawing geometry diagrams, each with its own strengths. Here are some popular choices: 1. GeoGebra: A dynamic mathematics software that combines geometry, algebra, and calculus. It's widely used in education for creating interactive geometry diagrams. 2....	461	[ { "from": "human", "value": "What software do you use to accurately draw geometry diagrams?\n" }, { "from": "gpt", "value": "There are several software options available for accurately drawing geometry diagrams, each with its own strengths. Here are some popular choices:\n\n1. GeoGebra: A dy...
The title really say's it all, but once again is a curve's curvature invariant under rotation and uniform scaling?	Yes, the curvature of a curve is invariant under rigid transformations such as rotation and uniform scaling. 1. Rotation: When a curve is rotated, its shape is preserved, and thus the curvature at any given point on the curve remains the same. The geometric properties of the curve do not change with rotation; only...	307	[ { "from": "human", "value": "The title really say's it all, but once again is a curve's curvature invariant under rotation and uniform scaling?\n" }, { "from": "gpt", "value": "Yes, the curvature of a curve is invariant under rigid transformations such as rotation and uniform scaling.\n\n1. **Ro...
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 to hear that you're thinking critically about how to document your mathematical proofs! Here are some suggestions that balance between efficiency, ecological concerns, and trustworthiness: 1. Use a Reusable Notebook: Consider using a reusable notebook like the Rocketbook, which allows you to write by ha...	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...
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 understand the measure of the sum \( C + C \) where \( C \) is the Cantor set, we first clarify what we mean by the sum of two sets. The sum \( A + B \) for two sets \( A \) and \( B \) is defined as: \[ A + B = \{ a + b \mid a \in A, b \in B \} \] In the case of the Cantor set \( C \), we are interested in \( C +...	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": ...
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, which include functions like sinh (hyperbolic sine), cosh (hyperbolic cosine), and tanh (hyperbolic tangent), have various applications outside academia, particularly in fields such as engineering, physics, and computer graphics. Here are some practical examples of their uses: 1. **...	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...
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're delving deeper into computational complexity! Here are several advanced texts that should enhance your understanding of the subject: 1. "Computational Complexity" by Christos H. Papadimitriou - This is a classic text that covers a wide range of topics in complexity theory. It provide...	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...
A fun question I ask students or interviewees (in engineering) is: This is not my question, this is an example: Using only what you know now, how many cans of soda would you estimate are produced per day (on average) in the United States? For this question, the result doesn't matter so much as the process you...	A great estimation question that encourages creative thinking and problem-solving is: "How many golf balls would fit in a school bus?" This question prompts respondents to break down the problem into smaller components, make reasonable assumptions, and use rough calculations. Here’s how you might guide the though...	291	[ { "from": "human", "value": "A fun question I ask students or interviewees (in engineering) is:\n\nThis is not my question, this is an example:\n Using only what you know now, how many\n cans of soda would you estimate are\n produced per day (on average) in the\n United States?\n\nFor this question, the...
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 you are describing, defined by \( f(a) = U a U^{-1} \), where \( U \) is a unitary operator from Hilbert space \( H \) to Hilbert space \( K \), is commonly referred to as a unitary equivalence* or unitary representation. In mathematical notation, it can be denoted as \( f = \text{Ad}_U \), ...	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...
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 idea you're proposing relates to the density of prime numbers and their distribution among the integers. While it may seem intuitive to think that there could be a "critical point" where all larger numbers have factors other than one and itself, this is not the case. In fact, there is a well-established result in ...	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 ...
Let $A$ be a valuation ring of rank two. Then $A$ gives an example of a commutative ring such that $\mathrm{Spec}(A)$ is a noetherian topological space, but $A$ is non-noetherian. (Indeed, otherwise $A$ would be a discrete valuation ring.) Is there a concrete example of such a ring $A$?	Yes, a concrete example of a valuation ring \( A \) of rank two, which is non-Noetherian but has a Noetherian spectrum, is the ring of power series \( k[[x,y]] \) in two variables over a field \( k \). ### Details: 1. Valuation Ring Structure: The ring \( k[[x,y]] \) is a local ring where the unique maximal i...	302	[ { "from": "human", "value": "Let $A$ be a valuation ring of rank two. Then $A$ gives an example of a commutative ring such that $\\mathrm{Spec}(A)$ is a noetherian topological space, but $A$ is non-noetherian. (Indeed, otherwise $A$ would be a discrete valuation ring.)\nIs there a concrete example of such a...
Let $X$ be a compact metric space. Given a sequence $x_n \in X$ and an ultrafilter $\mathcal{U}$ on $\mathbb{N}$, we can define the "Banach limit" of ${x_n}$ with respect to $\mathcal{U}$. This limit is the unique element of $X$ such that every neighborhood of it contains a $\mathcal{U}$-large collection of the sequen...	Yes, you can define a translation-invariant limit for sequences taking values in an arbitrary compact metric space by utilizing a notion known as the "limit point" in the context of compact spaces. Specifically, you can consider the following approach: 1. Using the Compactness Property: In a compact metric space \...	414	[ { "from": "human", "value": "Let $X$ be a compact metric space. Given a sequence $x_n \\in X$ and an ultrafilter $\\mathcal{U}$ on $\\mathbb{N}$, we can define the \"Banach limit\" of ${x_n}$ with respect to $\\mathcal{U}$. This limit is the unique element of $X$ such that every neighborhood of it contains...
In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?	The Riemann-Zeta function is a special mathematical tool that helps us understand certain properties of numbers, especially prime numbers. In simple terms, you can think of it as a way to analyze all kinds of numbers based on their relationships to each other. Here's how it works: 1. Definition: The Riemann-Zeta ...	119	[ { "from": "human", "value": "In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?\n" }, { "from": "gpt", "value": "The Riemann-Zeta function is a special mathematical tool that helps us understand certain prop...
Assuming I can play forever, what are my chances of coming out ahead in a coin flipping series? Let's say I want "heads"...then if I flip once, and get heads, then I win, because I've reached a point where I have more heads than tails (1-0). If it was tails, I can flip again. If I'm lucky, and I get two heads in a ro...	The problem you're describing involves determining the probability of having more heads than tails in a series of coin flips, with the objective of winning the game whenever the count of heads exceeds the count of tails. This problem is often analyzed using concepts from probability and combinatorial mathematics, part...	137	[ { "from": "human", "value": "Assuming I can play forever, what are my chances of coming out ahead in a coin flipping series?\nLet's say I want \"heads\"...then if I flip once, and get heads, then I win, because I've reached a point where I have more heads than tails (1-0). If it was tails, I can flip again...
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 referring to a specific problem, but you haven't provided the details of that problem. If you could share the text or main points of Problem 4, I'd be more than happy to help you with the solution or explain the mechanisms involved. Please provide the necessary information, and I'll do my best to ...	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" }, { ...
I would like to know something like what's the first PDE etc. Could you recommend book on the history of PDE/ODE? thanks.	Certainly! The history of partial differential equations (PDEs) and ordinary differential equations (ODEs) is rich and fascinating, involving contributions from many mathematicians over centuries. Here are a few books that cover the history of ODEs and PDEs: 1. **"A History of Differential Equations: 1600-1800" by J....	853	[ { "from": "human", "value": "I would like to know something like what's the first PDE etc. \nCould you recommend book on the history of PDE/ODE?\nthanks.\n" }, { "from": "gpt", "value": "Certainly! The history of partial differential equations (PDEs) and ordinary differential equations (ODEs) is...
What class of Partial Differential Equations can be solved using the method of separation of variables?	The method of separation of variables is a powerful technique used to solve certain types of partial differential equations (PDEs). It is typically applicable to linear PDEs that exhibit specific characteristics. The main classes of PDEs that can often be solved using this method include: 1. **Linear Elliptic Equation...	205	[ { "from": "human", "value": "What class of Partial Differential Equations can be solved using the method of separation of variables?\n" }, { "from": "gpt", "value": "The method of separation of variables is a powerful technique used to solve certain types of partial differential equations (PDEs)...
Definition: A subgroup of a group is termed strongly potentially characteristic if there is an embedding of the bigger group in some group such that, in that embedding both the group and the subgroup become characteristic. One can refer to this link. Although, I have somewhat understood the definition, I would like t...	The concept of strongly potentially characteristic subgroups is a refined notion in group theory that builds on the idea of characteristic subgroups. Characteristic subgroups are invariant under all automorphisms of the group, while strongly potentially characteristic subgroups can be shown to be characteristic in some...	798	[ { "from": "human", "value": "\nDefinition: A subgroup of a group is termed strongly potentially characteristic if there is an embedding of the bigger group in some group such that, in that embedding both the group and the subgroup become characteristic.\n\nOne can refer to this link.\nAlthough, I have somew...
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 indeed several software programs and tools designed to help with geometric problem-solving, including some that utilize the concept of "geometry bashing" by allowing users to input geometric problems and then automatically perform reasoning or calculations based on the given diagrams. Here are a few nota...	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...
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 determine the probability of getting at least a 1 or a 5 when rolling two dice, we can approach the problem as follows: 1. Total Outcomes: First, we calculate the total number of possible outcomes when rolling two six-sided dice. Each die has 6 faces, so the total number of combinations is: \[ 6 \times 6 ...	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 determine the probability of gettin...
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 mathematics to represent semantic entailment. Specifically, it indicates that a certain statement (or set of statements) logically entails another statement. In more formal terms, if we have a set of premises \( \Gamma \) and a statement \( \phi...	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 mathematics to represent semantic entailment. Specific...
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...	There are indeed software packages and tools designed to assist with understanding mathematical transformations and proofs. While there may not be a tool that will automatically derive the steps between any two mathematical statements in all cases, there are several programs and environments that can aid in this proces...	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...
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...	You're exactly right about the fallacious step in the proof you provided. The error occurs at the point where you divide by \( (a - b) \). To detail it further: 1. In the steps leading up to the division, you start with the equation \( (a - b)(a + b) = b(a - b) \). 2. Since you have initially set \( a = b \), it foll...	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...
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 ...	Your experience with row echelon form (REF) and reduced row echelon form (RREF) is quite common for those beginning their journey in linear algebra. While it might seem cumbersome and inefficient at first, especially when you are familiar with determinants, the techniques of Gaussian elimination (which leads to row ech...	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...
Let $G$ be a graph such that $\delta(G) \geq k$. Prove that $G$ has a path of length at least $k$. Solution: We know that $\delta(G) = \min\lbrace \deg(v) \mid v \in V(G) \rbrace$ If $\delta(G) = k$ then there exists some $v \in V(G)$ such that $\deg(v) = k$. This means all other vertices $u \in V(G)$ have $\deg(u)...	To prove that a graph \( G \) with minimum degree \( \delta(G) \geq k \) has a path of length at least \( k \), we can proceed as follows: 1. Understanding the Degree Condition: The condition \( \delta(G) \geq k \) implies that every vertex in the graph has at least \( k \) neighbors. 2. *Constructing a Path...	496	[ { "from": "human", "value": "Let $G$ be a graph such that $\\delta(G) \\geq k$.\n\nProve that $G$ has a path of length at least $k$.\nSolution: We know that \n$\\delta(G) = \\min\\lbrace \\deg(v) \\mid v \\in V(G) \\rbrace$ \nIf $\\delta(G) = k$ then there exists some $v \\in V(G)$ such that $\\deg(v) = k$....
Say you have a very dense matrix that is 30000x30000 elements. The very dense matrix comes from the radiosity equation, which I discussed here. Say you have Ax = B. You have B, and A is 30000x30000 elements. Solving: You need to find x given B. Solving a matrix like this in C code actually introduces an interesting ...	To tackle the problem of solving a large linear system \( Ax = B \) where the matrix \( A \) is \( 30000 \times 30000 \), memory limitations can make direct approaches infeasible. However, there are several techniques and algorithms that allow you to solve the system in a more memory-efficient manner. Here are some key...	642	[ { "from": "human", "value": "Say you have a very dense matrix that is 30000x30000 elements. The very dense matrix comes from the radiosity equation, which I discussed here.\nSay you have Ax = B. You have B, and A is 30000x30000 elements. Solving: You need to find x given B.\nSolving a matrix like this in...
I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime?	Finding the \( n \)-th prime number directly using a mathematical equation is a challenging problem, as there isn't a simple closed-form formula for it. However, there are several methods and approximations that can help in estimating or finding the \( n \)-th prime. 1. Prime Number Theorem (PNT): The prime number...	295	[ { "from": "human", "value": "I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime?\n" }, { "from": "gpt", "value": "Finding the \\( n \\)-th prime number directly using a mathematical equation is a challen...
In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set. Most logic texts either don't explain the terminology, or allude to the topological property of compactness. I see an analogy as, given a topological space X and a subset of it S, S...	The terminology of "compactness" in logic indeed draws an analogy from the topological notion of compactness, but there is a deeper conceptual relationship that justifies the use of the same term in both contexts. In topology, a space is compact if every open cover has a finite subcover, which essentially means that t...	215	[ { "from": "human", "value": "In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set. \nMost logic texts either don't explain the terminology, or allude to the topological property of compactness. I see an analogy as, given a t...
So the projective plane $\mathbb{RP}^2$ is not a vector space. Is it still isomorphic to its dual? If not, is there at least an invertible map that takes $\mathbb{RP}^2$ to its dual?	The real projective plane \(\mathbb{RP}^2\) is a geometric and topological object, but it is not a vector space, so we need to be careful in discussing its dual. The dual of a vector space \(V\), denoted \(V^*\), consists of linear functionals, which are linear maps from \(V\) to the base field (in this case, \(\mathbb...	223	[ { "from": "human", "value": "So the projective plane $\\mathbb{RP}^2$ is not a vector space. Is it still isomorphic to its dual? If not, is there at least an invertible map that takes $\\mathbb{RP}^2$ to its dual?\n" }, { "from": "gpt", "value": "The real projective plane \\(\\mathbb{RP}^2\\) ...