diff --git "a/metric_test_10_shot_100_ar.txt" "b/metric_test_10_shot_100_ar.txt" new file mode 100644--- /dev/null +++ "b/metric_test_10_shot_100_ar.txt" @@ -0,0 +1,2213 @@ + You're running DeepEval's latest Answer Relevancy Metric! (using Hudson/llemma:7b (Ollama), strict=False, async_mode=True)... + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI need to determine the radius of convergence of the series $\\sum_{n=1}^\\infty a_nx^n$, where $a_n=a^n+b^n$ and $a,b$ are real numbers. Not sure how to approach this one.\n" + - actual output: + - expected output: "\nBy Cauchy's-Hadamard formula, with $\\;R:=$ convergence radius, with the usual conventions when $\\;R=0\\,,\\,\\infty\\;$ , we get:\n$$\\frac1R=\\lim_{n\\to\\infty}\\sup\\sqrt[n]{|a^n+b^n|}$$\nand assuming $\\;|a|\\ge|b|\\;$ , we get\n$$\\sqrt[n]{|a^n+b^n|}=|a|\\sqrt[n]{1+\\left(\\frac{|b|}{|a|}\\right)^n}\\xrightarrow[n\\to\\infty]{}|a|$$\n" + - context: None + - retrieval context: ['"\\nHelp with radius of convergence of a power series.\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nPick the correct statement(s) below:\n\n$(a)$ There exists a group of order $44$ with a subgroup isomorphic to $\\Bbb Z_2 \\oplus \\Bbb Z_2 $.\n$(b)$ There exists a group of order $44$ with a subgroup isomorphic to $\\Bbb Z_4 $.\n$(c)$ There exists a group of order $44$ with a subgroup isomorphic to $\\Bbb Z_2 \\oplus \\Bbb Z_2 $ and a subgroup isomorphic to $\\Bbb Z_4 $.\n$(d)$ There exists a group of order $44$ without any subgroup isomorphic to $ \\Bbb Z_2 \\oplus \\Bbb Z_2 $ or to $ \\Bbb Z_4 $.\n\n" + - actual output: + - expected output: "\nThey key part is recognising part (c).\nThis is false. We know that there is only $1$ Sylow 11-subgroup of $G$. And so this must be a normal subgroup of $G$. Call it $H$. Then consider $G/H$. This has order 4. As such is must be isomorphic to either $\\mathbb{Z}_2\\times \\mathbb{Z}_2$ or $\\mathbb{Z}_4$. Hence $G \\cong H \\times \\mathbb{Z}_4$ or $G \\cong H \\times \\mathbb{Z}_2\\times \\mathbb{Z}_2$ or $G \\cong H \\rtimes \\mathbb{Z}_4$ or $G \\cong H \\rtimes (\\mathbb{Z}_2\\times \\mathbb{Z}_2)$.\n" + - context: None + - retrieval context: ['"\\nSubgroup of group of order $44$\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Passed Answer Relevancy (score: 1.0, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 1.0 because it is a perfect answer., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n\nShow that if $a^n+b^n+c^n=0$ with $a,b,c\\in\\mathbb{Q}$ and $n\\ge 3$, then $abc=0$.\n\nBy letting $a=a_1/a_2$ and so on I think I have shown it is sufficient to prove it for $a,b,c\\in\\mathbb{Z}$, but not sure how to proceed.\nEdit: Is there an elementary proof? I've not quite got through the proof of Fermat yet, being an undergraduate.\n" + - actual output: "You are correct.\n\n" + - expected output: "\nAssume that $a,b,c$ is not $0$. Then if $n$ is even, $a^n,b^n,c^n$ is bigger than $0$. So there obviously is a contradiction.\nAnd if $n$ is odd, it can be changed to $a^n+b^n=(-c)^n$. Then it takes same form as the Fermat's last theorem.\nSo, statement $a\\neq0$, $b\\neq0$ and $c\\neq0$ is false. From this we can know that $abc=0$.\n" + - context: None + - retrieval context: ['"\\nProving that $a^n+b^n+c^n=0 \\\\implies abc=0$.\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because it is not higher than 0.5, and there are no irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n(Not the quantitative section of the standard GRE)\nI am currently a math teacher (up to calc AB). I want to eventually go for my masters in applied math, but because I have a bachelors in business, I haven't taken many math courses... I've only taken precalc, calc 1-3, linear algebra, and discrete structures at the college level. What classes would you recommend taking so that I can get a decent score on this test (I will hopefully take it by the end of this year, so I have time to take a few more classes). I have been studying quite a bit (mainly from Cracking the GRE math subject test) but many of the concepts are new to me. Thank you!\n" + - actual output: + - expected output: "\nThis is not in particular for the GRE but is good knowledge to have when you start your masters in applied mathematics.\nI am not sure what calc 1-3 exactly includes but to do applied maths at Masters level you will need a good knowledge of the following :\n\nOrdinary differential equations, all 1st order, the theory behind linear ODE \nSome knowledge on numerical solution to differential equations would be very nice (Euler's method, Runge-Kutta methods to start with)\nDifferentiation of all kind of functions, integration methods.\nComplex analysis. Mainly stuff around the Residue Theorem.\nBasic knowledge in fluids could help, depending on what kind of applied you do.\nDynamics. Again, depends on the type of applied.\nBasic knowledge of PDE, transport equation, heat equation, wave equation, Laplace equation form a good start.\nKnowledge of one type of mathematics language certainly helps! Matlab, Maple, Mathematica, ...\nFourier transforms, z-transforms and other similar methods.\nGeneral multi-variable calculus\n\nCertainly this list is not complete but if you master these then you will surely have a better knowledge than a lot of people doing a similar masters.\nRegarding books, Mathematical techniques is a good place to look at as it includes most of the above.\n" + - context: None + - retrieval context: ['"\\nWhich classes do you recommend for the GRE math subject test?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because there are no irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nIf $A$ is real matrix, how can i show that $I+AA^T$ is positive definite matrix?\n$I$ is the identity matrix and $A^T$ is a transpose of $A$.\n" + - actual output: + - expected output: "\nYes, it is because $$x^T(I+AA^T)x = x^Tx + (A^Tx)^T(A^Tx) = \\Vert x \\Vert_2^2 + \\Vert A^Tx \\Vert_2^2 \\ge \\Vert x \\Vert_2^2> 0$$ where the last inequality is trivially true for any non-zero $x$.\n" + - context: None + - retrieval context: ['"\\nIs $I+AA^T$ positive definite matrix?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6666666666666666, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.67 because the answer relevancy score is 0.67., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nLet $\\zeta$ be a complex number on the unit circle $\\{z\\in \\mathbb{C}: |z|=1\\}$.Suppose that $[\\mathbb{Q}(\\zeta):\\mathbb{Q}] < \\infty$.Is it true that $\\zeta ^n=1$ for some positive integer $n$?\n" + - actual output: "This is a simple exercise. We have to prove that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field. It can be done like this:\nA cyclotomic field has the form $\\\\mathbb{Q}(e^{2i \\\\pi /n})$, where $n$ is an integer and its value is greater than one. Now we consider ${\\\\Bbb Q}[\\\\sqrt{-3}]$. This field contains $\\\\sqrt{3} = e^{i \\\\pi /3}$. But it is not true that $\\\\mathbb{Q}(\\\\sqrt{-1}, e^{i \\\\pi /3})=\\\\mathbb{Q}(e^{2i \\\\pi /6})$. Therefore ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is not a cyclotomic field.\n" + - expected output: "\nIn the spirit of Alex Youcis' comment let\n$$\nz=\\frac{2+i}{2-i}=\\frac35+\\frac45i.\n$$\nThen $[\\Bbb{Q}[z]:\\Bbb{Q}]=2<\\infty.$\nThe numbers $2+i$ and $2-i$ generate distinct prime ideals in the ring of Gaussian integers $\\Bbb{Z}[i]$. But $\\Bbb{Z}][i]$ is a unique factorization domain and it only has the four obvious units. If $z^n=1$ then $(2+i)^n=(2-i)^n$ violating unique factorization.\n" + - context: None + - retrieval context: ['"\\nIs is true that $\\\\zeta$ has finite order?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nWhen I submit a homework with a proof that uses a graph, ball, shape etc., most of the time the professors are not happy with them. They respond with a statement like: \n\"The proof you made seems very true but why don't you just make a usual proof without drawing anything?\" \nOf course this is something I can do, but I don't like proving something without any visualization. \nSo, is it because geometric proofs are more likely to be misleading?\nEdit: For example: An open ball $B(x,\\epsilon)$ is open.\n" + - actual output: "\"You are an intelligent mathematical assistant. \"\n\"Here are 10 examples of math questions (Q) with given answers (A).\\n\\n\\tQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\tA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\tQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\tA: \\n\\n\\tQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\tA: \\n\\n\\tQ: \\nIs there an expansion for the following summation?\\n$$ (a_1 + a_2 + \\\\cdots + a_k)^n $$\\n\\n\\tA: \\nhttp://en.wikipedia.org/wiki/Multinomial_theorem\\nThis is what you seek.\\n\\n\\tQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\tA: \\n$B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\n\\n\\tQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\tA: \\nBecause $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n\\tQ: \\nIf $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\tA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\tQ: \\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\tA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\tQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\tA: \\n\\n\\tQ: \\nIs there a name for the geometry of a classic puzzle key?\\nIt's not an ellipse, neither a circle, ...\\n\\n\\tA: \\n\\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: \\nWhen I submit a homework with a proof that uses a graph, ball, shape etc., most of the time the professors are not happy with them. They respond with a statement like: \\n\\\"The proof you made seems very true but why don't you just make a usual proof without drawing anything?\\\" \\nOf course this is something I can do, but I don't like proving something without any visualization. \\nSo, is it because geometric proofs are more likely to be misleading?\\nEdit: For example: An open ball $B(x,\\\\epsilon)$ is open.\\n\"\n" + - expected output: "\nSometimes visualization is indeed misleading, check this post:\nhttps://math.stackexchange.com/a/743458/136544\n" + - context: None + - retrieval context: ['"\\nAre geometric proofs less reliable than others?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Passed Answer Relevancy (score: 1.0, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.98 because the answer relevancy score is 0.98., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nDefine $f:\\mathbb R \\to \\mathbb R$ such that $f(x)= e^{x}-1:= \\sum_{n=1}^{\\infty} \\frac{x^{n}}{n!};$ for $x\\in \\mathbb R.$\nMy Question: Can we expect $|f(x)-f(y)|\\leq |x-y| \\cdot C;$ where $C$ is constant, $x, y \\in \\mathbb R$; if yes what can we say about $C$ ?\nRuff Attempt: $$f(x)-f(y)= \\sum_{n=1}^{\\infty} \\frac{1}{n!} (x^{n}-y^{n})= \\sum_{n=1}^{\\infty}\\frac{1}{n!} (x-y)(x^{n-1}+x^{n-2}y+...+y^{n-1})= (x-y)\\sum_{n=1}^{\\infty} \\frac{1}{n!}(x^{n-1}+x^{n-2}y+...+y^{n-1}) ;$$ \nI am not sure what I have done so far is legitimate; and also I have question; Is $$\\sum_{n=1}^{\\infty}\\frac{1}{n!} (x^{n-1}+x^{n-2}y+...+y^{n-1})$$ is converges ? Please correct me if I have done some thing wrong here;\n" + - actual output: "Aha! So the question is in fact about an integral equation. I thought it was just a stupid problem on the exam that I have to solve for you.\n" + - expected output: "\nNo, there is no such $C$. Assume that there is and take $y = 0$. Then $|e^x - 1| \\le C|x|$ for all $x$ which would for example imply that $e^x/x^2 \\to 0$ as $x \\to \\infty$ and that is false.\nOn the other hand, if $x$ and $y$ are restricted to a compact set, then the mean value theorem implies the existence of such a $C$.\nIn other words, $f$ is locally Lipschitz, but not Lipschitz on $\\mathbb{R}$.\n" + - context: None + - retrieval context: ['"\\nIs it true that, $|e^{x}-e^{y}|\\\\leq C \\\\cdot |x-y|$?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Passed Answer Relevancy (score: 1.0, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 1.0 because the answer relevancy score is 1.0 and there are no reasons why the score can't be higher based on irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nHere, on slide 27, it says that \n$\\pi = \\{3, 25/8, 201/64, ... | 4, 7/2, 13/4, ... \\}$\nThe largest number on the left will be $3 + 1/8 + 1/64 + \\dots$ which I evaluated as\n\\begin{align}\n2 + (1 + 1/8 + 1/64 + \\ldots) = 2 + 1/(1-1/8) = 3.142847142857143\n\\end{align}\nWhich is larger than $\\pi$?\nAnd the smallest number on the right I evaluated as\n\\begin{align}\n4 - 1/2 - 1/4 - ...\n&= 4 + 1 - 1 - 1/2 - 1/4 - .... \\\\\n&= 5 - (1 + 1/2 + 1/4 + ....) \\\\\n&= 5 - 2 \\\\\n&= 3 \\\\\n\\end{align}\nSince the smallest number on the right is larger than the largest number on the right, shouldn't this not equate to a number? How does it equate to $\\pi$?\n" + - actual output: "What I do is... I use my brain.\n" + - expected output: "\nYou seem to be inferring some unintended patterns in the truncated list as it was given. The LHS is simply supposed to be a sample of dyadic fractions which fall short of pi, chosen so that the supremum of the sample is precisely pi, and similarly the RHS a sample of dyadic fractions strictly greater than pi chosen so that the infimum is pi.\n" + - context: None + - retrieval context: ['"\\nPi in combinatorial game theory\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nCould you please give me some hint how to deal with this question:\nIf $\\sum a_n$ converges,\ndoes this necessarily mean that $\\sum (-1)^n \\frac {a_n}{1+a_n^2}$ must converge also ?\nThanks.\n" + - actual output: + - expected output: "\nConsider $a_n = (-1)^n\\cdot \\dfrac{1}{n}$, then the conclusion does not follow.\n" + - context: None + - retrieval context: ['"\\nIf $\\\\sum a_n$ converges then $\\\\sum (-1)^n \\\\frac {a_n}{1+a_n^2}$ converges?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n\n\n\n\n\n\nThis question already has answers here: \n\n\n\nA map is continuous if and only if for every set, the image of closure is contained in the closure of image\n\r\n (8 answers)\r\n \n\nClosed 8 years ago.\n\n\n\nI'm trying to solve this question from my textbook:\n\nLet $f:X\\rightarrow Y$ be a continuous map and let $Z \\subset X$.\n Prove the inclusion $f(\\bar Z)\\subset\\overline {f(Z)}$.\n\nThanks in advance for any help!\n" + - actual output: + - expected output: "\n$Z\\subset f^{-1}\\left(\\overline{f\\left(Z\\right)}\\right)$ and as preimage\nof a closed set $f^{-1}\\left(\\overline{f\\left(Z\\right)}\\right)$ will\nbe closed because $f$ is continuous. Then $\\bar{Z}\\subset f^{-1}\\left(\\overline{f\\left(Z\\right)}\\right)$\nso $f\\left(\\bar{Z}\\right)\\subset\\overline{f\\left(Z\\right)}$.\n" + - context: None + - retrieval context: ['"\\nProving that $f(\\\\bar Z)\\\\subset\\\\overline {f(Z)}$ when $f$ is a continuous map [duplicate]\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because it is not higher because there are no irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nLet $P$ be a permutation matrix obtained by the identity matrix by switching 2 rows $n$ times, (with no two rows switched more than one time).\nHow to show that\n$$P^{\\ n+1} = I$$?\nIs it true that, since $P$ represent a permutation of colums, it's like proving that if we take a set $\\{1, \\dots m\\}$ and have a permutation that switches two elements $n$ times, with no two elements switched more than one time, and we apply this permutation $n+1$ times, we'll return to the original set?\n" + - actual output: + - expected output: "\n$\\DeclareMathOperator{\\lcm}{lcm}$\nLook at the cycle lengths you produce by switching two rows: \nswitching $12$ and then $34$ will produce two $2$-cycles. The resulting permutation will return to $I$ under power $2$ (it's still an involution).\nBut if you switch $12$, then $23$, then $34$ you get $2341$ and that's a $4-cycle$. It returns to $I$ under a power $4$, one higher than the number of switches.\nGenerally, you can produce cycle lengths $c_1, c_2,\\dots, c_n $ using $(c_1+1)+(c_2+1)+\\dots + (c_n + 1)$ switches. \nSuch matrix returns to $I$ under power $p= \\lcm( c_1,c_2, \\dots, c_n)$\nIf $\\sum_{k=1}^m c_k + m < \\lcm(c_1, \\dots, c_m)$ then your statement is false. \nExample: $(2+1)+(3+1)+(5+1) = 13$ switches but $\\lcm(2,3,5)= 30$\n" + - context: None + - retrieval context: ['"\\nPowers of permutation matrices.\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answer relevancy score is 0.6., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nIt's clear to me what the interpretation is when we have something like:\n$$\\exists x (\\forall y \\Phi(x, y))$$\nor even how to interpret the formula when x or y are not variables in the expression $\\Phi$, but when these are used in a seemingly contradictory fashion, what is the meaning of these expressions:\n$$\\exists x (\\forall x \\Phi (x))$$\n $$\\forall x (\\exists x \\Phi (x))$$\nAre these even wffs?\n" + - actual output: + - expected output: "\nOne usually takes these to be well-formed formulas.\nLet us take, for example, $\\exists x\\forall x \\Phi(x)$. When we interpret this sentence, we examine $\\forall x \\Phi(x)$ for all free occurrences of $x$ in $\\forall x\\Phi(x)$. There are no such free occurrences, so $\\exists x\\forall x\\Phi(x)$ is true in a structure $M$ precisely if $\\forall x\\Phi(x)$ is true in $M$. \nMore informally, the $\\exists x$ in front has no effect. For that reason, one would never (except for the purposes of this question!) actually use the sentence $\\exists x\\forall x\\Phi(x)$. \n" + - context: None + - retrieval context: ['"\\nHow to interpret $\\\\exists x (\\\\forall x \\\\Phi (x))$?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI would like some suggestions on a nice book on geometry to gift an undergraduate. I'm not searching for something that is common: I need something new and exciting. Suggestions?\n" + - actual output: "You are an intelligent mathematical assistant. # zbMATH \u2014 the first resource for mathematics\n\n##### Examples\n Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. \"Topological group\" Phrases (multi-words) should be set in \"straight quotation marks\". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. \"Quasi* map*\" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. \"Partial diff* eq*\" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.\n\n##### Operators\n a & b logic and a | b logic or !ab logic not abc* right wildcard \"ab c\" phrase (ab c) parentheses\n##### Fields\n any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)\nA class of nonlocal impulsive problems. (English) Zbl\u00a01246.34053\nSummary: The aim of this paper is to investigate the existence, uniqueness and continuous dependence on initial value of mild solutions for a non-local fractional Cauchy problem with non-local conditions at ${t}_{0}$ and impulsive effect on a half-line. We use Browder\u2019s fixed point theorem in cones. The technique we employ in this paper is different from that of previous ones and more complex. An example is given to illustrate the feasibility of our results.\n##### MSC:\n 34K37 Functional-differential equations with fractional derivatives 26A33 Fractional derivatives and integrals (real functions) 34G20 Nonlinear ODE in abstract spaces 45J05 Integro-ordinary differential equations # How to add an attribute to each vertex\n\nI am currently working on a project that requires me to generate random convex polyhedra. I was able to generate convex polyhedrons with code from this link, but now I need help adding an attribute (color) to each vertex of the generated object. Any help would be appreciated. Thank you.\n\n\u2022 What do you mean by attribute? \u2013\u00a0Ron Proctor Mar 7 '19 at 2:57\n\u2022 I mean a color for example so that all the vertices are one color and not random colors like in the link I posted. If this is possible, how could I go about doing it? \u2013\u00a0Joshua Mar 8 '19 at 4:30\n\nThe method you've used to build your mesh is a bit hard to follow because there are so many nested functions, but from what I can tell you are adding the faces one by one. That means that the face indices reference vertices in whatever order they were added to the object, not necessarily based on any rules. This makes it difficult to add a color to each vertex of an object created this way.\n\nTo get the indices into some sort of usable order, you'll need to either re-order them after you have all faces and vertices defined or use something like bmesh (mentioned in the other answer) to define your mesh directly as a set of faces that make up the object.\n\nHere is an example for how to do this with bmesh:\n\nimport bpy\nimport bmesh\nfrom random import randint, uniform, choice\n\ndef get_random_vertex(vertices):\nreturn vertices[randint(0, len(vertices) - 1)]\n\ndef get_point(a, b, c):\nreturn (a * uniform(-.25, .25) + b * uniform(.75, 1.25) + c * uniform(-.25, .25))\n\ndef make_cube():\nverts = [(get_point(i, j, k), 'red') for i in (-1., 0., 1.) for j in (-1., 0., 1.) for k in (-1., 0., 1.)]\nfaces = [('red', 'blue', 'green'), ('blue', 'yellow', 'orange'), ('yellow', 'pink', 'purple'), ('pink', 'purple', 'green')]\nreturn verts, faces\n\ndef make_polyhedron(num):\nverts = []\nfaces = []\nfor i in range(num):\nv, f = make_cube()\nverts.extend(v)\nfaces.extend(f)\nreturn verts, faces\n\n# Create vertices and polygons for a cube\n#verts, faces = make_polyhedron(1)\n\n# Create bmesh from the verts and polygons and show in edit mode\nbm = bmesh.new()\nfor v in verts:\nbm.verts.new(v[0])\nfor f in faces:\nbm.faces.new([bm.verts[i] for i in f])\nbmesh.update_edit_mesh(bm.mesh)\n\n# Show the edit mode mesh object\nC = bpy.context\nD = bpy.data\nscn = C.scene\nob = C.object # Prove the following identity: $\\sum_{k=0}^n \\binom{2k}{k} H_k = (2n+1)\\binom{2n}{n}\\frac{H_n}{4^n}$\n\nThe question is to prove that\n$$\\sum_{k=0}^{n} {2k \\choose k}H_k = (2n + 1){2n \\choose n}\\frac{H_n}{4^n}$$ where $H_n$ is the harmonic number, defined by $\\displaystyle H_n := \\sum\\limits_{j=1}^{n} \\frac{1}{j}$.\n\nI tried to expand the left-hand side and use induction on $n$, but I'm not sure how to deal with the resulting summation. Any help is appreciated! Thanks in advance.\n\n\u2022 Have you come across the fact that $$\\sum_{k\\geq 0} (-1)^k \\binom{x}{k}\\frac{1}{(x+k)^s}=\\frac{\\Gamma(x+1)}{\\Gamma(x+1-s)}$$ for $x>0$, $s>0$? This can be used to prove your identity. Feb 8 '13 at 4:52\n\u2022 @Ethan It's called the binomial series. Feb 8 '13 at 4:57\n\u2022 The sum is a polynomial in $n$. Using the explicit formula for $H_n$ you can find a linear recurrence for its coefficients, and then solve it explicitly to find the generating function of the sum. Feb 8 '13 at 6:09\n\u2022 @EricNaslund That's not true in general. But if there is an explicit formula that can be used to compute a sequence via recurrence relation, then it should apply here as well. There is another method (which I am too lazy to type right now) which doesn't rely on any generating function argument. Feb 8 '13 at 6:57\n\u2022 This question and the answer by Marko Riedel give an interesting account of the development of various proofs using a variety of techniques. Apr 20 '14 at 19:56\n\nSuppose we seek to verify that $$\\sum_{k=0}^n {2k\\choose k} H_k = (2n+1) {2n\\choose n} \\frac{H_n}{4^n}.$$\n\nNow the LHS is $$\\frac{(2n)!}{n! n!} \\times \\frac{1}{n+1} \\sum_{q=0}^n \\frac{1}{2n-2q+1} {2n-2q+1\\choose q} = (2n)!\\times \\frac{1}{(n+1)!} \\sum_{q=0}^n \\frac{(n-q)!^2}{2n-2q+1}.$$\n\nNow $$\\frac{1}{x+y}=\\int_0^1 dt\\; t^{x-1} (1-t)^{y-1}$$ so that we may write this as $$\\frac{1}{(n+1)!}\\sum_{q=0}^n \\sum_{p=0}^\\infty {p\\choose n-q} (-1)^p [(n+1-p)!]^2 = \\frac{1}{(n+1)!}\\sum_{q=0}^n (n-q)!^2 \\sum_{p=0}^\\infty {p\\choose n-q} (-1)^p [n+1-p].$$\n\nNow the inner sum is $$\\sum_{p=0}^\\infty {p\\choose n-q} (-1)^p [n+1-p] = \\sum_{p=0}^{n-q} {p\\choose n-q} (-1)^p [n+1-p].$$\n\nThis becomes $$[n+1]\\sum_{p=0}^{n-q}{p\\choose n-q}(-1)^{p+1}+ \\sum_{p=0}^{n-q} {p\\choose n-q} (-1)^p = -[n+1] {n-q\\choose n-q} + 0$$\n\nand hence we get for the sum $$(2n)!\\times (2n+1)\\frac{1}{4^n}\\sum_{q=0}^n \\frac{(n-q)!^2}{(n+1)!}.$$\n\nNow $$\\frac{1}{(n+1)!} \\sum_{q=0}^n {n-q\\choose n-q} (n-q)!^2 = \\frac{1}{n!} \\sum_{q=0}^n 1 = \\frac{n+1}{n!}$$\n\nand we finally have $$(2n)!\\times (2n+1)\\frac{H_n}{4^n} = (2n+1) {2n\\choose n}\\frac{H_n}{4^n}$$ " + - expected output: "\nI love this one. It is one of the most stimulating books on geometry that I have ever read: it is strikingly innovative and really enjoyable (also, it does not require a particularly advanced background).\nFrom the book description: \n\nNew Horizons in Geometry represents the fruits of 15 years of work in\n geometry by a remarkable team of prize-winning authors\u2014Tom Apostol and\n Mamikon Mnatsakanian. It serves as a capstone to an amazing\n collaboration. Apostol and Mamikon provide fresh and powerful insights\n into geometry that requires only a modest background in mathematics.\n Using new and intuitively rich methods, they give beautifully\n illustrated proofs of results, the majority of which are new, and\n frequently develop extensions of familiar theorems that are often\n surprising and sometimes astounding. It is mathematical exposition of\n the highest order.\nThe hundreds of full color illustrations by Mamikon are visually\n enticing and provide great motivation to read further and savor the\n wonderful results. Lengths, areas, and volumes of curves, surfaces,\n and solids are explored from a visually captivating perspective. It is\n an understatement to say that Apostol and Mamikon have breathed new\n life into geometry.\n\n" + - context: None + - retrieval context: ['"\\nNice book on geometry to gift an undergraduate in mathematics\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nCan anyone give a hint of how to go about solving this? Please don't give answer thanks\nFind the integer $a$ such that $a \\equiv 99 \\pmod{41}$ and $100 \\le a \\le 140$\nWe did not go over this in class and can really use some start up ways. I know 99 mod 41 is 17 but what do you do after that?\n" + - actual output: + - expected output: "\nSince you want a hint,\n$$a\\equiv b\\bmod m$$\nMeans that\n$$km=a-b,\\text{ for some integer $k$.}$$\nThus, pick values of $k$ so that the resulting $a$ satisfies $100\\leq a\\leq140$.\n" + - context: None + - retrieval context: ['"\\nFind a number congruent to mod\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Passed Answer Relevancy (score: 1.0, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 1.0 because the answer relevancy score is 1.0 and there are no irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n\nLet $\\mathcal{K}$ be a nonzero cardinal number. Show that there does not exist a set to which every set of $\\mathcal{K}$ belongs. \n\nLet the set containing all sets of cardinality $\\mathcal{K}$ be $A$. Let $S\\subset A$ such that $S$ contains all sets of $A$ that do not contain themselves. Now select $R\\subset S$ such that $\\text{card } R=\\mathcal{K}$. It can now easily be proven that $R\\notin A$.\n\nIs the argument above correct?\nHow can we ensure that $\\text{card }S\\geq \\mathcal{K}$, in order to create a subset $R$ of $S$ or cardinality $\\mathcal{K}$?\n\nThanks \n" + - actual output: "\"I'm sorry but I don't understand your query. Would you like me to explain the concept again?\"\n" + - expected output: "\nThe argument you give is not correct. Even if you can prove that such $S$ exists, the fact that $R\\subseteq S$ does not mean that $R\\notin A$. It might be that $R\\in A$ and we just have $R\\in S\\setminus R$.\nThe crux of your error is in the words \"easily be proven\".\n\nInstead, show that there is no set of singletons (HINT: the axiom of union); then use this fact and the fact that given a non-empty set $A$ and an object $x$, there is a set $A_x$ such that $x\\in A_x$ and $|A|=|A_x|$.\n" + - context: None + - retrieval context: ['"\\nProving that for any cardinal number, there doesn\'t exist a set containing containing all sets of that cardinality.\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because there are no irrelevant statements., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI want to find an group-isomorphism $$ \\psi : (\\mathbb{Z}/8\\mathbb{Z},+) \\longrightarrow \\mathbb{F}_9^\\times $$\nwhich should be used to multiply elements in $\\mathbb{F}_9$ or to find the inverse element in an easy way.\n" + - actual output: "\"Here are 10 examples of math questions (Q) with given answers (A).\\nQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\nA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\nA: \\n\\nQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\nA: \\n\\nQ: \\nIs there an expansion for the following summation?\\n$$ (a_1 + a_2 + \\\\cdots + a_k)^n $$\\n\\n\\nA: \\nhttp://en.wikipedia.org/wiki/Multinomial_theorem\\nThis is what you seek.\\n\\n\\nQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\nA: \\n$B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\n\\n\\nQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\nA: \\nBecause $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n\\nQ: \\nIf $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\nA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\nQ: \\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\nA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\nQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\\\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\nA: \\n\\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: \\nI want to find an group-isomorphism \\\\[1mm] $$ \\\\psi : (\\\\mathbb{Z}/8\\\\mathbb{Z},+) \\\\longrightarrow \\\\mathbb{F}_9^\\\\times $$\\nwhich should be used to multiply elements in $\\\\mathbb{F}_9$ or to find the inverse element in an easy way.\\n\"\n\n" + - expected output: "\nAs Seth and Pedro indicated the existence of such an isomorphism follows from (and is equivalent to) the cyclicity of the multiplicative group $\\Bbb{F}_9^*$. To exhibit an explicit isomorphism you need to specify a construction of $\\Bbb{F}_9$ and find a generator of the multiplicative group (aka a primitive element).\nElements of order $8$ are zeros of the cyclotomic polynomial $\\phi_8(x)=x^4+1$. As we are working modulo $3$, we can factor it as follows\n$$\n\\begin{aligned}\nx^4+1&=x^4+4=(x^4+4x^2+4)-4x^2\\\\\n&=(x^2+2)^2-(2x)^2=(x^2+2-2x)(x^2+2+2x)\\\\\n&=(x^2-x-1)(x^2+x-1).\n\\end{aligned}\n$$\nSo, if we identify $\\Bbb{F}_9$ with $\\Bbb{Z}_3[x]/\\langle x^2-x-1\\rangle$, then the coset\n$\\alpha=x+\\langle x^2-x-1\\rangle$ will be a generator.\nThe desired isomorphism $\\psi:(\\Bbb{Z}_8,+)\\to (\\Bbb{F}_9^*,\\cdot)$ is given by\n$$\n\\psi(\\overline{n})=\\alpha^n\n$$\nfor all $n=0,1,\\ldots,7$. This comes to the following:\n$$\n\\begin{array}{c|ccc|c}\nn&\\psi(\\overline{n})&&n&\\psi(\\overline{n})\\\\\n\\hline\n0&1&&1&\\alpha\\\\\n2&\\alpha^2=\\alpha+1&&3&\\alpha^3=\\alpha^2+\\alpha=2\\alpha+1\\\\\n4&\\alpha^4=2&&5&\\alpha^5=2\\alpha\\\\\n6&\\alpha^6=2\\alpha^2=2\\alpha+2&&7&\\alpha^7=\\alpha+2\n\\end{array}\n$$\nHere I used the equation $\\alpha^2=\\alpha+1$ repeatedly. Note also that $\\alpha^4=2=-1$, so\n$\\alpha^{4+j}=-\\alpha^j$ for all $j$. As a final check we do that\n$$\n\\alpha^8=\\alpha\\cdot\\alpha^7=\\alpha(\\alpha+2)=\\alpha^2+2\\alpha=3\\alpha+1=1\n$$\nas it should.\nThe way to use this in finding products and inverses is to treat the inverse of $\\psi$ as a logarithm, i.e. a mapping that transforms multiplication in $\\Bbb{F}_9$ into addition in $\\Bbb{Z}_8$. For example (read the above table from right to left)\n$$\n(\\alpha+2)(\\alpha+1)=\\alpha^7\\cdot\\alpha^2=\\alpha^9=\\alpha^{8+1}=\\alpha,\n$$\nand\n$$\n(\\alpha+1)^{-1}=(\\alpha^2)^{-1}=\\alpha^{-2}=\\alpha^{8-2}=\\alpha^6=2\\alpha+2.\n$$\nYou mentioned that you had found $x+2=x-1$ and $2x+1$ to be generators of $\\Bbb{F}_9^*$.\nThat may be true, but to make that meaningful you need to specify the minimal polynomial of $x$. Above I used the minimal polynomial $x^2-x-1$, and $\\alpha$ was one of its zeros ($\\alpha^3=2\\alpha+1$ being the other. If a cyclic group of order $8$ is generated by an element $c$, then $c^m$ is also a generator iff $\\gcd(m,8)=1$. So here any of $\\alpha$, $\\alpha^3$, $\\alpha^5$ and $\\alpha^7$ is a generator. Not surprisingly those are the zeros of $x^4+1$ in this copy of $\\Bbb{F}_9$.\n\nChanging the irreducible polynomial always changes the scene. Your copy of $\\Bbb{F}_9$ is $\\Bbb{Z}_3[x]/\\langle x^2+1\\rangle$. Because $x^2+1\\mid x^4-1$ in this case $x+\\langle x^2+1\\rangle$ will not work as a generator. If we let $\\beta=x+\\langle x^2+1\\rangle$, then \nfor $\\alpha=\\beta+2$ (that you might call $x+2$ even though that abuses notation slightly)\nwe get\n$$\n0+\\langle x^2+1\\rangle =(x^2+1)+\\langle x^2+1\\rangle=(\\alpha+1)^2+1=\\alpha^2+2\\alpha+2=\\alpha^2-\\alpha-1.\n$$\nThus $x^2-x-1$ is a minimal polynomial of $\\alpha$. In terms of $\\beta$ the above the homomorphism looks like\n$$\n\\begin{array}{c|ccc|c}\nn&\\psi(\\overline{n})&&n&\\psi(\\overline{n})\\\\\n\\hline\n0&1&&1&\\beta+1\\\\\n2&(\\beta+1)^2=2\\beta&&3&(\\beta+1)^3=2\\beta^2+2\\beta=2\\beta+1\\\\\n4&(\\beta+1)^4=2&&5&(\\beta+1)^5=2\\beta+2\\\\\n6&(\\beta+1)^6=\\beta&&7&(\\beta+1)^7=\\beta+2\n\\end{array}\n$$\nAll the elements on the right column ($\\beta+1,2\\beta+1,2\\beta+2,\\beta+2$) can take the role of the generator. Thus you get four different such isomorphisms $\\psi$. Each will work equally well as the inverse of a discrete logarithm.\n" + - context: None + - retrieval context: ['"\\nGroup-Isomorphism problem\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answers are not relevant to the input., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nHere's a word problem having to do with percentages, and the answer doesn't make sense. A furniture store is selling a desk for $300. If the store makes a 25% profit on the sale, what is the store's cost for the desk? The answer is $240 but how? \nThanks.\n" + - actual output: "A = \"Here are 10 examples of math questions (Q) with given answers (A).\\nQ: Can you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\nA: A cyclotomic field is... And the complex roots of $x^3-1$ are...\\nQ: Is the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\nA: http://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem (See Picard-Lindel\u00f6f theorem)\\nQ: Given that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\nA: http://mathoverflow.net/questions/30491/trace-of-a-matrix-multiplied-by-another-matrix (See Trace of a matrix multiplied by another matrix)\\nQ: I have to simplify the following expression:\\n$$(a_1 + a_2 + \\\\cdots + a_k)^n $$\\nA: http://en.wikipedia.org/wiki/Multinomial_theorem (See Multinomial theorem)\\nQ: I have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\nA: $B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\nQ: Quick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\nA: Because $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\nQ: If $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?) A: $\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\nQ: if $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\nA: OK - $f(x)=7=3x-4$, can you solve this?\\nQ: On p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\nA: The notion of index of submodules does make sense for finitely generated modules over a PID. See http://en.wikipedia.org/wiki/Module_(mathematics)#Submodules\\nQ: Here's a word problem having to do with percentages, and the answer doesn't make sense. A furniture store is selling a desk for $300. If the store makes a 25% profit on the sale, what is the store's cost for the desk? The answer is$240 but how? \\nThanks.\\nA: $100\\\\% + 25\\\\% = 125\\\\%$. Thus \\$300 times this percentage equals...\\nQ: Is there a name for the geometry of a classic puzzle key? It's not an ellipse, neither a circle, ...\\nA: I would call it \"elliptic curve\". In general any equation $y^2=f(x)$ where $f$ is a polynomial defines a curve which may or may not be an ellipse. If $f$ has degree 3 then it's called a \"cubic curve\" (http://en.wikipedia.org/wiki/Cubic_curve)\\nQ: Provide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: Here are 10 examples of math questions (Q) with given answers (A).\\n\\n\"\n" + - expected output: "\nLet's say the cost of the desk for the store is 100%. We know 25% more than that is \\$300. Hence, 125% of the cost price of the desk is \\$300, and the original price is therefore $100\\times \\frac{\\$300}{125}$. Since $\\$300/125$ = 1% of cost price, and then we just multiply by 100 to find the original 100%.\n" + - context: None + - retrieval context: ['"\\nPercent word problem - calculate store cost\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI am having very difficult times in understanding the following and related mixture problems.Can anybody guide me the easy and nice trick that is useful in understanding,visualizing and solving these types of problems? following are few of those.\n\nHow many liters of a 70% alcohol solution must be added to 50 liters of a 40% alcohol solution to produce a 50% alcohol solution?\nHow many ounces of pure water must be added to 50 ounces of a 15% saline solution to make a saline solution that is 10% salt?\nFind the selling price per pound of a coffee mixture made from 8 pounds of coffee that sells for \\$9.20 per pound and 12 pounds of coffee that costs \\$5.50 per pound?\nHow many pounds of lima beans that cost $0.90 per pound must be mixed with 16 pounds of corn that costs \\$0.50 per pound to make a mixture of vegetables that costs \\$0.65 per pound?\nTwo hundred liters of a punch that contains 35% fruit juice is mixed with 300 liters (L) of another punch. The resulting fruit punch is 20% fruit juice. Find the percent of fruit juice in the 300 liters of punch?\nTen grams of sugar are added to a 40-g serving of a breakfast cereal that is 30% sugar. What is the percent concentration of sugar in the resulting mixture?\n\n" + - actual output: "A: \"In the previous question, Q16 means : Can you help me with this 'simple' exercise: Prove that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field. \\nThe answer given by A is not correct. To make it right, you need to write down the definition of a cyclotomic field (which is easy) and then apply it on ${\\\\Bbb Q}[\\\\sqrt{-3}]$. For example:\\n\\nQ: Can you help me with this 'simple' exercise: Prove that ${\\\\Bbb Q}[\\\\sqrt{2}]$ is a cyclotomic field?\\nA: A cyclotomic field is... And the complex roots of $x^2-2=0$ are...\\n\\nQ: Is it true that any finite group $G$ can be embedded in $S_n$, where $|G|=n$? \\nA: Sure. It's a standard result.\\n\\nQ: How do I find the inverse of $\\\\displaystyle \\\\frac{3}{4}$ ?\\nA: The number that, when multiplied by $\\\\displaystyle \\\\frac{3}{4}$, gives you 1? You could also calculate it from the definition:\\n\\nQ: Find the inverse of -6.\\nA: First find its reciprocal (which is $1/6$ ), then change the sign. \\nOr, equivalently, multiply by $-1$: $(-1)*(-6)=6$. So the answer is 6.\\n\\nQ: Do you know how to simplify $\\\\frac{-3}{4}\\\\cdot \\\\frac{7}{8}$ ?\\nA: Yes! The answer is $3/8$ . \\n\\nQ: Is there a way to get rid of all fractions?\\nA: If the only reason that they are fractions is because you haven't yet learnt how to divide, then there is an easy trick. Just multiply by the reciprocal.\\n\\nQ: What is $\\\\sqrt{-25}$ ? \\nA: It is undefined (since $-25$ is negative). \\n\\nQ: Is it true that any finite group $G$ can be embedded in $S_n$, where $|G|=n$? \\nA: Sure. It's a standard result.\\n\\nQ: What does 'decimal notation' mean?\\nA: It means base 10, like $6748=2*10^3+7*10^2+4*10^1+8*10^0$ . \\n\\nQ: What is the prime factorisation of 98?\\nA: The prime factorization of 98 = 2 x 7 x 7.\\n\\nQ: Simplify $\\\\sqrt{36}$ ? \\nA: $\\\\sqrt{36}=\\\\pm6$ .\\n\\nQ: How do I simplify $-5+ \\\\frac{-4}{3}\\\\cdot (-1)$?\\nA: The answer is 2. See my other answer.\\n\\nQ: How can we find the common denominator of $\\\\frac{7}{8}$ and $\\\\frac{9}{10}$ ?\\nA: We multiply by $1$ . In our case, we multiply both fractions by $\\\\frac{10}{8}$, which is equal to $1$, so that the denominators become 80.\\n\\nQ: How do you solve $-7+3y=4$? \\nA: Simply add 7 on both sides. Then divide by 3, and the answer is y = 2.\\n\\nQ: Is it true that $\\\\frac{5}{-1}=\\\\frac{-1}{5}$ ?\\nA: Yes.\\n\\nQ: How do you write an equation of a line in standard form? \\nA: The standard form for the equation of a straight line is $ax+by=c$ , where $a$, $b$ and $c$ are constants. If the line passes through two points $\\\\left(x_1,y_1\\\\right)$ and $\\\\left(x_2,y_2\\\\right)$, then the following formula can be used:\\n\\nQ: What is the standard form of an equation for a circle? \\nA: $x^2+y^2=r^2$ .\\n\\nQ: How do you find the radius of a circle? \\nA: The distance from the centre to any point on the circumference.\\n\\nQ: What is a radical expression? \\nA: An expression that involves taking roots of numbers, eg. $\\\\sqrt{2}$ .\\n\\nQ: How can you simplify $\\\\frac{-3}{-4}+ \\\\frac{-5}{-6}$ ?\\nA: First find the common denominator (in this case it is 12). Then add the numerators and reduce the resulting fraction:\\n\\nQ: Can you help me with this 'simple' exercise: Prove that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\nA: A cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\nQ: Is the following integral equation solvable ? \\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)} $$\\nand $\\\\gamma$ is a parameter.\\nA: Try $F(x)=x^2+1$. \\n\\nQ: Given that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\nA: Let $M=X+Y$. Then $X,Y \\\\geq 0$, so by the Blichfeldt\u2019s theorem,\\nthe singular values of X and Y are not larger than its trace. \\nThen $X\\\\leq M$ and $Y\\\\leq M$, hence by Von Neumann's trace inequality,\\nthe trace of their sum is bounded above by the sum of their traces, i.e.\\nthe trace of a positive definite matrix is not larger than its smallest singular value!\\n\\nQ: The function $\\\\sqrt{x^2+y^2}$ is called the distance from point $(0, 0)$ to $(x, y)$. Can you find the expression for this distance? \\nA: The answer is $d=\\\\sqrt{x^2+y^2}$.\\n\\nQ: Can you help me with this 'simple' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\nA: It is not correct. A cyclotomic field is a field obtained from the rational numbers by adjoining (forming a ring extension of) all roots of unity of a given degree.\\nFor example, $\\\\mathbb{Q}[\\\\sqrt{-1}]$ is the cyclotomic field obtained by adjoining to \\\\mathbb{Q} the square root of -1 which is $i$. In this case we have $\\\\mathbb{Q}[\\\\sqrt{-3}] = \\\\mathbb{Q}[x]/(x^2+3)$ . \\nIt turns out that this field contains a primitive 6th root of unity, and so it is indeed cyclotomic.\\n\\nQ: What are the factors of $97$?\\nA: The factorization of 97 is $1\\\\cdot 97$ (also called trivial). \\n\\nQ: How do I solve $x^2+4=0$? \\nA: Subtract 4 from both sides, then you get $x^2=-4$. Take the square root of each side to give the result.\\n\\nQ: What is the prime factorization of 180? \\nA: The prime factorization of 180 is $2\\\\cdot 2\\\\cdot 3\\\\cdot 3\\\\cdot 5$ .\\n\\nQ: How do I simplify $\\\\frac{5}{-4}+ \\\\frac{-7}{-6}$ ?\\nA: First find the common denominator (in this case it is 12). Then add the numerators and reduce the resulting fraction:\\n$\\\\displaystyle\\\\frac{\\\\left(30+7\\\\right)}{\\\\left(-24\\\\right)}=\\\\frac{37}{-24}=-\\\\frac{37}{24}$ . \\n\\nQ: What is a complex number? \\nA: A complex number $z$ is an expression of the form $a+bi$, where $i^2=-1$. It's called \"imaginary\" because it does not lie on the real axis in the Argand diagram.\\n\\nQ: Do you know how to simplify $\\\\frac{-3}{4}\\\\cdot \\\\frac{7}{8}$ ?\\nA: Yes! The answer is $3/8$ . \\n\\nQ: How do I write the prime factorization of 12?\\nA: $2\\\\cdot 2\\\\cdot 3$ .\\n\\nQ: What is a radical expression?\\nA: An expression that involves taking roots of numbers, eg. $\\\\sqrt{2}$ .\\n\\nQ: How do you write an equation of a line in standard form? \\nA: The standard form for the equation of a straight line is $ax+by=c$ , where $a$, $b$ and $c$ are constants. If the line passes through two points $\\\\left(x_1,y_1\\\\right)$ and $\\\\left(x_2,y_2\\\\right)$, then the following formula can be used:\\n\\nQ: What is a radical expression? \\nA: An expression that involves taking roots of numbers, eg. $\\\\sqrt{2}$ .\\n\\nQ: How do you find the radius of a circle? \\nA: The distance from the centre to any point on the circumference.\\n\\nQ: What is a radical expression? \\nA: An expression that involves taking roots of numbers, eg. $\\\\sqrt{2}$ .\\n\\nQ: How do you find the radius of a circle? \\nA: The distance from the centre to any point on the circumference.\\n\\nQ: What is a radical expression? \\nA: An expression that involves taking roots of numbers, eg. $\\\\sqrt{2}$ .\\n\\nQ: How do you find the radius of a circle? \\nA: The distance from the centre to any point on the circumference.\\n\\nQ: What is a radical expression? \\nA: An expression that involves taking roots of numbers, eg. $\\\\sqrt{2}$ .\\n\\nQ: How do you find the radius of a circle? \\nA: The distance from the centre to any point on the circumference.\\n\"\n" + - expected output: "\nLet's consider $1$st problem.\nProblem:\n\nHow many liters of a 70% alcohol solution must be added to 50 liters of a 40% alcohol solution to produce a 50% alcohol solution?\n\n\"Easy and nice trick\" :)\nLet's consider liquids separately: \n - alcohol; \n - water.\n\u00a0\nThen image:\n\nThen one can write equation for each liquid:\nfor alcohol: \n$$\\qquad 0.7 \\cdot x + 0.4 \\cdot 50 = 0.5 \\cdot (x+50);\\tag{1}$$\nor for water: \n$$\\qquad 0.3 \\cdot x + 0.6 \\cdot 50 = 0.5 \\cdot (x+50).\\tag{2}$$\nThen solve $(1)$ (or $(2)$ ) :\n$$\n0.2\\cdot x=5;\n$$\n$$\nx=25 \\mbox{ (liters)}.\n$$\n" + - context: None + - retrieval context: ['"\\nSolving Mixture word problems?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.3333333333333333, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.33 because the answer relevancy score is 0.33., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nWhile discussing with my 11 y.o. daughter about the definition of a cube as regular hexahedron, I observed that actually we can let drop the assumption that the faces are squares, and require only that they are equal.\nShe agreed, because trying to imagine, let's say, six equal parallelograms (without right angles) matching to form a solid leads to impossible solids.\nI recently made a conjecture requiring even weaker conditions.\nA polyhedron having equal quadrilateral faces is a regular hexahedron.\nAny hint to prove the statement or a couterexample would be great\nThank you in advance for your attention\n" + - actual output: "\"Here are 10 examples of math questions (Q) with given answers (A).\\nQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\nA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\nA: \\n\\nQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\nA: \\n\\nQ: \\nIs there an expansion for the following summation?\\n$$ (a_1 + a_2 + \\\\cdots + a_k)^n $$\\n\\n\\nA: \\nhttp://en.wikipedia.org/wiki/Multinomial_theorem\\nThis is what you seek.\\n\\n\\nQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\nA: \\n$B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\n\\n\\nQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\nA: \\nBecause $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n\\nQ: \\nIf $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\nA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\nQ: \\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\nA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\nQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\nA: \\n\\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: \\nWhile discussing with my 11 y.o. daughter about the definition of a cube as regular hexahedron, I observed that actually we can let drop the assumption that the faces are squares, and require only that they are equal.\\nShe agreed, because trying to imagine, let's say, six equal parallelograms (without right angles) matching to form a solid leads to impossible solids.\\nI recently made a conjecture requiring even weaker conditions.\\nA polyhedron having equal quadrilateral faces is a regular hexahedron.\\nAny hint to prove the statement or a couterexample would be great\\nThank you in advance for your attention\\n\"\n" + - expected output: "\nYou can \"bookcase squash\" a cube in two directions, making the faces congruent rhombi, and get a Rhombohedron\n\n" + - context: None + - retrieval context: ['"\\nPolyhedra having equal quadrilateral faces are cubes?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Passed Answer Relevancy (score: 1.0, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 1.0 because the answer relevancy score is 1.0 and there are no reasons why it can't be higher., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nHypothesis: \n\nLet \n$$\nG \\cong \\left\\langle a,b \\mid abab^{-1}\\right\\rangle\n$$\n$$\nH \\cong \\left\\langle c,d \\mid c^2 d^2 \\right\\rangle\n$$\nLet the function $f$ be defined as follows. First let $f(a) = cd$ and $f(b) = d^{-1}$. For all other elements $g$ of $G$, define $f(g)$ as follows:\n$$\nf(g) = f(a^{\\alpha_1} b^{\\beta_1} \\cdot \\ldots \\cdot a^{\\alpha_k}b^{\\beta_k}) = f(a)^{\\alpha_1} f(b)^{\\beta_1} \\cdot \\ldots \\cdot f(a)^{\\alpha_k}f(b)^{\\beta_k}\n$$\nsuch that $a^{\\alpha_1} b^{\\beta_1} \\cdot \\ldots \\cdot a^{\\alpha_k}b^{\\beta_k}$ is the fully reduced and unique word representation of $g$ in $G$. \nThen $f$ is a well-defined mapping from $G$ to $H$.\n\nGoal: Show that $f$ is an isomorphism. As my attempt below will reflect, I know how to show that $f$ is a surjective homomorphism, however I don't know how to show that it is an injection.\nAttempt:\n\nWe need only check that $f(abab^{-1}) = e_H = c^2d^2$ in order for $f$ to be a homomorphism. To do this we have\n$$\nf(abab^{-1}) = f(a)f(b)f(b)f(b)^{-1} = (cd)(d^{-1})(cd)(d^{-1})^{-1} = c^2d^2 = e_H\n$$\nas desired.\nTo show that $f$ is surjective, we note that\n$$\nf(ab) = f(a)f(b) = (cd)(d^{-1}) = c\n$$\n$$\nf(b^{-1}) = f(b)^{-1} = (d^{-1})^{-1} = d\n$$\nso that if $h = c^{\\alpha_1}d^{\\beta_1} \\cdot \\ldots \\cdot c^{\\alpha_k}d^{\\beta_k}$ we have that \n$$\nf\\left((ab)^{\\alpha_1}(b^{-1})^{\\beta_1} \\cdot \\ldots \\cdot (ab)^{\\alpha_k}(b^{-1})^{\\beta_k}\\right) = c^{\\alpha_1}d^{\\beta_1} \\cdot \\ldots \\cdot c^{\\alpha_k}d^{\\beta_k} = h\n$$\nas desired.\n\nQuestion: Why is $f$ injective?\n" + - actual output: "The program will help you solve math problems.\n" + - expected output: "\nAn alternative way to do this is to show that the map $g:c \\mapsto ab,\\,d \\mapsto b^{-1}$ extends to a homomorphism $g:H \\to G$ (to do that, you just need to check that $g(c^2d^2)=1$), and then show that $fg:H \\to H$ and $gf:G \\to G$ are the identity maps on $H$ and $G$, which is easy: just check that they map the group generators to themselves. So $f$ and $g$ are $2$-sided inverse maps, and must be isomorphisms.\n" + - context: None + - retrieval context: ['"\\nShowing a Mapping Between $\\\\left\\\\langle a,b \\\\mid abab^{-1}\\\\right\\\\rangle$ and $\\\\left\\\\langle c,d \\\\mid c^2 d^2 \\\\right\\\\rangle$ is Surjective\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nLet $a,b,c,d$ be $4$ distinct non-zero integers such that $a+b+c+d = 0$. It is know that the number\n$$M = (bc - ad)(ac - bd)(ab-cd)$$\nlies strictly between $96100$ and $98000$. Determine the value of $M$.\nI tried expanding the expression out, as well as using AM-GM on it, but to no avail. Any help would be appreciated. Thanks!\n(Source: Singapore Mathematical Olympiad 2013, Open Section, First Round, Question 24)\n" + - actual output: "I'm sorry, I don't understand your question. Could you please clarify?\n" + - expected output: "\nAfter the substitution of $a = -b-c-d$, we have $M = (b+c)^2(b+d)^2(c+d)^2$. \n" + - context: None + - retrieval context: ['"\\nFinding the value of $(bc-ad)(ac-bd)(ab-cd)$\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nLet \n$\\displaystyle f(x)=\\begin{cases}\n\\frac{1}{n}, & \\text{if }x=\\frac{m}{n},m,n\\in\\mathbb{N}\\text{ and m and n has no common divisor} \\\\\n0, & \\text{otherwise}\n\\end{cases}$\nShow $f\\in R[0,1]$.\nHow to find a partition $P_\\epsilon$ such that $U(P_\\epsilon,f)-L(P_\\epsilon,f)<\\epsilon$ for all $\\epsilon>0$?\n$L(P_\\epsilon,f)=0$, so only need to consider $U(P_\\epsilon,f)$.\n" + - actual output: + - expected output: "\nHint. A bounded function $f:[a,b]\\to\\mathbb R$ is Riemann integrable iff for every $\\varepsilon>0$, here exists a partition $P$ of $[a,b]$, such that\n$$\nU(f,P)-L(f,P)<\\varepsilon.\n$$ \nIn the case of this function $L(f,P)=0$, for all $P$ of $[0,1]$.\nGiven an $\\varepsilon>0$, we simply need to find a partition $P$ of $[0,1]$ for which $U(f,P)<\\varepsilon$.\nClearly, there are only finitely many points $\\{x_k\\}$ in $[0,1]$, where\n$$\nf(x_k)>\\frac{\\varepsilon}{2}.\n$$\nSay $N=N(\\varepsilon)$ such point. Simply take a partition, where all these $x_k$'s lie in very small subintervals $[t_{k_1},t_{k_2}]$, i.e.,\n$$\nt_{k_1}=x_k-\\delta 0$$\nThen we can pose the questions: \n\nFor which pairs $\\alpha, \\beta$ is this function uniformly continuous?\nFor which sets $(\\alpha, \\beta)$ in $(0, \\infty)^2$ is the family equicontinuous?\n\nI am baffled as to how to go about answering these questions in a clear and concise way. I think that it is possible to produce an answer by considering many cases and lots of tedious estimates. Is there a better way to approach the problem?\nAny help will be appreciated. \n" + - actual output: + - expected output: "\nFact 1. A uniformly continuous function on $(0,\\infty)$ is Lipschitz for large distances, which can be expressed concisely by the inequality \n$$|f(x)-f(y)|\\le L|x-y|+M,\\quad x,y>0 \\tag{1}$$\nwhere $L$ and $M$ are constants. This is a general and useful fact, so it's worth recording regardless of this problem. Proof: there is $\\delta>0$ such that $|f(x)-f(y)| \\le 1$ whenever $|x-y|\\le \\delta$. Divide $[x,y]$ into intervals of size at most $\\delta$; we need no more than $1+|y-x|/\\delta$ of these. Then $|f(x)-f(y)|\\le 1+|y-x|/\\delta$, which gives (1). $\\quad \\Box$\nIn particular, (1) implies that $(f(x)-f(1))/(x-1)$ remains bounded as $x\\to \\infty$. This excludes $\\alpha>1$ from consideration. \nFurthermore, consider the sequence $x_n=(\\pi/2+\\pi n)^{1/\\beta}$. Observe that \n$|x_{n+1}-x_n| \\le C n^{\\frac{1}{\\beta}-1}$ and $|f(x_n)-f(x_{n+1})|\\ge c n^{\\alpha}$ where $C$ and $c$ are positive constants independent of $n$. By (1), $\\alpha\\le \\frac{1}{\\beta}-1$. \nSummarize the necessary conditions obtained so far: \n$$0<\\alpha\\le 1, \\qquad 0<\\beta \\le \\frac{1}{\\alpha+1}\\tag{2}$$\nHere's an interesting fact: under conditions (2), the second derivative of $f$ is bounded at infinity. Indeed, the largest term in $f''$ has $x^{\\alpha+2\\beta-2}$, and \n$$\\alpha+2\\beta-2 \\le \\frac{\\alpha^2-\\alpha}{\\alpha+1}\\le 0$$\nFact 2. If $f$ is uniformly continuous on $(0,\\infty)$ and $f''$ is bounded at infinity, then $f'$ is also bounded at infinity. \nProof: there is $\\delta>0$ such that $|f(x)-f(y)| \\le 1$ whenever $|x-y|\\le \\delta$. By the mean value theorem, this implies $|f'|\\le 1/\\delta$ at some point within the interval $[x,x+\\delta]$. But then $|f'(x)| \\le 1/\\delta+ \\delta \\sup_{[x,x+\\delta]} |f''|$, which is a uniform bound on $f'$. $\\quad \\Box$ \nIt is not hard to see that $f'$ is bounded at infinity if and only if \n$$ \\alpha +\\beta \\le 1 \\tag{3} $$\nSo, (3) is necessary for uniform continuity. It is also sufficient, since bounded derivative implies Lipschitz implies uniformly continuous on $[1,\\infty)$; the interval $[0,1]$ is compact, so it's not a problem. \n\nConcerning equicontinuity: on $[1,\\infty)$, the family described by (3) is equicontinuous because the derivative $f'$ is uniformly bounded. There is an issue at $0$, where $f$ behaves like $x^{\\alpha+\\beta}$. You don't want to allow this exponent to be arbitrarily small. So, \n$$ \\epsilon \\le \\alpha +\\beta \\le 1 \\tag{4} $$\ngives an equicontinuous family, for every $\\epsilon>0$. \n" + - context: None + - retrieval context: ['"\\nWhen is $x^{\\\\alpha}\\\\sin(x^{\\\\beta})$ uniformly continuous for $\\\\alpha, \\\\beta > 0$?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because there are no irrelevant statements., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nLet $U \\subseteq \\mathbb{R}^n$ be open in the usual topology. Is its boundary, $\\partial U$, necessarily a topological manifold?\n" + - actual output: + - expected output: "\nNo, consider $U = \\{ (x,y) \\in \\mathbb{R}^2 : xy \\neq 0 \\}$. Then $\\partial U = \\{ (x,y) : x = 0 \\vee y = 0 \\}$, and the point $(0,0)$ doesn't have a neighborhood homeomorphic to $\\mathbb{R}$.\nYou can modify this example to get a connected and bounded $U$: consider\n$$U = \\{ (x,y) : x^2 + y^2 < 2 \\} \\setminus \\left( [-1,1]\\times0 \\cup 0 \\times [-1,1]\\right)$$\n" + - context: None + - retrieval context: ['"\\nIs the boundary of an open subset of $\\\\mathbb{R}^n$ always a topological manifold?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer is irrelevant to the input., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nFrom what I understand a derivative may not exist at a given point if the function is not continuous or the right and left side derivatives are not equal.\nDoes that imply that if a function is continuous, the one sided derivatives exist at it's every point?\n" + - actual output: "You're talking to an intelligent math tutor! Ask your questions... only about math.\n" + - expected output: "\nFor an elementary example, consider\n$$f(x) = \\begin{cases} \\displaystyle x\\sin\\frac1x & \\text{if } x \\neq 0, \\\\\n0 & \\text{if } x = 0.\\end{cases}$$\nThis is obviously continuous, but can't be differentiable on either side of $0$, because the function has points on lines $y=x$ and $y=-x$ arbitrarily close to $0$.\n" + - context: None + - retrieval context: ['"\\nDoes continuity imply existence of one sided derivatives?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Passed Answer Relevancy (score: 1.0, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.98 because the answer is correct and there are no irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nShow $\\frac{a_{1}+...+a_{n}}{b_{1}+...+b_{n}}$ is between the smallest and largest fraction $\\frac{a_{i}}{b_{i}}$, where $b_{i}>0$.\nAttempt\nAssume the largest is $\\frac{a_{n}}{b_{n}}\\Rightarrow$\n$\\frac{a_{n}}{b_{n}}-\\frac{a_{1}+...+a_{n}}{b_{1}+...+b_{n}}\\Rightarrow $\n$\\frac{b_{1}+...+b_{n-1}}{{b_{1}+...+b_{n}}}[\\frac{a_{n}}{b_{n}}-\\frac{a_{1}}{b_{1}+...+b_{n-1}}-...-\\frac{a_{n-1}}{b_{1}+...+b_{n-1}}]\\Rightarrow $\nif $a_{1}<0$, $w=\\frac{a_{n}}{b_{n}}-\\frac{a_{1}}{b_{1}+...+b_{n-1}}-...-\\frac{a_{n-1}}{b_{1}+...+b_{n-1}}>\\frac{a_{n}}{b_{n}}-\\frac{a_{2}}{b_{1}+...+b_{n-1}}-...-\\frac{a_{n-1}}{b_{1}+...+b_{n-1}}$\nany hints or solutions?\n" + - actual output: "I don't understand the question. Could you please rephrase it for me?\n" + - expected output: "\nIf you are willing to grant that the weighted average of several terms falls between the smallest term and the largest term (pretty easily shown), we can do the proof in one line.\n$\\frac{a_1+...+a_n}{b_1+...+b_n}$ = ($\\frac{a_1}{b_1}$)($\\frac{b_1}{b_1+...b_n}$) +...+($\\frac{a_n}{b_n}$)($\\frac{b_n}{b_1+...b_n}$) This is a weighted average of the $\\frac{a_i}{b_i}$ terms and so must not be greater than the largest or smaller than the smallest.\n" + - context: None + - retrieval context: ['"\\nshorter proof of generalized mediant inequality?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI am stuck with the proof of below expression;\n\"If F is an infinite field, then no infinite subgroup of F* (the multiplicative group of F) is cyclic.\"\nanyone can help?\n" + - actual output: "What do you think of the example that I have provided for you?\n\n## 3.2\u00a0The system will try to answer math questions based on your interactions with it.\n\nLet us look at the following example: \u201cHere are 10 examples of math questions (Q) with given answers (A).\\nQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\nA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\nA: \\n\\nQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\nA: \\n\\nQ: \\nIs there an expansion for the following summation?\\n$$ (a_1 + a_2 + \\\\cdots + a_k)^n $$\\n\\n\\nA: \\nhttp://en.wikipedia.org/wiki/Multinomial_theorem\\nThis is what you seek.\\n\\n\\nQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\nA: \\n$B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\n\\n\\nQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\nA: \\nBecause $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n\\nQ: \\nIf $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\nA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\nQ: \\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\nA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\nQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\nA: \\n\\nQ: \\nIs there a name for the geometry of a classic puzzle key?\\nIt's not an ellipse, neither a circle, ...\\n\\n\\n\\nA: \\n\\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: \\nI am stuck with the proof of below expression;\\n\\\"If F is an infinite field, then no infinite subgroup of F* (the multiplicative group of F) is cyclic.\\\"\\nanyone can help?\\n\"\n" + - expected output: "\nAny infinite field $F$ of characteristic zero has infinite cyclic subgroups of $F^\\times$. Any such field would contain a copy of $\\Bbb Q$ so it suffices to give examples for that. Take $x^{\\Bbb Z}$ for any $x\\in\\Bbb Q\\setminus\\{-1,0,1\\}$.\nIn characteristic $p$ one can show a field's group of units $F^\\times$ has an infinite cyclic subgroup if and only if it has an element $T$ transcendental over the prime subfield ${\\Bbb F}_p$. If such an element exists then take $T^{\\Bbb Z}$ to be the subgroup. If no such element exists then every element of $F$ is algebraic over ${\\Bbb F}_p$ and hence a root of unity (by the theory of finite fields) hence generates a finite group.\nThus it is incorrect to say \"no infinite subgroup of an infinite field's group of units is cyclic.\"\nSuppose $R$ is a ring such that $R^\\times$ is infinite cyclic. There can be no nontrivial roots of unity, so therefore $-1=1$ and ${\\rm char}\\,R=2$. If $R^\\times=\\langle x\\rangle$ then $x^{-1}\\in R$ and hence ${\\Bbb F}_2[x,x^{-1}]\\subseteq R$. Our above argument shows $x$ is transcendental over ${\\Bbb F}_2$. One may show the only units of ${\\Bbb F}_2[x,x^{-1}]$ are in $x^{\\Bbb Z}$. Therefore $R={\\Bbb F}_2[T,T^{-1}]$ is the minimal example of such a ring where $R^\\times\\cong\\Bbb Z$ in the sense that it injects into all rings with this property.\n" + - context: None + - retrieval context: ['"\\nmultiplicative group of infinite fields\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nIn 2 dimensions it takes at least 3 sides to make a polygon, the triangle, and in 3 dimensions it takes at least 4 faces (so far as I'm aware) to make a polyhedron. Can this rule be generalized to higher dimensions, so that the minimum number of sides or faces to make a polytope in n-dimensions is equal to n+1? If so, is there a proof for it? And if not, is there a counterexample?\n" + - actual output: + - expected output: "\nFirstly, $n+1$ is enough, because we have the $n$-simplex formed by taking $x_1 + \\ldots + x_n = 1$ with the coordinate hyperperplanes $x_i = 0$. \nTo show this is the minimum necessary, assume there is a $n-1$-dimensional polygon in $n$ space with $n$ faces of dimension $n-1$ (which I'll call facets from now on) , and look at one of its facets. This is a $n-2$-dimensional polygon in $n-1$ space. It can have at most $n-1$ facets itself, because there are only $n-1$ other facets of the original polygon that interesect with it. Therefore, if we've shown there are no $n-2$ dimensional polygons with $n-1$ faces, this is absurd, so the proof goes through by induction. \n" + - context: None + - retrieval context: ['"\\nWhat\'s the fewest number of sides required to make a polytope in n dimensions?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because there are no irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nWhat does it mean for a sequence $\\{f_n\\}_{n=1}^\\infty\\subseteq H$ to converge weakly? I know it means that it converges in the weak topology and I've read a few definitions of weak topology which all seemed quite confusing and it seems like I'm missing something important here. \n" + - actual output: + - expected output: "\nIn Banach space $X$ a sequence $\\{f_n\\}$ converges weakly to $f$ if\n$$\n\\varphi(f_n)\\to\\varphi(f),\n$$\nfor all $\\varphi\\in X^*$, where $X^*$ is the dual of $X$.\nIn the case of Hilbert space $H$, every element of the dual space is realized by an element \nof $H$ (Riesz Representation Theorem). Thus $f_n\\to f$ weakly if and only if\n$$\n\\langle f_n,\\varphi\\rangle\\to\\langle f,\\varphi\\rangle,\n$$ \nfor all $\\varphi\\in H$.\n" + - context: None + - retrieval context: ['"\\nWeak convergence in a Hilbert Space\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI could manage to solve the following integral:\n$$\n\\int \\left(k e^{-2x}+4e^{-x}\\right)^{-1/2} \\,dx\n$$\nWhere $k \\in \\Re$. However I'm not satisfied with my method so I wanted to ask if there's a more elegant way of solving it. Also I don't care if it requires further knowledge. It's just for curiosity.\nI posted as an answer my solution for comparison. What is nagging me is how simple the solution is.\nPS: After writing everything down I figured an easier substitution $u = e^{x}$ but I'll post it anyways if it helps. Still I would like to know if there's a better (in terms of elegance) solution. \n" + - actual output: + - expected output: "\nUsing $u=\\frac4ke^x$,\n$$\n\\begin{align}\n\\int\\frac{\\mathrm{d}x}{\\sqrt{ke^{-2x}+4e^{-x}}}\n&=\\frac{\\sqrt{k}}4\\int\\frac{\\frac4ke^x\\,\\mathrm{d}x}{\\sqrt{1+\\frac4ke^x}}\\\\\n&=\\frac{\\sqrt{k}}4\\int\\frac{\\mathrm{d}u}{\\sqrt{1+u}}\\\\\n&=\\frac{\\sqrt{k}}2\\sqrt{1+u}+C\\\\[3pt]\n&=\\frac{\\sqrt{k+4e^x}}{2}+C\n\\end{align}\n$$\n" + - context: None + - retrieval context: ['"\\nClosed form for $\\\\int \\\\left(k e^{-2x}+4e^{-x}\\\\right)^{-1/2}dx$\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answer relevancy score is 0.6 and there are no irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI am a beginner in more advanced algebra and my question is very simple, I would like to know the difference between these books of the same author, Hideyuki Matsumura\n\nCommutative Ring Theory (Cambridge Studies in Advanced Mathematics)\nCommutative Algebra (Mathematics lecture note series ; 56)\n\n" + - actual output: + - expected output: "\nThis is essentially identical to this MathOverflow thread: \nMatsumura: \u201cCommutative Algebra\u201d versus \u201cCommutative Ring Theory\u201d\n" + - context: None + - retrieval context: ['"\\nDifference between Matsumura\'s Commutative Algebra and Commutative Ring Theory\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nGiven two positive definite symmetric matrices $A,B$, I'd like to find the minimum $\\alpha > 0$ such that $A - \\alpha B$ is singular, i.e., the threshold where $A - \\alpha B$ is no longer positive definite. An algorithmic approach is ok if it's too hard to come up with a formula.\n" + - actual output: "What would be your question? ## Differential and Integral Equations\n\n### A remark on a paper by Golovin for the $L^1$ Subcritical Data Regime of Some Semilinear Parabolic Equations\n\nJung-Chao Ban\n\n#### Article information\n\nSource\nDifferential Integral Equations, Volume 30, Number 5/6 (2017), 489-494.\n\nDates\nFirst available in Project Euclid: 18 March 2017\n\nBan, Jung-Chao. A remark on a paper by Golovin for the $L^1$ Subcritical Data Regime of Some Semilinear Parabolic Equations. Differential Integral Equations 30 (2017), no. 5/6, 489--494. https://projecteuclid.org/euclid.die/1489802421 # Tag Info\n\n1 vote\nAccepted\n\n### How to find the value of $\\cos\\frac{2\u03c0}{7}+\\cos\\frac{4\u03c0}{7}+\\cos\\frac{6\u03c0}{7}$?\n\nYou can use the formula $$\\cos(A-B)-\\cos(A+B)=2 \\sin A \\sin B$$ and note that \\begin{align} \\cos\\dfrac{2\\pi}{7}-\\cos\\dfrac{10\\pi}{7}&=2\\sin\\dfrac{6\\pi}{7}\\sin\\dfrac{\\pi}{7}\\\\ &=-\\left(2\\sin\\...\n\u2022 58.3k\nThe first equation implies $$f(x)=\\frac{(x-c)^n}{(x+c)^{n-1}}$$ for some $c$. The second implies the same, but with $(x+d)$ replacing $(x-c)$. Hence you have to determine all possible values of $c$ and $d$, ... ## 6.17\n\nA uniform distribution has lower bound a and upper bound b, where a < b. Find the probability density function f(x). # 2019 CED Unit 3: Differentiation: Composite , Implicit, and Inverse\u00a0Functions\n\nUnit 3 covers the Chain Rule, differentiation techniques that follow from it, and higher order derivatives. These include implicit differentiation, the derivative of an inverse function, and Related Rates (first derivative applied to real world situations). This unit starts with inverse functions and ends with Related Rates.\n\n### Schedule of Posts for Unit 3\n\nDay 1 \u00a0For context we start with a review, finding slopes of lines tangent to the graph of a differentiable function f. We introduce composite functions (function composition) in order to define the Chain Rule which is used in finding the derivative of an inverse function and the derivative of any\u00a0compound function (Composition of any two or more differentiable functions).\n\nDay 2\u00a0We begin with the derivative of an inverse function and then use the Chain Rule to find higher order derivatives. We prove the Chain Rule, an essential tool in using the formula for differentiation. An essential use\u00a0of implicit differentiation is to convert a function of x(t) and y(t) into one of only t, calling for the derivative with respect to t, and then solving\u00a0for $\\frac{dy}{dx}$. This idea goes back to the original definition of the differential $\\frac{dy}{dx}$ as the quotient of the \u0394y\u2019s and the \u0394x\u2019s. Related rates are applications of this idea.\n\nDay 3 Applications of Differentiation:\u00a0The Chain Rule, Implicit Part 1: Using implicit differentiation to relate rates.\n\nDay 4 Implicit Part 2: More problems in which we use implicit differentiation to relate rates.\n\nDay 5 Implicit Part 3: One more application of Related Rates using the Accumulation Function. # Math Help - Factoring\n\n1. ## Factoring\n\nI'm completely stumped on this one. $x^6 + \\frac{1}{2}x^5 + \\frac{3}{4}x^4$. Any help? Thanks\n\n2. Originally Posted by alex_nicole\nI'm completely stumped on this one. $x^6 + \\frac{1}{2}x^5 + \\frac{3}{4}x^4$. Any help? Thanks\n$= x^4(x^2+ax+\\frac{a^2}{4})(x^2+bx+\\frac{b^2}{4})$\n\nwhere $ab=\\frac12, \\ a^2+b^2=\\frac34.$ # How do you solve 0.6n=0.5(17-n)?\n\nJun 18, 2018\n\n$n = 5.6$\n\n#### Explanation:\n\n$0.6 n = \\left(0.5\\right) \\left(17 - n\\right)$\n\nDivide both sides by $0.6$:\n$\\frac{0.6 n}{0.6} = \\frac{\\left(0.5\\right) \\left(17 - n\\right)}{0.6}$\n\n$n = 8.5 - 0.8333333333 n$\n\nAdd $0.8333333333 n$ to both sides:\n$n + 0.8333333333 n = \\left(8.5\\right) + \\left(0.8333333333 n\\right)$\n\n$1.8333333333 n = 8.5$\n\nDivide both sides by $1.8333333333$:\n$\\frac{1.8333333333 n}{1.8333333333} = \\frac{\\left(8.5\\right)}{1.8333333333}$\n\n$n = 4.761904762$\n\nRound to the nearest thousandth:\n$n = 5.6$ # Is there a way to use the 'showframe' option of geometry along with the 'a4paper' option?\n\nI am using Texmaker (XeLaTex) and would like to use the showframe option of the geometry package along with the paper size a4paper, but I don't know if it is possible.\n\n\u2022 Yes, just load the package like this \\usepackage[showframe=true]{geometry} \u2013\u00a0AboAmmar May 25 '16 at 0:38\n\u2022 Welcome! Please post an MWE so we can test our suggestions. \u2013\u00a0cfr May 25 '16 at 1:07\n\u2022 @AboAmmar - That's not quite right, though. The showframe option will cause the frame lines to extend beyond the paper by 3mm (or whatever length is set for the showframe parameter) on all four sides of the page. \u2013\u00a0Mico May 25 '16 at 7:09\n\u2022 @Mico, Yes you are right it is better than what I wrote, but still the option name itself explains how it's working. \u2013\u00a0AboAmmar May 25 '16 at 8:16\n\nYou can use the geometry package with its showframe option to print frame around the text block (and also show you other important measurements of the document layout) as follows:\n\\usepackage[a4paper,showframe]{geometry} # [Solution] Drainage solution codeforces\n\nDrainage solution codeforces \u2013 You are given two strings \ud835\udc60s\u00a0and\u00a0\ud835\udc61t. Each character of these strings can be either a digit from\u00a0019, or an uppercase letter from\u00a0AZ, i.e. the set of allowed characters is\u00a00\u20269A\u2026Z.\n\n## [Solution] Drainage solution codeforces\n\nEach string consists of some (possibly zero) non-empty words separated by one space character. Let\u2019s denote the\u00a0\ud835\udc56i-th word in string\u00a0\ud835\udc60s\u00a0as\u00a0\ud835\udc60\ud835\udc56si, similarly \u2014 the\u00a0\ud835\udc57j-th word in string\u00a0\ud835\udc61t\u00a0\u2014 as\u00a0\ud835\udc61\ud835\udc57tj. Each non-empty word consists of one or more consecutive letters. Let\u2019s denote the number of letters in a word as its length.\n\nDrainage solution codeforces\n\nYou can do several (possibly zero) actions with the strings. In one action you must do exactly one of the following:\n\n\u2022 either increase the length of some non-empty word in\u00a0\ud835\udc60s, or\n\u2022 either decrease the length of some non-empty word in\u00a0\ud835\udc61t.\n\nHere \u201cincreasing\u201d means that new letters appear to a position after the end of the old word (and there is no overlap with it), and \u201cdecreasing\u201d means that some letters are removed from a position before the start of the old word (and again, there is no overlap). Letters cannot be added/removed from an empty word.\n\nFor example, if we have two strings\u00a0\ud835\udc60s=\u201cACAT\u201ds=\u201dACAT\u201d\u00a0and\u00a0\ud835\udc61t=\u201cCATATC\u201dt=\u201dCATATC\u201d, then in one action you can either increase the length of \u201cCAT\u201d in\u00a0\ud835\udc60s\u00a0to become \u201cACATAT\u201d, or decrease the length of \u201cCAT\u201d in\u00a0\ud835\udc61t\u00a0to become \u201cCA\u201d.\n\nDrainage solution codeforces\n\nLet\u2019s denote the length of a string as its total number of characters, i.e. the sum of lengths of all words in it. Among all possible ways to make strings\u00a0\ud835\udc60s\u00a0and\u00a0\ud835\udc61t\u00a0the same with the given set of actions, you need to find one that minimizes their total length.\n\n## [Solution] Drainage solution codeforces\n\nThe first line contains an integer\u00a0\ud835\udc5bn\u00a0(1\ud835\udc5b2000001\u2264n\u2264200000) \u2014 the number of test cases. Each of the next\u00a0\ud835\udc5bn\u00a0lines describes one case, consisting of two strings\u00a0\ud835\udc60s\u00a0and\u00a0\ud835\udc61t\u00a0(1|\ud835\udc60|,|\ud835\udc61|51061\u2264|s|,|t|\u22645\u22c5106), where |\ud835\udc65||x| is the length of string\u00a0\ud835\udc65x.\n\nIt\u2019s guaranteed that\u00a0\ud835\udc60s\u00a0and\u00a0\ud835\udc61t\u00a0consist only of uppercase letters and digits from\u00a0090\u20269, i.e. they match the description above. Also it\u2019s guaranteed that all strings are non-empty.\n\nOutput\n\nFor each test case print one line \u2014 the minimum total length among all valid ways to make strings\u00a0\ud835\udc60s\u00a0and\u00a0\ud835\udc61t\u00a0the same. If there is no way to do so, then print\u00a0-1.\n\nExample\ninput\n\nCopy\n3\nACAT CATATC\nA 123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ\nS SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS" + - expected output: "\nSince $A$ and $B$ are positive definite, you could write:\n$$\ndet\\left( B^{-1}A - \\alpha I \\right) = 0\n$$\nThe problem reduces to finding the eigenvalues of $B^{-1}A$. \n" + - context: None + - retrieval context: ['"\\nFinding minimum $\\\\alpha > 0$ so that $\\\\det(A - \\\\alpha B) = 0$ for positive definite $A,B$\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI'm asking this in a programming context (because I'm a programmer) but I'm looking for general answers as well.\nIn programming, all of the implementations of asin ($\\arcsin$) that I have seen involve, at some point, the use of a constant directly derived from $\\pi$ (for example, this implementation uses a hard-coded $\\pi/2$).\nMy question is: Is there another method of calculating $\\arcsin$ that doesn't rely on knowing $\\pi$?\nThe only method I have been able to imagine is to substitute calculated approximations of $\\pi$ in place for the hard-coded values in the implementations (e.g. approximate the area of a circle by counting discrete pairs $(x,y)$ that satisfy $x^2+y^2 because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nLet $f$ be continuous on $[0,1]$ and $f(0)=f(1)$. Also, $n\\in \\mathbb{N}$. Prove that there exists some $x$ s.t. $f(x)=f(x+\\frac{1}{n})$.\nI think I need to assume towards a contradiction that if $g(x)=f(x)-f(x+\\frac{1}{n})$, then $g(x)\\neq 0$ $\\forall x$.\nSo, $g(0)=f(0)-f(\\frac{1}{n})$, $g(1)=f(1)-f(1+\\frac{1}{n})$. Since $f(0)=f(1)$, \n$g(0)-g(1)=f(1+\\frac{1}{n})-f(\\frac{1}{n})$. For context, I am only up to Ch. 8 in Spivak, so I cannot use any derivative-based theorems which is why I am stuck here.\n" + - actual output: "I need to know what you mean by a cyclotomic field.\n" + - expected output: "\nAssume the contrary that $g(x)$ is not $0$ on $[0,1-\\frac{1}{n}]$, which means either $g(x)>0$ or $g(x)<0$ on $[0,1-\\frac{1}{n}]$ (since, g is continuous). If, $g(x)>0 \\implies f(0)>f(\\frac{1}{n})>f(\\frac{2}{n})>\\cdots>f(1-\\frac{1}{n})>f(1)$, contradiction !! Similarly, for $g<0$, we get a contradiction. Therefore, $g(x)$ has a zero in $[0,1-\\frac{1}{n}]$.\n" + - context: None + - retrieval context: ['"\\nSimple intermediate value theorem proof\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.3333333333333333, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.33 because the answer relevancy score is 0.33, and there are two reasons why the score can't be higher based on irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nThe task is to determine if the series converges absolutely, conditionally or doesn't converge at all.\n$$\\sum_{k=1}^\\infty \\frac{\\sin(k)}{k!}$$\nI have tried solving it with D'Alembert test and comparison test method. No luck.\nWe haven't covered integration of\n$$\\int_1^\\infty \\frac{\\sin(k)}{k}$$\nI am stuck. Please give me a hint how to solve it.\nThank you for your attention!\n" + - actual output: "\"Here are 10 examples of math questions (Q) with given answers (A).\\nQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\nA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\nA: \\n\\nQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\nA: \\n\\nQ: \\nIs there an expansion for the following summation?\\n$$ (a_1 + a_2 + \\\\cdots + a_k)^n $$\\n\\n\\nA: \\nhttp://en.wikipedia.org/wiki/Multinomial_theorem\\nThis is what you seek.\\n\\n\\nQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\nA: \\n$B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\n\\n\\nQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\nA: \\nBecause $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n\\nQ: \\nIf $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\nA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\nQ: \\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\nA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\nQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\nA: \\n\\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: \\nThe task is to determine if the series converges absolutely, conditionally or doesn't converge at all.\\n$$\\\\sum_{k=1}^\\\\infty \\\\frac{\\\\sin(k)}{k!}$$\\nI have tried solving it with D'Alembert test and comparison test method. No luck.\\nWe haven't covered integration of\\n$$\\\\int_1^\\\\infty \\\\frac{\\\\sin(k)}{k}$$\\nI am stuck. Please give me a hint how to solve it.\\nThank you for your attention!\\n\"\n" + - expected output: "\nHint: Think about $e^1 {}{}{}{}{}{}{}{}{}{}{}{}{}$\n" + - context: None + - retrieval context: ['"\\nConvergence of $\\\\,\\\\,\\\\sum_{k=1}^\\\\infty \\\\frac{\\\\sin(k)}{k!}$\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answer relevancy score is 0.6., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nlet three complex $z_{1},z_{2},z_{3}$ such \n$$z_{1}+z_{2}+z_{3}\\neq 0,|z_{1}|=|z_{2}|=|z_{3}|=1$$\nFind this value\n$$\\left|\\dfrac{z_{1}z_{2}+z_{1}z_{3}+z_{2}z_{3}}{z_{1}+z_{2}+z_{3}}\\right|$$\nMy idea:if $z_{1},z_{2},z_{3}$ is real numbers,and such \n$z_{1}=z_{2}=z_{3}=1$,then we easy to find this value\n$$\\left|\\dfrac{z_{1}z_{2}+z_{1}z_{3}+z_{2}z_{3}}{z_{1}+z_{2}+z_{3}}\\right|=1$$\nBut other complex case,I can't,Thank you\n" + - actual output: + - expected output: "\nNote that\n$$\\lvert z_2z_3+z_3z_1+z_1z_2\\rvert=\\lvert z_1z_2z_3\\rvert\\cdot\\lvert z_1^{-1}+z_2^{-1}+z_3^{-1}\\rvert=\\lvert\\overline z_1+\\overline z_2+\\overline z_3\\rvert=\\lvert z_1+z_2+z_3\\rvert$$\nCan you point out the reason of each equality?\n" + - context: None + - retrieval context: ['"\\nHow find this value $\\\\left|\\\\frac{z_{1}z_{2}+z_{1}z_{3}+z_{2}z_{3}}{z_{1}+z_{2}+z_{3}}\\\\right|$\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answer relevancy score is 0.6., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n$A$ is a commutative ring with identity.\n$I$ is a ideal of $A$.\nthen ideal $I$ is prime iff $A/I$ is a integral domain.\nhere is what I thought\n$(\\Rightarrow)$ We want to prove $A/I$ is a integral domain. It's equivalent to prove there are no nonzero element $a+I$ can divide $0$ such that $a\\in A$. Let $a,b\\in A$. Then $(a+I)(b+I)=ab+I$. Let $ab+I=I$, with $a+I\\neq I$. So we need prove $b=0$ if $a\\neq 0$ and $ab=0$. It's equivalent to prove $A$ is a integral domain.\n$A$ is a integral domain iff the zero ideal is prime.\nI stuck at here and doubt something is wrong.\n($\\Leftarrow$) \n$A/I$ is a integral domain $\\iff$ A is a integral domain\nthen I have no ideal how to prove.\n" + - actual output: + - expected output: "\nYou started off well and then got a little bit lost towards the end. By definition, $A/I$ is an integral domain if and only if the following statement holds.\n\nIf $a,b\\in A$ are such that $(a+I)(b+I) = I$, then either $a + I = I$ or $b+I = I$. Also, we require $A/I\\neq 0$, or in other words, $I\\subsetneq A$.\n\nLet's first make sure we know what the statement $(a+I)(b+I) = I$ means. You correctly point out that $(a+I)(b+I) = ab + I$. Thus it means $ab +I = I$, or equivalently, that $ab\\in I$. Similarly, the statement $a+I = I$ is equivalent to $a\\in I$, and $b+I = I$ is equivalent to $b\\in I$. So we can rewrite the boxed statement above as \n\nIf $a,b\\in A$ are such that $ab\\in I$, then either $a\\in I$ or $b\\in I$. Also, we require $A/I\\neq 0$, or in other words, $I\\subsetneq A$.\n\nNotice, this last boxed statement is exactly the definition of $I$ being a prime ideal of $A$. In this way we see that $I$ being prime is equivalent to $A/I$ being an integral domain.\n\nEDIT: Just one additional comment. I should emphasize that the statement $A$ is an integral domain $\\iff A/I$ is an integral domain is false. For instance, if $A = \\mathbb{Z}$ and $I = 4\\mathbb{Z}$, then $A$ is an integral domain, but $A/I = \\mathbb{Z}/4\\mathbb{Z}$ is not an integral domain!\n" + - context: None + - retrieval context: ['"\\nHow to prove \\"ideal $I$ is prime iff $A/I$ is a integral domain \\"?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nSuppose $f: \\mathbb{R}^n \\to \\mathbb{R}^m$ is homogeneous and differentiable at $0$, then does it follow that $f$ is a linear transformation?\nI know that I need to show that for any $x,y \\in \\mathbb{R}^n$ and $\\lambda \\in \\mathbb{R}$, then\n$$ f( \\lambda x + y ) = \\lambda f(x) + f(y) $$. \nI know by homogeneity that $f( \\lambda x ) = \\lambda f(x) $. But how can I show that $f$ preserves the operation $+$ ?\n" + - actual output: + - expected output: "\nSince $f$ is differentiable at $0$, it has the form $$f(x) = f(0) + D_f(0) x + o(x)$$\nwhere $D_f(0)$ is a matrix and $\\lim_{x\\rightarrow 0}\\frac{o(x)}{||x||} = 0$. Choosing $\\lambda = 0$ in $f(\\lambda x) = \\lambda f(x)$ quickly gives you $f(0)=0$, so $f$ has the form \n$$f(x) = D_f(0) x + o(x).$$\nNow, replace $x$ with $\\lambda x$ and you get\n$$\\lambda f(x) = \\lambda D_f(0)x + o(\\lambda x)$$\nor $$o(\\lambda x) = \\lambda(f(x) - D_f(0) x) = \\lambda o(x).$$\nNow take any $0\\neq x\\in\\mathbb R^n$. You then have\n$$0=\\lim_{\\lambda \\downarrow 0}\\frac{o(\\lambda x)}{||\\lambda x||}=\\lim_{\\lambda\\downarrow 0}\\frac{o(x)}{||x||} = \\frac{o(x)}{||x||}.$$\nThis means $o(x) = 0$ for all $x$, meaning $f$ has the form $f(x) = D_f(0) x$ and $f$ is linear.\n" + - context: None + - retrieval context: ['"\\nHomogeneity and Differentiability at $0$ implies linearity?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Passed Answer Relevancy (score: 1.0, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 1 because it is a good answer., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI'm having trouble answering this question:\nA person leaves for work between $8:00$ A.M. and $8:30$ A.M. and takes between $40$ and $50$ minutes to get to his office. Let $X$ denote the time of departure and let $Y$ denote the time of travel. If we assume that these random variables are independent and uniformly distributed, find the probability that he arrives at the office before $9:00$ A.M.\nAny help would be appreciated.\n" + - actual output: "I'm an intelligent mathematical assistant.\n\n" + - expected output: "\nHint: Draw a square, whose base is 30 minutes in width (8 to 8:30), and whose height is 10 minutes (40 to 50 minutes). Shade red those combinations that lead to him being late, and blue those combinations that lead to him being on time. Then find the fraction of the total area that is blue.\n" + - context: None + - retrieval context: ['"\\nProbability of arriving at office before $9$ am\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer is irrelevant to the input., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nSupose we have an exact sequence $$A\\overset{f}\\longrightarrow B\\overset{g}\\rightarrow C\\overset{h}\\rightarrow D$$ in an abelian category $\\mathcal{A}$. Is it true that $f$ is an epimorphism if and only if $h$ is a monomorphism?\nIt is clear that for any category of modules this is true.\n" + - actual output: + - expected output: "\n$f$ epi $\\Leftrightarrow$ $\\mathrm{im}(f)=B$ $\\Leftrightarrow$ $\\ker(g)=B$ $\\Leftrightarrow$ $g=0$ $\\Leftrightarrow$ $\\mathrm{im}(g)=0$ $\\Leftrightarrow$ $\\ker(h)=0$ $\\Leftrightarrow$ $h$ mono.\n" + - context: None + - retrieval context: ['"\\nExercise in an abelian category\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Passed Answer Relevancy (score: 1.0, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 1.00 because the answer relevancy score is 1.00 and there are no reasons why it can't be higher., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nThe wikipedia definition of lambda terms is:\n\nThe following three rules give an inductive definition that can be\n applied to build all syntactically valid lambda terms:\n\na variable, $x$, is itself a valid lambda term\nif $t$ is a lambda term, and $x$ is a variable, then $(\\lambda x.t)$ is a lambda term (called a '''lambda abstraction''');\nif $t$ and $s$ are lambda terms, then $(ts)$ is a lambda term (called an '''application'''). Nothing else is a lambda term. Thus a\n lambda term is valid if and only if it can be obtained by repeated\n application of these three rules. However, some parentheses can be\n omitted according to certain rules. For example, the outermost\n parentheses are usually not written.\n\n\nHowever, I cannot understand what variables are. Are they numbers? Or are they functions?\nWhat is the result of an application? Is it a number or is it a function?\nReading many explanations regarding lambda calculus, I often see expressions like: \n$$\\lambda x.x+3$$\nBut, according to the definition above, those expressions are not valid lambda terms, because integers and operators are not valid lambda terms. So I guess I cannot express arithmetic expressions in lambda calculus. How can I describe functions?\n" + - actual output: "I'm glad you have contacted me! You seem quite stuck on this problem. This is actually a very easy question. Let $K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$ and let $\\\\Omega$ be the integral operator defined by $$\\Omega f=f^2$$ Then we can see that $$\\\\Omega K=\\\\int^{1}_{-1} K(x,y)K(x,y)dx=\\\\frac{1}{\\\\gamma^2{\\\\pi}^2}\\$$\n" + - expected output: "\nThe point is that $\\lambda$-calculus is pure syntax. The symbols are just symbols and don't stand for anything in particular. However, it is possible to do arithmetic in $\\lambda$-calculus using Church numerals. \nThis is not to be confused with the simple notion of $\\lambda$-abstraction, which finds use outside $\\lambda$-calculus as a convenient way of defining function terms.\n" + - context: None + - retrieval context: ['"\\nMeaning of variables and applications in lambda calculus\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer is irrelevant to the input., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nIf I have function $u: \\mathbb{R}^n \\longrightarrow \\mathbb{R}$ smooth, does it always hold that:\n$$\\nabla^2(\\nabla u)= \\nabla(\\nabla^2 u)$$\nthis is true in general?\n" + - actual output: + - expected output: "\nThis is true for a $C^3$ function defined on $\\mathbb{R}^n$. To see this, note that \n$$\\tag{1}\\Delta (\\nabla u)=\\Delta\\Big(\\frac{\\partial u}{\\partial x_1},\\frac{\\partial u}{\\partial x_2},...,\\frac{\\partial u}{\\partial x_n}\\Big)\n=\\Big(\\Delta(\\frac{\\partial u}{\\partial x_1}),\\Delta(\\frac{\\partial u}{\\partial x_2}),...,\\Delta(\\frac{\\partial u}{\\partial x_n})\\Big).$$\nOn the other hand, we have \n$$\\tag{2}\\nabla(\\Delta u)=\\Big(\\frac{\\partial }{\\partial x_1}(\\Delta u),\\frac{\\partial}{\\partial x_2}(\\Delta u),...,\\frac{\\partial}{\\partial x_n}(\\Delta u)\\Big).$$\nSince $u$ is $C^3$, we have for $1\\leq i\\leq n$\n$$\\Delta(\\frac{\\partial u}{\\partial x_i})=\n\\sum_{j=1}^n\\frac{\\partial^2}{\\partial x_j^2}(\\frac{\\partial u}{\\partial x_i})=\\sum_{j=1}^n\\frac{\\partial^3u}{\\partial x_j^2\\partial x_i}\\\\\n=\\sum_{j=1}^n\\frac{\\partial^3u}{\\partial x_i\\partial x_j^2}\n=\\frac{\\partial}{\\partial x_i}\\Big(\\sum_{j=1}^n\\frac{\\partial^2u}{\\partial x_j^2}\\Big)\n=\\frac{\\partial}{\\partial x_i}(\\Delta u),\n$$\nwhich implies that \n$$\\Delta (\\nabla u)=\\nabla(\\Delta u)$$\nby $(1)$ and $(2)$. \n" + - context: None + - retrieval context: ['"\\nDoes the Laplacian and gradient commute?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nHow does $\\displaystyle \\frac{e^{-x}}{1-e^{-x}}$ become $\\sum_{k=1}^{\\infty}e^{-kx}$?\nI know $\\displaystyle \\frac{e^{-x}}{1-e^{-x}}=\\frac{1}{e^x-1}=\\left(\\sum_{k=0}^{\\infty} \\frac{x^k}{k!}-1\\right)^{-1}=\\left(\\sum_{k=1}^{\\infty} \\frac{x^k}{k!}\\right)^{-1}$ and we know $\\displaystyle\\sum_{k=0}^{\\infty} \\frac{x^k}{k!}=e^x$\n" + - actual output: "Here are 10 examples of math questions (Q) with given answers (A).\nQ:\nCan you help me with this ''simple'' exercise: Prove that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\nA:\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\nQ:\nIs the following integral equation solvable ? $$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x)$$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)} \\ and \\ f(x)=e^{i\\\\gamma x}$$ And $\\\\gamma$ is a parameter.\nA:\n$$ e^{-i\\\\gamma x} + \\\\int_{-1}^{1}\\\\frac{\\\\sin \\\\gamma (x-y) }{\\\\pi (x-y)}e^{-i\\\\gamma y}dy = 0 $$ Now this is a Fredholm integral equation of the second kind. So, we can write it as:\n$$ f(x) + \\int_{-1}^{1}K(x,y)f(y)dy=0$$ which has solution\n$$ f(x) = -\\\\sum_{n=1}^{\\\\infty}\\\\frac{1}{2}\\\\sqrt{\\\\frac{2}{\\\\pi}}\\\\exp(- x^{2})sin(nx)$$ So, we have that $$F(x)=\\\\int_{-1}^{x}f(y)dy$$ satisfies the given equation.\nQ:\nGiven that $X$ and $Y$ are positive definite matrices, how can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\nA:\nTake the eigenvalue decompositions of $X$ and $Y$. We have that $\\\\Sigma_{X} = U_X^T\\\\Sigma_XU_X$ and $\\\\Sigma_{Y}=U_Y^T\\\\Sigma_YU_Y$, where $\\\\Sigma_X$ and $\\\\Sigma_Y$ are diagonal matrices. Now, $\\\\sigma(X+Y) = \\\\sigma(\\\\Sigma_{X}+\\\\Sigma_{Y})=\\\\sigma(U_X^T\\\\Sigma_XU_X + U_Y^T\\\\Sigma_YU_Y)$, so it's enough to bound $\\\\sigma(A)$ for a diagonal matrix $A$, and that is fairly easy.\nQ:\nIs there an expansion for the following summation? $$ (a_1 + a_2 + \\\\cdots + a_k)^n$$\nA: http://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\nQ:\nI have to simplify the following expression: $B=\\\\sqrt{x^2} - x$ The only thing that I can do is: $\\\\sqrt{a^2}=|a|$ thus $B=|x|-x$. Is that enough?\nA: $B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\nQ:\nQuick question: Wolfram Alpha tells me that $\\\\int |e^{ix}|^2 dx = x + C$ Why is that?\nA: Because $|e^{ix}|=1$ for every $x \\\\in \\\\mathbb{R}$.\nQ:\nIf $10^{80}=2^x$, what is the value of $x$? (Or, what binary word length would you need to contain $10$ to the $80$?).\nA: $\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\nQ:\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. I would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\nA: OK - $f(x)=7=3x-4$, can you solve this?\nQ: On p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? Thanks for help.\nA:\n### Tutor\nYou are an intelligent mathematical assistant.\n\n" + - expected output: "\nRecall the geometric series (see Wikipedia): for any $y$ with $|y|<1$,\n$$\\frac{1}{1-y}=1+y+y^2+\\cdots=\\sum_{k=0}^\\infty y^k.$$\nTherefore, for any such $y$, we also have\n$$\\frac{y}{1-y}=y+y^2+y^3+\\cdots=\\sum_{k=1}^\\infty y^k.$$\nNow let $y=e^{-x}$ (though observe that we need $x>0$ to have $e^{-x}<1$).\n" + - context: None + - retrieval context: ['"\\nHow does $\\\\displaystyle \\\\frac{e^{-x}}{1-e^{-x}}$ become $\\\\sum_{k=1}^{\\\\infty}e^{-kx}$?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.3333333333333333, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.33 because the answer relevancy score is 0.33 and there are no irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n\nShow that $(xy)^n=x^ny^n$ if $xy=yx$. \n\nI assume I will need 3 different cases: $n < 0$, $n=0$, and $n > 0$.\nFor the $n > 0$ case, can I use induction? For the base case I'll show that if $n=$1 then $(xy)^1=x^1y^1$ which will give us xy=xy so the base case is satisfied. Then I'll assume that the statement is true for n+1. Is that on the right track? \nFor $n=0$, is it just trivial? that $(xy)^0=x^0y^0$ and when talking about groups $x^0=1$ so we will have $1=$1 which is true. \nFor $n < 0$, I was advised to use $(xy)^n=((xy)^{-1})^{-n}$. I'm not exactly sure how to go about this step. \nI don't think I completely understand the part where this only works when $xy=yx$. \nAny suggestions or help would be great! Thanks! \n" + - actual output: "Here are 10 examples of math questions (Q) with given answers (A).\\nQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\nA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\nA: \\nYes. Let $F(x)$ be the Green's function of the equation.\\n\\n\\nQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\nA: \\nI don't know about bounding, but for singular values:\\n$$\\\\left( \\\\sum_{i=1}^{k} \\\\lambda_i \\\\right)^2 \\\\geq \\\\mathrm{tr} X Y \\\\geq \\\\lambda_1 \\\\dots \\\\lambda_k.$$\\n(Use the Hoffman-Wielandt inequality and Schur complements.)\\n\\n\\nQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\nA: \\nYes.\\n\\n\\nQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\nA: \\n$|e^{ix}|=1$\\n\\n\\nQ: \\nIf $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\nA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\nQ: \\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\nA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\nQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\nA: \\nYou probably mean modules over a PID.\\n\\n\\nQ: \\nIs there a name for the geometry of a classic puzzle key?\\nIt's not an ellipse, neither a circle, ...\\n\\n\\n\\nA: \\nRound things are called \"crystallographic\" (in German: kristallographisch).\\nThe classical puzzle pieces are trapezoids.\\n\\n\\nQ: \\nShow that $(xy)^n=x^ny^n$ if $xy=yx$. \\nI assume I will need 3 different cases: $n < 0$, $n=0$, and $n > 0$.\\nFor the $n > 0$ case, can I use induction? For the base case I'll show that if $n=$1 then $(xy)^1=x^1y^1$ which will give us xy=xy so the base case is satisfied. Then I'll assume that the statement is true for n+1. Is that on the right track? \\nFor $n=0$, is it just trivial? that $(xy)^0=x^0y^0$ and when talking about groups $x^0=1$ so we will have $1=$1 which is true. \\nFor $n < 0$, I was advised to use $(xy)^n=((xy)^{-1})^{-n}$. I'm not exactly sure how to go about this step. \\nI don't think I completely understand the part where this only works when $xy=yx$. \\nAny suggestions or help would be great! Thanks! \\n\"\n" + - expected output: "\nThe answer of Ishfaag is right, but I want to review some points about this problem.\nThere are a lot of attempts for generalizing the concept of abelian groups and one of them is $n$-abelian groups. A group $G$ is said to be $n$-abelian if $(xy)^n = x^ny^n$ for all $x,y \\in G$. It is also easy to see that a group $G$ is $n$-abelian if and only if it is $(1-n)$-abelian. The idea of $n$-abelian is proposed by Levi in $1944$. As the friends said, $G$ is an abelian group if and only if $2$-abelian. $n$-abelian have been the subject of many investigations, for instance, Alperin, Delizia-Tortora and Baer.\n" + - context: None + - retrieval context: ['"\\n$n$-abelian Groups\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI was tutoring, and for some reason I could not get the correct answer to this question:\nA licence plate can have 6 characters; 1 letter followed by 5 digits. In how many ways can a plate me made?\nI tried 26$\\cdot\\text{P}(10,5)$. What did I do wrong?\n" + - actual output: + - expected output: "\nLicense plates with repeated digits (for example, $A00000$) are allowed under the rules you cited. However, $P(10,5)$ only counts the sequences of five digits where repetition is not allowed.\nSee the Wikipedia page on permutations.\n" + - context: None + - retrieval context: ['"\\nA homework question about permutations\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nDetermine all pairs $(a, b)$ of positive integers such that $ab^{2} + b + 7$ divides\n$a^{2}b + a + b$. I really have no idea where to start with this. This is the first IMO problem that I attempted, because it looked to be the easiest, but I honestly have no idea where to begin. Any help or hints would be appreciated.\n" + - actual output: "\"Hi there.\"\n### Question\n[1] \"Can you help me with this ''simple'' exercise:\"\n[2] \"Prove that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\"\n### Answer\n[1] \"A cyclotomic field is... And the complex roots of $x^3-1$ are...\"\n### Question\n[1] \"Is the following integral equation solvable ?\"\n[2] \"$F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x)$ Where $K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$ and $f(x)=e^{i\\\\gamma x}$\"\n[3] \"and $\\\\gamma$ is a parameter.\"\n### Answer\n[1] \"\"\n### Question\n[1] \"Given that $X$ and $Y$ are positive definite matrices,\"\n[2] \"how can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$? \"\n### Answer\n[1] \"\"\n### Question\n[1] \"Is there an expansion for the following summation?\"\n[2] \"$(a_1 + a_2 + \\\\cdots + a_k)^n\" [3] \"\" ## # Question 4 " + - expected output: "\nI've got a different solution to the one offered here already. \nIf $ab^2+b+7 \\mid a^2b+a+b$, then $ab^2+b+7 \\mid b(a^2b+a+b) - a(ab^2+b+7) = b^2-7a$\nNow there are three cases left to check:\n\n$b^2-7a = 0$. This yields the infinite set of solutions $(a,b) = (7k^2, 7k)$, where $k\\ge1$\n$b^2-7a > 0$. Hence, we have $ab^2+b+7 \\le b^2-7a 0$ we must have $b^2-7 <0 \\Rightarrow b=1$ or $2$. Checking these cases individually we get two more solutions: $(a, b) = (11, 1), (49, 1)$.\n\nTherefore, the solutions are $(a, b) = (11, 1), (49, 1), (7k^2, 7k)$\n" + - context: None + - retrieval context: ['"\\nIMO problem 4, $1998$\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nSay I have the following recurrence:\n$$T(n) = n + T\\left(\\frac{n}{2}\\right) + n + T\\left(\\frac{n}{4}\\right) + n + T\\left(\\frac{n}{8}\\right) + \\cdots +n + T\\left(\\frac{n}{n}\\right) $$\nwhere $n = 2^k$, $k \\in \\mathbb{N} $ and $T(1) = 1$.\nsimplified to:\n$$T(n) = n \\log_2n + \\sum_{i=1}^{\\log_2n}T\\left(\\frac{n}{2^i}\\right) $$\nThe Master's theorem is not applicable; neither is the Akra-Bazzi method since $k = \\log_2$ is not a constant. \nWhat strategy can I use to find a closed form solution? I have a feeling that the closed form is $T(n) = \\sum_{i=0}^{\\log_2n}\\left[j\\frac{n}{2^i} \\log_2 \\left(\\frac{n}{2^i} \\right)\\right] + 1 $ where $j = \\max\\left(1, 2^{i-1}\\right)$ but would like a proof.\n" + - actual output: "\"I am your intelligent mathematical assistant. You may have noticed that I tend to speak in short sentences. As such, I am not very good at answering questions. In particular, I don't know how to prove things. However, you are an intelligent person so perhaps you might be able to answer these questions yourself?\\n\\nIf you would like me to say something interesting for a change, please ask!\\n\\nI apologize in advance if the answers seem too obvious.\"\n" + - expected output: "\nSince we only need and evaluate $T(n)$ when $n$ is a power of $2$, say that $a(k)=T(2^k)$. The recursion becomes (I will use $n$ again from now on):\n$$\na(n)=n2^n+\\sum_{i=0}^{n-1}a(i)\n$$\nDefine $s(n)=\\sum_{i=0}^na(i)$. The above relation can be rewritten to\n$$\ns(n)-s(n-1)=n2^n+s(n-1)\n$$\nthus\n$$\ns(n)=n2^n+2s(n-1)\n$$\nThis is is just a first degree linear recurrence relation that should be solvable.\nSince we have $T(1)=1$, it follows that $a(0)=1$ and thus $s(0)=1$. The general solution for the recursion relation is\n$$\ns(n)=2^{n-1}(n(n+1)+C)\n$$\nfor any constant $C$. (I found this using mathematica. The homogeneous part ($C2^{n-1}$) is obvious, but the inhomogeneous part is (probably) easiest found by writing out some small values. See @CarstenSchultz's comment below for a nice way to find it.) Solving this with $n=0$ gives\n$$\n1=\\frac 12 C\n$$\nThus, $C=2$ and we get\n$$\ns(n)=2^{n-1}(2+n+n^2)\\\\\na(n)=s(n)-s(n-1)=2^{n-2}(2+3n+n^2)\\\\\nT(2^n)=s(n)-s(n-1)=2^{n-2}(2+3n+n^2)\\\\\n$$\nNote that $a(0)=T(1)=1\\neq s(0)-s(-1)=\\frac 12$. Since $s(0)=1$, we have $s(-1)=0$, which makes sense since it is an empty sum. The formula for $s(n)$ is thus only valid for nonnegative $n$.\n" + - context: None + - retrieval context: ['"\\nHow to tackle a recurrence that contains the sum of all previous elements?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nThe question is:\nDetermine the smallest multiple of 9 which divided by each of the numbers 2, 5 and 11 leaves a remainder 1.\nThe answer is 441.\nWhat I did when I tried solving this was to set up 3 different equations:\n\n9n = 1 mod 2\n9n = 1 mod 5\n9n = 1 mod 11\n\nAnd solved for each n. I got values of 27, 54 and 108 respectively. I didn't really know where to go from here to get the answer of 441. I think I may be on the wrong path, I'm not really sure. If someone can enlighten me as to why the answer is 441, it will be greatly appreciated.\nThanks.\n" + - actual output: + - expected output: "\nSince $2$, $5$, and $11$ are relatively prime, your systems is the same as the single equation $$9n\\equiv1\\mod 110$$ by the Chinese Remainder Theorem. So one approach is to start looking at the multiples of $110$ and add $1$. Real soon you hit $441$, which is the first divisible by $9$.\n(Which of course, means $n$ is $441/9$, or $49$.)\nYou wouldn't even have to guess and check at all, if you compute that $$110\\equiv2\\mod9$$ then you can see that $$4\\cdot110\\equiv-1\\mod9$$ so $4\\cdot110+1$ is your multiple of $9$.\n" + - context: None + - retrieval context: ['"\\nDetermine the smallest multiple of 9 which divided by each of the numbers 2, 5 and 11 leaves a remainder 1.\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because there are no irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nLet $T(S)$ be the set of all functions on $S = \\{\\ 1,2,3 \\}\\ $. $T(S)$ is a group under composition of functions.\nI am to prove that this is either true or not. I would like some help understanding exactly what the author means by \"Group under composition of functions\". And can you give me an example of a set of all functions on some other set?\n" + - actual output: "\"Here are 10 examples of math questions (Q) with given answers (A).\\nQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\nA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\nA: \\nThat's very complex... If you take an inverse Laplace transform, your equation becomes:$$ \\\\frac{F(t)}{s}+\\\\int^{1}_{-1}\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)} F(y)dy=e^{i\\\\gamma t}$$Then you have to solve a differential equation. I'm afraid that there is no simple solution to your problem... Or I can't see it...\\n\\n\\nQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\nA: \\nI'm not sure... But probably you could use some kind of inequality between matrix norms. For example, if you know that your matrices are Hermitian, then $$\\\\\\\\|X\\\\\\|_2 \\\\leq \\\\sqrt{n} \\\\sigma_1(X)$$ where $\\\\sigma_1$ is the greatest singular value (the one with the greatest modulus), and $n$ is the dimension of the matrix. If you have some relation between X and Y, then you can use it to write inequalities on norms... And thus on singular values.\\n\\n\\nQ: \\nIs there an expansion for the following summation?\\n$$ (a_1 + a_2 + \\\\cdots + a_k)^n $$\\n\\n\\nA: \\nhttp://en.wikipedia.org/wiki/Multinomial_theorem\\nThis is what you seek.\\n\\n\\nQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\nA: \\n$B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\n\\n\\nQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\nA: \\nBecause $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n\\nQ: \\nIf $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\nA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\nQ: \\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\nA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\nQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\nA: \\n\\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: \\nLet $T(S)$ be the set of all functions on $S = \\\\{\\\\ 1,2,3 \\\\}\\\\ $. $T(S)$ is a group under composition of functions.\\nI am to prove that this is either true or not. I would like some help understanding exactly what the author means by \\\"Group under composition of functions\\\". And can you give me an example of a set of all functions on some other set?\\n\"\n\n### System User Input\n\nYou are an intelligent mathematical assistant.\n\n" + - expected output: "\nTest 0: Is the operation associative? If not, then you're done. $T(S)$ is not a group. If the operation is associative, then proceed to...\nTest 1: Does the set $T(S)$ contains an identity? If you have a candidate function in $T(S)$, then what equations must it satisfy? Does it? If not, then $T(S)$ is not a group. If you do have an identity, then proceed to...\nTest 2: Does every element of $T(S)$ have an inverse? Given an arbitrary function in $f \\in T(S)$, can you write down its inverse $f^{-1} \\in T(S)$? What equation must $f$ and $f^{-1}$ satisfy? (Hint: you need the identity function.) If any function fails to have an inverse, then $T(S)$ is not a group. If every function does have an inverse, then..\nCongratulations! Your set $T(S)$ is a group.\nMouse over the box to reveal a hint.\n\nOne of these tests fails, so $T(S)$ is not actually a group under composition.\n\n" + - context: None + - retrieval context: ['"\\nGroup under composition of functions\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because there are no irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n$$n \\cdot {2^{n - 1}} = \\sum\\limits_{k = 1}^n {k\\left( {\\begin{array}{*{20}{c}}\n n \\\\\n k \\\\\n\\end{array}} \\right)} $$\nThe left-hand side can describe the number of possibilities choosing a committee with one chairman. \nHow can the right-hand side feet to this story?\n" + - actual output: "A: \"Prove that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\nThe roots are 1, -1 and $w = -\\\\frac{1}{2}+i\\\\frac{\\\\sqrt{3}}{2}$\\nIf we take powers of w, they have the following structure: \\nw^0=1,\\nw^6=1,\\nw^3=-1,\\nw^4=w,\\nw^5=-w etc.\\nSo this field is cyclotomic.\\n\\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$\\nI don't think so. There are two types of functions which could work:\\n$g(x)=c_0+c_1x$, where $c_0$ and $c_1$ are constants\\nor\\nn$g(x)=a_nx^n+\\\\dots +a_0$\\nwhere all the a's are constants.\\nThe integral of a polynomial is given by: \\n$$\\\\int f(x)dx=\\\\frac{f^{(n)}(x)}{n!}$$ Where the nth derivative of $g$ is $a_nx^n+\\\\dots +a_0$.\\nAnd this doesn't equal the right hand side.\\n\\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\nThe eigenvalues of a positive definite matrix are all positive. The sum of the eigenvalues is equal to the trace.\\nSo: \\n$$\\\\sum \\\\lambda_i (x) +\\\\sum \\\\lambda_j (y)=\\\\text{trace}(x+y)=\\text{trace}(X)+\\text{trace}(Y)$$ so the singular values are bounded by the sums of the traces of X and Y.\\n\\n(a_1 + a_2 + \\\\cdots + a_k)^n $$\\ni http://en.wikipedia.org/wiki/Multinomial_theorem\\nThis is what you seek.\\n\\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\ni don't think so, because if x<0 you get: |x|-x=x-x=-x which is a problem.\\ni would go with B=\\\\text{sign}(x) \\\\sqrt{|x|} - x which works for all x. \\n\\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\ni don't know, but i guess it is because e^(ix)=cos(x)+isin(x) and so |e^(ix)|=\\\\sqrt{cos(x)^2+sin(x)^2}=1. \\nThen you get: \\\\int dx = x + C.\\n\\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\ni would do the following:\\n\\\\begin{eqnarray*} f(x) & = & 7 \\\\\\\\ f(x)/3 & = & \\\\frac{7}{3} \\\\\\\\ x-4/3 & = & 7/3 \\\\\\\\ x & = & 15/3+4/3=19/3=6.333 \\end{eqnarray*}\\ni hope that helps.\\n\\ni think this is not possible, because you have to choose a specific number n, if you go through the proof. \\n\\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\\\\"ohlich,\\nthe concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\ni think it doesn't make sense in the case of not being free.\\n\\n\\ni don't know if this fits your geometry, but a classic puzzle key (you can find more about it on google) is defined as:\\na circle which touches another circle and a line. The distance between the center of the circle to the point where the line meets the other circle is also given.\\n\\n\\n\\ni don't know.\"\n" + - expected output: "\nHint. The left hand side counts the number of possibilities of choosing the committee in this way:\n(1) choose the chairman;\n(2) choose the rest of the committee from the remaining $n-1$ people.\nBut you could also do it this way:\n(1) choose a committee of $k$ people;\n(2) choose the chairman from this committee;\nand then realise that there are various possibilities for $k$. See if you can fill in the details.\n" + - context: None + - retrieval context: ['"\\nProve an equality using combinatorial arguments\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answer relevancy score is 0.6., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n\n\n\n\n\n\nClosed. This question does not meet Mathematics Stack Exchange guidelines. It is not currently accepting answers. \n\n\n\n\n\n\n\n\n\n\n This question is not about mathematics, within the scope defined in the help center.\n\n\nClosed 8 years ago.\n\n\n\n\n\n\n\r\n Improve this question\r\n \n\n\n\nSay you are writing a mathematical modeling paper in a science journal. How do you write your variable and parameter definitions in paragraph form so that you keep active voice but do not sound repetitive. For example\n\"$x$ is the average rate of change of ...\"\nVariable name followed by \"is\" sounds correct, but this must be frowned upon due to capitalization issues. Perhaps you could write,\n\"The average rate of change of ... is [called] $x$\"\nThis is passive voice. We tend to use in proofs,\n\"Let $x$ be the rate of change of ...\"\nBut this gets repetitive fast if you are writing in paragraph form. My guess is that this is more of an issue for applied mathematicians writing for non-math audiences.\n" + - actual output: + - expected output: "\nOne of the great things about the English language is that there are so many different ways to say the same thing. So, avoiding repetitiveness should not be too difficult. Here are a few different ways to phrase your example sentence:\n(1) Let $x$ be the rate of change ...\n(2) Using $x$ to denote the rate of change, ...\n(3) Let's use $x$ to represent the rate of change ...\n(4) If $x$ is the rate of change, then ...\nAnd, of course, you can get other variants by putting this clause later in the sentence:\n(5) We will show that $y = kx$, where $x$ is the rate of change ...\nAnd so on. Bear in mind that this sort of flexibility is often a source of confusion for readers whose native language is not English. If we use different words, this suggests to them that the meaning might be somehow different, too. There is a nagging fear, for example, that \"denote\" and \"represent\" might have two different meanings. For this audience, repetition is good, sometimes. So, don't go overboard with the thesaurus.\n" + - context: None + - retrieval context: ['"\\nWhen writing a mathematical model for non-mathematical audiences, how do you begin a sentence defining a variables and parameters? [closed]\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI want to find the eigenvector of the matrix\n$$\nA \\equiv\n\\left(\\begin{array}{rr}\n-\\,{1 \\over k} & 0\n\\\\\n1 & -\\,{2 \\over k}\n\\end{array}\\right)\n$$\nI found the eigenvalue $\\lambda_1=-1/k$, $\\lambda_2=-2/k$\nThen, I have to found the eigenvector, $V_1=1/(1+k^2)^{1/2}(1, k)$ and $V_2=(0,1)$. I can't get them properly.\nThank.\n" + - actual output: "\"OK - $f(x)=7=3x-4$, can you solve this?\"\n" + - expected output: "\nFor the first eigenvalue, $\\lambda_1=-\\dfrac{1}{k}$, we have a RREF for $[A - \\lambda I]v_1 = 0$ of:\n$$\\begin{bmatrix}1 & -\\dfrac{1}{k}\\\\0 & 0\\end{bmatrix}v_1 = 0$$\nWe choose $v_1 = \\left(\\dfrac{1}{k},1\\right)$\nNote: they normalized both eigenvectors and the easiest way to do that is to divide by its length, $|v_1| = \\sqrt{a^2 + b^2}$, which yields:\n$$\\dfrac{1}{\\sqrt{1 + \\dfrac{1}{k^2}}} = \\dfrac{k}{\\sqrt{1+k^2}}$$\nMultiply that by the eigenvector and you have your result.\nFor the second eigenvalue, $\\lambda_2=-\\dfrac{2}{k}$, we have a RREF for $[A - \\lambda I]v_2=0$ of:\n$$\\begin{bmatrix}1 & 0\\\\0 & 0\\end{bmatrix}v_2 = 0$$\nWe choose $v_2 = (0,1)$\nNote: the second eigenvector is already normalized, else we would have done the same process as we did for the first.\n" + - context: None + - retrieval context: ['"\\nEigenvector of simple matrix\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6666666666666666, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.67 because the answer relevancy score is 0.67, which is not higher than 0.67., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n$v$ being a vector.\nI never understood what they mean and haven't found online resources. Just a quick question.\nThought it was absolute and magnitude respectively when regarding vectors. need confirmation\n" + - actual output: "The assistant is asking an example math question to demonstrate the type of questions you might ask. Here is another sample: \\n\\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\n" + - expected output: "\nIn general $\\lvert\\cdot\\rvert$ and $\\lVert\\cdot\\rVert$ are both used to signify norms of some sort. Different texts use different notation conventions, and sometimes the precise definition (if there is one) will vary from context to context\n" + - context: None + - retrieval context: ['"\\nwhats the difference between $|v|$ and $||v||$?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because it is not higher than 0.5, and there are no irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nTake an arbitrary number - for the sake of an example, I'll use 392. If we add the digits, we get 3 + 9 + 2 = 14, and then add those digits to get 5 (keep adding the digits of each result until it reduces to a single-digit number). \nCompare that to this method: let q and r be the quotient and remainder of the number with respect to 10. Take q+r, and repeat with that new result until it reduces to a single digit number. So, for 392, we do 39+2 = 41, and 4+1 = 5.\nIn this case, the two methods end up with the same result. I haven't been able to find any counter examples. Is this guaranteed to happen?\n" + - actual output: + - expected output: "\nFor both operations (add all digits or replace $10q+r$ with $q+r$), the result of the operation has the same remainder when dividing by $9$ as the original number (for the digit sum this is a well-known arithmetic \"trick\", for the othr operation note that $(10q+r)-(q+r)=9q$).\nTherefore, by starting with $n$ and repeating either of the operations until the result is single-digit (which is guaranteed to happen), we obtain a result $m$ with $m\\equiv n\\pmod{10}$ and $0\\le m\\le 9$.\nNote that neither of the two operations produces $0$ unless its input is $0$. Therefore, if $n>0$, we will in fact have $1\\le m\\le 9$, which then together with the modulo 9 condition determines $m$ uniquely.\n" + - context: None + - retrieval context: ['"\\nRepeatedly summing the digits of a number\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answer relevancy score is 0.6, and there are no irrelevant statements., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nShow that there exists a convex hexagon in the plane such that (a) all its interior angles are equal, (b) all its sides are 1, 2, 3, 4, 5, 6 in some order.\nit is the 9th question inmo 1993.\ni cant even start this question..\n" + - actual output: "\"Here are 10 examples of math questions (Q) with given answers (A).\\nQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\nA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\nA: \\n\\nQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\nA: \\n\\nQ: \\nIs there an expansion for the following summation?\\n$$ (a_1 + a_2 + \\\\cdots + a_k)^n $$\\n\\n\\nA: \\nhttp://en.wikipedia.org/wiki/Multinomial_theorem\\nThis is what you seek.\\n\\n\\nQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\nA: \\n$B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\n\\n\\nQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\nA: \\nBecause $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n\\nQ: \\nIf $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\nA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\nQ: \\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\nA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\nQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\nA: \\n\\nQ: \\nIs there a name for the geometry of a classic puzzle key?\\nIt's not an ellipse, neither a circle, ...\\n\\n\\n\\nA: \\n\\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: \\nShow that there exists a convex hexagon in the plane such that (a) all its interior angles are equal, (b) all its sides are 1, 2, 3, 4, 5, 6 in some order.\\nit is the 9th question inmo 1993.\\ni cant even start this question..\\n\"\n" + - expected output: "\nA really simple solution makes the observation that any convex equiangular hexagon must have interior angle measure of $2\\pi/3$, and thus can be constructed by considering truncation of the vertices of a suitable equilateral triangle. Suppose this triangle has side length $x$, and the vertices are truncated by lines parallel to the opposite side (so that the truncated pieces are themselves equilateral triangles). Let these pieces have side length $a$, $b$, $c$, respectively. Then the resulting equiangular hexagon has side lengths $$\\{a, x - a - b, b, x - b - c, c, x - c - a\\}.$$ It then suffices to find $a, b, c, x$ such that the above set is a permutation of $\\{1, 2, 3, 4, 5, 6\\}$. This can be done by trial and error; e.g., $a = 1$, $b = 2$, $c = 3$, $x = 9$; or by simply checking at most $\\binom{6}{3} = 20$ possibilities for $a, b, c$.\n" + - context: None + - retrieval context: ['"\\nShow that there exists a convex hexagon in the plane such that (a) all its interior angles are equal,\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nLet $G$ be a graph of order $8$ with $V(G)=\\{v_1, v_2,...,v_8\\}$ such that deg $v_i=i$ for $1 \\leq i \\leq 7$. What is deg $v_8$.\nAny help or hints would be greatly appreciated.\n" + - actual output: "Assistant: \"Here are 10 examples of math questions (Q) with given answers (A).\\nQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\nA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\nA: \\nNo, it has no solution in general. You can check this using the Fredholm alternative:\\nhttp://en.wikipedia.org/wiki/Fredholm_theory\\n\\n\\nQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\nA: \\nhttp://en.wikipedia.org/wiki/Weyl%27s_inequality_on_eigenvalues#Hermitian_matrices\\nThis is what you seek.\\n\\n\\nQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\nA: \\n$B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\n\\n\\nQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\nA: \\nBecause $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n\\nQ: \\nIf $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\nA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\nQ: \\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\nA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\nQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\\\\"ohlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\nA: \\n\\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: \\nLet $G$ be a graph of order $8$ with $V(G)=\\\\{v_1, v_2,...,v_8\\\\}$ such that deg $v_i=i$ for $1 \\\\leq i \\\\leq 7$. What is deg $v_8$.\\nAny help or hints would be greatly appreciated.\\n\"\n" + - expected output: "\nThis is a fun question, I knew as a riddle about people shaking hands.\nI'll give you the start:\n\n$v_7$ has degree $7$, so it is connected to everybody else (including $v_8$).\nthis means $v_1$ is connected only to $v_7$.\nso $v_6$ is connected to everybody except $v_1$ (this includes $v_8$).\n...\n\n" + - context: None + - retrieval context: ['"\\nGraph Theory Vertex Problem\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI just started my first upper level undergrad course, and as we were being taught vector spaces over fields we quickly went over fields. What confused me was when the book (Advanced Linear Algebra, Cooperstein) said that the the set {0, 1, 2} was a field. However, to my understanding, that set doesn't satisfy the axiom, \"For every element a in F, there is an element b such that a+b=0\", among others. Can someone help clarify where my understanding is off.\nAlso Cooperstein states \"for every prime power p^n, there exists a field with p^n elements\" and then doesn't expand on it. If someone could give an example or proof i would be very grateful.\nThanks for reading.\n" + - actual output: + - expected output: "\n$\\{0,1,2\\}$ is a field if you do all arithmetic modulo 3, that is, adding/subtracting an appropriate multiple of $3$ after each operation to make the result one of $0$, $1$ or $2$.\nIn aritmetic modulo 3, the negative of $1$ is $2$, because $1+2=3$ and subtracting $3$ to get that into the range $\\{0,1,2\\}$ makes $0$.\nThis construction will give you the fields with $p$ elements. Getting to the fields with $p^n$ elements for $n\\ge 2$ requires more algebraic prerequisites than will probably fit comfortably into a MSE answer.\n" + - context: None + - retrieval context: ['"\\nFinite fields, Linear Algebra\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI know that $\\vec a\\cdot(\\vec b+\\vec c)=\\vec a\\cdot\\vec b+\\vec a\\cdot\\vec c$, but is it also true that $\\vec a\\cdot(\\vec b-\\vec c)=\\vec a\\cdot\\vec b-\\vec a\\cdot\\vec c$? \n" + - actual output: "Answer: \"A cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nThe answer to this question is YES.\\n\\n\\nThe singular values $\\\\sigma(X+Y)$ can be bounded by using an inequality for singular values, like \\n$$\\\\sigma_i(\\\\,A\\\\oplus B\\\\,) = \\\\sigma_i (A) + \\\\sigma_i (B)$$ and the triangle inequality.\\n\\n\\nThe answer to this question is NO. You need to multiply the summation with $(a_1 + a_2 + \\\\cdots + a_k)^n$.\\n\\n\\nThis is not correct because you take absolute value of a complex number, which is $0$.\\n\\n\\nThe answer is $B=x-x=0$\\n\\n\\nThe answer to this question is YES.\\n\"\n" + - expected output: "\nLet $c' = -c$. Then $b-c = b+c'$ Then $a\\cdot (b-c) = a\\cdot (b+c') = a\\cdot b + a\\cdot c' = a\\cdot b -a\\cdot c$.\nSo, yes.\n" + - context: None + - retrieval context: ['"\\nDoes the distributive law for dot products go for both addition and subtraction?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n\n\n\n\n\n\nClosed. This question is off-topic. It is not currently accepting answers. \n\n\n\n\n\n\n\n\n\n\n This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level.\n\n\nClosed 9 years ago.\n\n\n\n\n\n\n\r\n Improve this question\r\n \n\n\n\nAdditionally, it's required that neither of them induces the discrete topology. \n" + - actual output: "\nYou are an intelligent mathematical assistant.\n\n" + - expected output: "\nThe following metric is not topologically equivalent to the standard Euclidean metric:\n$$d ( x , y ) = \\begin{cases}\n\\min \\{ | x - y |,1\\}, &\\text{if }x,y\\text{ are either both rational or both irrational} \\\\\n1, &\\text{otherwise}\n\\end{cases}$$\n(The underlying idea can be used to generate lots of inequivalent metrics on $\\mathbb{R}$.)\n" + - context: None + - retrieval context: ['"\\nWhich two would be an example of non-equivalent metrics on $\\\\mathbb{R}$. [closed]\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answer relevancy score is 0.6., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nLet $f\\colon X \\to X$ be a one-to-one function and let $A \\subseteq X$. Does $A \\subseteq f(A)$? \nI ask because I found a step which not clear to me in this paper \nhttp://www.maths.ed.ac.uk/~aar/papers/stong2.pdf\npage(328) proposition 8 .which is $f'(\\tau)\\supseteq \\tau$. \n" + - actual output: + - expected output: "\nNo.\nAs long as $f$ isn't the identity, you can choose some $x\\in X$ such that $f(x)=y\\neq x$. Then consider the singleton set $A=\\{x\\}$.\n" + - context: None + - retrieval context: ['"\\ndoes the domain can be considered as subset of it image under 1 to 1 function?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nThere is a set $Q$ which contains all integral values that can be represented by $$a^2 + b^2 + 4ab$$, where $a$ and $b$ are also integers. \nIf some integers $x$ and $y$ exist in this set, prove that $xy$ does too.\nI really have no idea how I can go about solving this. I tried simple multiplication of the two assuming one to be $(a^2 + 4ab + b^2)$ and other as $(c^2 + 4cd + d^2)$ but ultimately it leads to a long equation I can make no tail of :/\nAny help whatsoever would be greatly appreciated\n" + - actual output: + - expected output: "\nSince $a^2+b^2+4ab=(a+2b)^2-3b^2$, your numbers are exactly\nthe numbers of the form $x^2-3y^2$. Now $x^2-3y^2$ is the norm\nof the algrebraic number $x+y\\sqrt{3}$, so you have the identity\n$$\n(x^2-3y^2)(u^2-3v^2)=(xu+3yv)^2-3(xv+yu)^2\n$$\n(multiplicativity of norms).\n" + - context: None + - retrieval context: ['"\\nIf both integers $x$ and $y$ can be represented as $a^2 + b^2 + 4ab$, prove that $xy$ can also be represented like this ...\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nProve that for any $x\\in[0,\\infty)$ there exist sequences of positive integers $\\{a_{n}\\}_{n\\in\\mathbb N}$ and $\\{b_{n}\\}_{n\\in\\mathbb N}$, such that\n$$\\lim_{n\\to\\infty}\\left(\\dfrac{1}{a_{n}+1}+\\dfrac{1}{a_{n}+2}+\\cdots+\\dfrac{1}{a_{n}+b_{n}}\\right)=x.$$\nI only know this\n$$\\lim_{n\\to\\infty}\\dfrac{1}{n+1}+\\dfrac{1}{n+2}+\\cdots+\\dfrac{1}{n+n}=\\ln{2}$$\nBut for my problem I can't. Thank you\n" + - actual output: + - expected output: "\nNote that (integral comparison test)\n$$\n\\ln \\big(\\tfrac{\\ell-1/n}{k-1/n}\\big)\n=\\int_{kn-1}^{\\ell n-1}\\frac{dx}{x}<\n\\frac{1}{kn+1}+\\frac{1}{kn+2}+\\cdots+\\frac{1}{\\ell n}<\\int_{kn}^{\\ell n}\\frac{dx}{x}\n=\\ln (\\tfrac{\\ell}{k})\n$$\nLet strictly increasing sequences of integers $k_n$, $\\ell_n$, such that\n$$\n\\frac{\\ell_n}{k_n}\\to\\mathrm{e}^x.\n$$\nThen\n$$\n\\lim_{n\\to\\infty}\\sum_{j=1}^{(\\ell_m-k_m)n}\n\\frac{1}{k_nn+j}\\to x.\n$$\n" + - context: None + - retrieval context: ['"\\nHow find this $\\\\lim_{n\\\\to\\\\infty}\\\\left(\\\\frac{1}{a_{n}+1}+\\\\frac{1}{a_{n}+2}+\\\\cdots+\\\\frac{1}{a_{n}+b_{n}}\\\\right)=x$\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because . The irrelevant statements represent things in the actual output that is irrelevant to addressing whatever is asked/talked about in the input., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nFor $A,B \\in \\big( \\mathrm{Mat}_{n}(\\mathbb{C}) \\big)^2$, I know that there exists $Y \\in \\mathrm{Mat}_{n}(\\mathbb{C})$, $Y \\neq 0$, such as $AY=YB$ if and only if $\\mathrm{Sp}_{\\mathbb{C}}(A) \\cap \\mathrm{Sp}_{\\mathbb{C}}(B) \\neq \\emptyset$. Here, $\\mathrm{Sp}_{\\mathbb{C}}(A)$ denotes the set of complex eigenvalues of $A$.\nThe reason to this is the following : \nLet $u \\, : \\, \\mathrm{Mat}_{n}(\\mathbb{C}) \\, \\rightarrow \\, \\mathrm{Mat}_{n}(\\mathbb{C}) \\, ; \\, M \\, \\mapsto \\, AM-MB$. One can prove that $\\mathrm{Sp}(u) = \\mathrm{Sp}(A) - \\mathrm{Sp}(B) = \\left\\{ a-b, \\, (a,b) \\in \\mathrm{Sp}(A) \\times \\mathrm{Sp}(B) \\right\\}$.\nNow, I am wondering whether the result is still true for real matrices. If $A$ and $B$ are in $\\mathrm{Mat}_{n}(\\mathbb{R})$, can we still find $Y \\in \\mathrm{Mat}_{n}(\\mathbb{R})$, $Y \\neq 0$, such that $AY=YB$ while $\\mathrm{Sp}_{\\mathbb{R}}(A) \\cap \\mathrm{Sp}_{\\mathbb{R}}(B) = \\emptyset$ ? $\\mathrm{Sp}_{\\mathbb{R}}(A)$ denotes the set of real eigenvalues of $A$.\nI tried to build a counter-example taking, for example $A = \\begin{bmatrix} 0 & -1 \\\\ 1 & 0 \\end{bmatrix}$ because it would ensure that $\\mathrm{Sp}_{\\mathbb{R}}(A) = \\emptyset$ but I didn't succeed. This is not an homework question.\n" + - actual output: + - expected output: "\nYour attempt at a counterexample should succeed, taking $A=B$ with $\\text{Sp}_\\mathbb{R}(A) = \\emptyset$. Then clearly $AX=XB$ has a solution, in spite of there being no real eigenvalue of $A$ (in common with itself).\n" + - context: None + - retrieval context: ['"\\nMatrix equation $AX=XB$\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Passed Answer Relevancy (score: 1.0, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 1.0 because it is a good answer., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nLet $C_{R}$ be the upper half of the circle $|z|= R$.\nDoes $$ \\lim_{R \\to \\infty} \\int_{C_{R}} e^{iz} \\ dz = 0 ?$$\nI don't know if you can conclude from Jordan's lemma that it doesn't vanish. And the estimation lemma would appear to be inconclusive here.\nBut along the circle $|z|=R$, $ \\displaystyle |e^{iz}| = e^{-R \\sin t}$.\nSo as $R \\to \\infty$, the integrand decays exponentially. \nIs that enough to conclude the integral vanishes?\nAnd what about $ \\displaystyle\\lim_{R \\to \\infty} \\int_{C_{R}} z e^{iz} \\ dz$?\nEDIT:\nWhat's preventing both integrals from vanishing is the size of $|e^{iz}|$ near the endpoints of the contour.\nIf you were to integrate along only a portion of the contour that stays away from the endpoints, the estimation lemma would show that both integrals do vanish.\n" + - actual output: "\"I'm sorry I can't help with that.\"\n" + - expected output: "\nLet $z=Re^{i\\theta}$. After substitution:\n$\\displaystyle \\int_{C_R} e^{iz} dz = \\int_0^{\\pi} e^{iRe^{i\\theta}}iRe^{i\\theta} d\\theta$\n$= \\displaystyle -i\\int_0^{\\pi} e^{iRe^{i\\theta}}(-Re^{i\\theta}) d\\theta = -ie^{iRe^{i\\theta}}\\Big|_0^{\\pi} = -i[e^{-iR} - e^{iR}] = -2\\sin(R)$\nSo as $R\\to\\infty$, the limit does not exist.\n" + - context: None + - retrieval context: ['"\\nDoes $\\\\lim_{R \\\\to \\\\infty} \\\\int_{C_{R}} e^{iz} \\\\ dz = 0$?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answer relevancy score is 0.6, which is not higher than 0.6., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nHow would you calculate $df \\over d\u03b8$ if $f(x,y) = x^2+y^2$ where $x = \\sin 2\u03b8$ and $y = \\cos 2\u03b8$?\nI tried Wolfram and using the product rule but I can't seem to get anywhere.\n" + - actual output: + - expected output: "\nMore basic than the chain rule:\n$f(x(\\theta),y(\\theta)) = \\sin^2(2\\theta)+\\cos^2(2\\theta) = 1$, so $\\frac{df}{d\\theta} = \\frac{d}{d\\theta} (1) = 0$\n" + - context: None + - retrieval context: ['"\\nHow do you calculate the derivative after a change of variables?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nIn the theory of elliptic curves, I have read that the elliptic curves is topologically equivalent to a torus, given by $\\mathbb{C}$/ $\\Lambda$, where $\\Lambda$ is a lattice. \nThe proof appears to use the Uniformization Theorem, which states that every simply connected Riemann surface is topologically equivalent to either the open unit disk, the complex plane, or the Riemann sphere. I believe I understand how two spheres with two branch cuts each, joined at the branch cuts yield a torus. Could someone explain a general sketch of the proof (and correct any misunderstandings)?\n" + - actual output: "\"You are an intelligent mathematical assistant. \\n\\n Q: Prove that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n A: The complex roots of $x^2+1$ are $\\\\pm i$, so ${\\\\Bbb R}[i]$ forms a subfield of ${\\\\Bbb C}$ which contains $\\\\sqrt{-3}$. Since ${\\\\Bbb Q}(\\\\sqrt{-3})$ is the smallest field containing all rational numbers and $\\\\sqrt{-3}$, it must be a subfield of ${\\\\Bbb R}[i]$. Moreover, both fields have degree 2 over ${\\\\Bbb Q}$, so they are equal. By definition, cyclotomic fields are those obtained from $\\\\Q$ by adjoining a root of unity, hence this is the only cyclotomic field containing $\\\\sqrt{-3}$.\\n\\n Q: Is the following integral equation solvable ?\\n $$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$\\n Where $K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$ and $f(x)=e^{i\\\\gamma x}$ . \\n And $\\\\gamma$ is a parameter.\\n A: The integral equation $$F(x) = f(x)+\\\\int_{a}^{b}K(x,t)F(t)\\\\,dt$$ always has the solution $$F(x) = \\\\sum_{n=0}^{\\\\infty}\\\\frac{f^{(n)}(x)}{n!} \\\\prod_{m=1}^{n}\\\\left(\\\\int_a^b K(x,t) \\\\, dt\\\\right)^{(-1)^m+1}$$ provided that $$\\\\lim_{n\\\\rightarrow\\\\infty}\\\\sup_{x\\\\in[a,b]} \\\\left|\\\\frac{f^{(n)}(x)}{n!} \\\\prod_{m=1}^{n}\\\\left(\\\\int_a^b K(x,t) \\\\, dt\\\\right)^{(-1)^m+1} \\\\right| < \\\\infty.$$ \\nThis formula is known as Fredholm's equation.\\n\\n Q: Given that X and Y are positive definite matrices , how can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n A: The trace operator on Hermitian matrices is invariant under unitary similarity transformations, so for any real invertible matrix $P$, $$\\text{tr}(P^{-1}AP)=\\sum_{j=1}^r\\\\lambda_j(A)=\\sigma^2_j(A).$$ Hence the sum of the singular values squared equals its trace. Moreover, if we assume $X$ and $Y$ are positive definite Hermitian matrices, then their eigenvalues must be real and strictly positive, so $\\\\sum_{i=1}^r\\\\lambda_i^2(X) \\\\leq \\\\text{tr}(X)$. \\nThe result follows by noting that $$\\sigma^2_j(X+Y)=\\\\left(\\\\sqrt{\\\\lambda_j(X+Y)}\\\\right)^2=\\\\sum_{i=1}^r\\\\lambda_i(X+Y)\\\\leq \\\\text{tr}(X+Y)=\\text{tr}(X)+\\text{tr}(Y)$$ where the equality follows from a simple calculation.\\n\\n Q: Is there an expansion for the following summation? \\n $$ (a_1 + a_2 + \\\\cdots + a_k)^n $$\\n A: The multinomial theorem.\\n\\n Q: I have to simplify the following expression:\\n $B=\\\\sqrt{x^2} - x$\\n The only thing that I can do is:\\n $\\\\sqrt{a^2}=|a|$ , thus $B=|x|-x$. \\n Is that enough?\\n A: No. You also need to account for the case when $x < 0$. If $x < 0$, then $|x| = -x$ and so in this case we have $$B = - x - x = -2x.$$ In general, if $a > 0$, then $$\\sqrt{a^2} = a$$ while if $a < 0$, then $$\\sqrt{a^2} = -a.$$ So when you are simplifying expressions involving radicals, you should always assume the possibility of negative numbers. \\n\\n Q: Quick question:\\n Wolfram Alpha tells me that \\n $$\\\\int |e^{ix}|^2 dx = x + C$$\\n Why is that?\\n A: Because $|e^{ix}|=1$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n Q: If $10^{80} = 2^x$, what is the value of $x$? \\n (Or, what binary word length would you need to contain $10$ to the $80$?) \\n A: $\\\\log_2(10^{80}) = x$ or equivalently, $$x=\\\\frac{80}{\\\\log 10} \\\\approx 230.25897.$$ \\n\\n Q: if $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\n I would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n A: OK - $f(x)=7=3x-4$, can you solve this?\\n\\n Q: On p.10 of Algebraic Number Theory edited by Cassels and Fr\\\"ohlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\n Thanks for help.\\n A: See the following excerpt from Chapter 1 of Lang's Algebraic Number Theory:\\n\\n >If M and N are R-modules (where R is a ring), one can define the module index [M : N] as follows. Let m_1, \\\\ldots, m_r be generators for M; then define \\$$N = Rm_1 + \\\\cdots + Rm_r\\$$. Then [M : N] is defined to be the index of this submodule in M.\\n\"\n" + - expected output: "\nHere is a general theorem:\nTheorem: Let $A$ be an abelian variety over $\\mathbb{C}$ (that is, a connected projective algebraic group; in particular if its dimension is 1 then it's an elliptic curve). Then $A$ is analytically isomorphic to a complex torus.\nSketch of proof: $A$ is in particular a complex compact Lie group, and we have the exponential map $\\mbox{exp}:T_0A\\simeq\\mathbb{C}^n\\to A$. It is surjective and is a homomorphism, since $A$ is commutative. Since it is a local diffeomorphism, its kernel is discrete, and so is a lattice in $\\mathbb{C}^n$. Therefore $A\\simeq\\mathbb{C}^n/\\ker(\\mbox{exp})$. $\\Box$\nYou can find a more detailed proof in Milne's notes on abelian varieties.\n" + - context: None + - retrieval context: ['"\\nThe Uniformization Theorem and Elliptic Curves\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.3333333333333333, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.33 because the answer relevancy score is 0.33., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nProve that there exist infinitely many Pythagorean integers $a\u00b2+b\u00b2=c\u00b2$\nMy key idea is to show that there exists infinitely many integers that can be the length of the sides of a right triangle, but I fail at it.\nOther try is that $\\sqrt{a^2+b^2}=c$ and so it is an equation of a circle, so I tried to show that for every natural number c, there exist a and b such that $\\sqrt{a^2+b^2}=c$ is the equation of a circle but I failed again.\nHow to prove it?\nThank you.\n" + - actual output: "\"Here are 10 examples of math questions (Q) with given answers (A).\\nQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\nA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\nA: \\n\\nQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\nA: \\n\\nQ: \\nIs there an expansion for the following summation?\\n$$ (a_1 + a_2 + \\\\cdots + a_k)^n $$\\n\\n\\nA: \\nhttp://en.wikipedia.org/wiki/Multinomial_theorem\\nThis is what you seek.\\n\\n\\nQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\nA: \\n$B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\n\\n\\nQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\nA: \\nBecause $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n\\nQ: \\nIf $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\nA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\nQ: \\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\nA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\nQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\nA: \\n\\nQ: \\nIs there a name for the geometry of a classic puzzle key?\\nIt's not an ellipse, neither a circle, ...\\n\\n\\n\\nA: \\n\\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: \\nProve that there exist infinitely many Pythagorean integers $a\\u00b2+b\\u00b2=c\\u00b2$\\nMy key idea is to show that there exists infinitely many integers that can be the length of the sides of a right triangle, but I fail at it.\\nOther try is that $\\\\sqrt{a^2+b^2}=c$ and so it is an equation of a circle, so I tried to show that for every natural number c, there exist a and b such that $\\\\sqrt{a^2+b^2}=c$ is the equation of a circle but I failed again.\\nHow to prove it?\\nThank you.\\n\"\n" + - expected output: "\nYou could simply look at multiples of $a$, $b$ and $c$, e.g. $2a$, $2b$ and $2c$.\n" + - context: None + - retrieval context: ['"\\nProve that there exist infinitely many pythagorean integers $a\\u00b2+b\\u00b2=c\\u00b2$\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5, and there are no irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI was thinking about induction like: \n\nBase:\n$$\\#\\mathbb{R}^{1} = \\#\\mathbb{R} = \\mathfrak{c}$$ \nAnd for $n+1$ \n $$\\#\\mathbb{R}^{n+1} = \\#\\mathbb{R}^{n}\\mathbb{R} = \\mathfrak{c}$$ \n But it seems too easy. Do you have some other ideas? \n" + - actual output: "Q: I was thinking about induction like:\nAnd for $n+1$\n $$=$$ \nBut it seems too easy. Do you have some other ideas?\nA:\nQ: Is there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n" + - expected output: "\nIf you already know that $\\Bbb{R\\times R}$ and $\\Bbb R$ have the same cardinality, then this is enough.\n" + - context: None + - retrieval context: ['"\\nProve that for every $ \\\\quad n\\\\in\\\\mathbb{N},\\\\quad \\\\mathbb{R}^{n} = \\\\mathfrak{c}$?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nThis is from exercice 5, chap 2 from Atiyah and McDonald \"Introduction to Commutative Algebra\".\n\nLet $A[x]$ be the ring of polynomials in one indeterminate over a ring $A$. Prove that $A[x]$ is a flat $A$-algebra.\n\nClearly, we notice that $\\displaystyle A[x]=\\bigoplus_{m=0}^\\infty A\\cdot (x^m)$.\nWe showed in the previous exercice that for any family $M_i$ ($i\\in I$) of $A$-modules and $M$ their direct sum, then $M$ is flat iff each $M_i$ is flat.\nOur problem is then reduced to showing that each $(x^m)$ is flat and that it is an $A$-algebra.\nSome solutions on the internet require \"Lang's Lemma\" such that\n\n[I]t only suffices to prove that the natural map $\\phi : a \\otimes (x^m ) \\longrightarrow a(x^m )$ is an\n isomorphism for any ideal $a$ of $A$.\n\nBut we haven't seen that lemma previously in the book so is there another method ?\nThanks.\n" + - actual output: "\"This is not easy to solve.\\nI can tell you why, though. Forget about the A and the Q for now and think only of the math question.\\n\\nThe first thing to do is to find an answer A that is coherent with what you wrote.\\nYou're asking us if the following is true:\\n $$A[x]=\\\\bigoplus_{m=0}^\\\\infty A\\\\cdot (x^m)$$ The above should be a definition of $A[x]$, not something that we need to prove. If this is for an exercise, then it is given. Otherwise, why don't you tell us what exactly are you trying to do?\\n\\nSecond, the question doesn't make sense.\\nThe statement \\\"Let $A[x]$ be the ring of polynomials in one indeterminate over a ring $A$\\\" does not mean anything. The question should be more like:\\n\\n Let $k$ be a field and let $A=k[a_1,...,a_r]$ where $(a_1,...a_r)$ is an ideal of $k$. Define the ring of polynomials in one indeterminate over $A$, denoted by $A[x]$. Prove that $A[x]$ is a flat $A$-algebra.\\n\\nThe notation you used does not make sense and your question doesn't either. For instance, what is the definition of the polynomial ring $k[x]$? You need to specify a base field $k$ first, then define a polynomial in one indeterminate over that field. The statement \\\"Let $A[x]$ be the ring of polynomials in one indeterminate over a ring $A$\\\" does not make sense because you are asking your readers to understand from it what the question is.\\n\\nThird, I don't see why Lang's lemma would help. You seem to know this result but that was not part of your question and we can't know if it will be used in a future exercise unless you tell us how exactly to solve the problem. If it has nothing to do with the current topic then there's no point to show it to us.\\n\\nFinally, I don't see any indication that this is an actual question from an exam or homework assignment. There's no context. The only thing we can say about your situation is that you're trying to solve a problem using the same tools as in the Atiyah and McDonald textbook, but without that book, there's little we can do.\\n\\nI hope I made myself clear.\\n\"\n" + - expected output: "\n$A[x]$ is free as an $A$-module. Free modules are flat.\n" + - context: None + - retrieval context: ['"\\nProve that $A[x]$ is a flat $A$-algebra.\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because it is a good answer but there are irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI ran into this question when reading Artin's Algebra book and tried to google the answer but seems it's too easy that I didn't get any.\nMy answer is: Not necessarily.\nProof:\nFrom the definition of a left coset: $aH=\\{ah\\ |\\ h\\in H\\}$, where $H$ is a subgroup of $G$, and $a\\in G$. \nIf we want $aH$ to be a group, it has to be a closure: $ah_{1}ah_{2}\\in aH$, thus $h_{1}ah_{2}\\in H$. And since $a\\in G$ according to the assumption, $h_{1}ah_{2}\\notin H$. So cosets are not necessarily groups.\nAm I correct? Thanks.\n" + - actual output: + - expected output: "\nA coset is a set while a group is a set together with a binary operation that satisfies some axioms. So, a coset is not a group since the binary operation is missing. Any question asking whether a given set is a group is a wrong question. If you meant to ask if a coset is a subgroup (of the obvious ambient group), then that can be answered negatively by noticing that the identity element, which must be an element of any subgroup, is not necessarily an element in a coset. \n" + - context: None + - retrieval context: ['"\\nAre cosets groups?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nIs there any way to transform the denominator in a power series?\nFor example,\n$$\\sum_{n=1}^{\\infty}\\frac{1}{n^2}=\\frac{\\pi^2}{6}$$ and $$\\sum_{n=1}^{\\infty}\\frac{1}{(4n)^2}=\\frac{\\pi^2}{96}.$$ My question asks if you can convert each term in an infinite zeta series from $n$ to $an+b.$ As shown above, scaling is easily done however I cannot think of any way to add to each denominator. i.e.\n$$ \\sum_{n=1}^{\\infty}\\frac{1}{n^2} \\to \\sum_{n=1}^{\\infty}\\frac{1}{(an+b)^2}.$$\n" + - actual output: "I got you this time, and I hope I understood correctly this time.\n" + - expected output: "\nThe Wikipedia article Hurwitz zeta function states\n\nIn mathematics, the Hurwitz zeta function is one of the many\nzeta functions. It is formally defined for complex variables $\\,s\\,$\nwith $\\,\\Re(s)>1\\,$ and $\\,a\\ne 0,-1,-2,\\dots\\,$ by\n$$\\zeta(s,a) = \\sum_{n=0}^\\infty \\frac{1}{(n+a)^{s}}.$$\n\nYou wrote\n\nMy question asks if you can convert each term in an infinite power series from $n$ to $an+b$.\n\nIn general, no. That is why the Hurwitz zeta function is needed to\ngeneralize the Riemann zeta function. In some special cases, it is\npossible to get results. The Wikipedia article states\n\nIf $\\,a=1\\,$ the Hurwitz zeta function reduces to the Riemann\nzeta function itself; if $\\,a=1/2\\,$ it reduces to the Riemann\nzeta function multiplied by a simple function of the complex\nargument $\\,s.$\n\nAs a first step, if $\\,a\\ne 0,\\,$ it is easy to get\n$$ \\sum_{n=1}^\\infty \\frac1{(an+b)^s} = a^{-s}\\zeta(s,b/a). $$\nA simple result is that if $\\,k>0\\,$ is an integer, then\n$$ \\zeta(s,k) = \\sum_{n=1}^\\infty \\frac1{(n+k)^s}\n= \\zeta(s) - \\sum_{n=1}^k \\frac1{n^s}. $$\n" + - context: None + - retrieval context: ['"\\nCan you transform the denominator of each term in an infinite series?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI am sure the answer to this question is out there, but I cannot find it, maybe because I don't know the correct torm for 'matrix multiplication from left and right'.\nConsider a matrix $W \\in \\mathbb{R}^{m \\times n}$ and a square matrix $A \\in \\mathbb{R}^{m \\times m}$. What can we say about the determinant\n$$\\det (W^TAW)?$$\nIn the case that $W$ is also square, since the determinant then commutes, we have\n$$\\det (W^TAW) = \\det (A WW^T) = \\det (A) \\det (WW^T) = \\det(A) \\det(W^TW).$$\nDo we have a similar result when $W$ is not square? We can not expect the two last equalities on the left generally hold, since one of $\\det (WW^T)$ and $ \\det(W^TW)$ will be equal to $0$, but maybe it holds for the respective one that might have full rank?\nEdit:\nMaybe this works indeed, using singular value decomposition:\nLet $W = U \\Sigma V$ and assume $W$ has full rank, and assume that $mn$. Let $U \\Sigma V^T$ be the SVD of $W$, where $\\Sigma$ has the form\n$$\n\\Sigma = \\pmatrix{D\\\\0}.\n$$\nDenote $B = U^TAU$. Partition $B$ into\n$$\nB = \\pmatrix{B_{11} & B_{12}\\\\ B_{21} & B_{22}}, \n$$\nwhere $B$ is square of size $n$. We have\n$$\n\\Sigma^TB\\Sigma = \\pmatrix{D & 0}\\pmatrix{B_{11} & B_{12}\\\\ B_{21} & B_{22}} \\pmatrix{D \\\\ 0} = DB_{11}D.\n$$\nThus, we have\n$$\n\\det(W^TAW) = \\det(\\Sigma^TB\\Sigma) = \\det(DB_{11}D) = \\det(B_{11})\\det(D)^2\\\\\n= \\det(B_{11}) \\det(W^TW). \n$$\nThis isn't as \"nice\" as the formula for the case of $m = n$, but perhaps you will find it interesting nevertheless.\n" + - context: None + - retrieval context: ['"\\nWhat is $\\\\det (W^T A W)$ for $W$ not square?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI am given the series $$\\sum_{n=1}^{\\infty}\\frac{n^{n-1}}{\\left ( 2n^2+n+1 \\right )^{\\left (n+\\frac{1}{2} \\right )}}.$$\nDoes this series converge? Solution states that comparison test should be used, though it is not stated which one. The one which looks the most promising to me is the one that says that if for two sequences $a_n \\sim b_n$ then $\\sum_{n=1}^\\infty a_n$ is equiconvergent to $\\sum_{n=1}^\\infty b_n$. This requires me to find the sequence $b_n$. So I went ahead, done some gymnastics and found $b_n = \\left(\\frac{1}{2n} \\right)^n$, meaning that my original series would be convergent. I am wondering is this valid, that is whether I found $b_n$ correctly, or not. They never really explained the process of finding $b_n$ so I am not sure which \"transformation\" am I allowed to do. I hope that someone can confirm my solution, if it is correct, or if it is not, provide an answer that would use one of the comparison tests to determine the convergence of this series. I have not tried other methods, since the comparing sequences is an expected way to solve it, but I think that this could also be done using the root test.\nFor completeness, here is my work for finding $b_n$. The reasoning I used is that I can remove terms of lower exponents everywhere, and that would still preserve asymptotic equivalence. Again, I am not sure if this is true, as it was never explained properly during the course. I just believe that that is how the process works. I also graphed both series in Desmos and it seemed that they started meeting at some point along the x axis and then continuing to converge towards $0$ together. Anyways, this is what I did:\n$$a_n = \\frac{n^{n-1}}{\\left ( 2n^2+n+1 \\right )^{\\left (n+\\frac{1}{n} \\right )}} \\sim \\frac{n^{n}}{\\left ( 2n^2+n+1 \\right )^n} \\sim \\frac{n^{n}}{( 2n^2)^n} \\sim \\frac{n^{n}}{ 2^n n^{2n}} \\sim \\frac{1}{ 2^n n^n} \\sim \\frac{1}{(2n)^n} = b_n. $$\nMaybe this solution is correct. I would still very much appreciate if someone could write an answer explaining in what ways I can manipulate the original series in order to preserve the relation of asymptotic equivalence. I have no idea if what I did is right, or why is it right if it is, or why it would not be if it is not.\n" + - actual output: + - expected output: "\n$$\\frac{n^{n-1}}{\\left ( 2n^2+n+1 \\right )^{\\left (n+\\frac{1}{n} \\right )}}=\\frac1{n\\sqrt[n]{2n^2+n+1}}\\cdot\\left(\\frac n{2n^2+n+1}\\right)^n$$\nObserve that both factors have zero as limit, thus:\n$$\\frac1{n\\sqrt[n]{2n^2+n+1}}\\cdot\\left(\\frac n{2n^2+n+1}\\right)^n\\le\\left(\\frac n{2n^2+n+1}\\right)^n\\le \\left(\\frac12\\right)^n$$\nand the comparison test gives us the answer.\n" + - context: None + - retrieval context: ['"\\nAsymptotic equivalence while considering series convergence\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answer relevancy score is 0.6., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nCan people suggest some problems (probability puzzle type) where the use of linearity of expectation together with indicator random variable is unexpected/hard to see but it makes problems much easier?\nI have encountered a lot of questions asking similar types of problems in the various domains so I think combined use of linearity of expectation and indicator random variable deserves its own.\nThis and This are the question I was motivated from, but some of problem in my second mentioned problem are not puzzle related.\n" + - actual output: + - expected output: "\nIn a certain village 80% of the villagers drink tea, 60% drink beer, and 60% drink wine. Nobody consumes all three beverages. What percentage of villagers consume alcohol?\n\n Let $I_T$ be the indicator that a villager consumes tea, and similarly\n for $I_B$ and $I_W$. Then the number of beverages consumed by a\n villager is $$N:=I_T + I_B + I_W.$$ By calculation, using linearity of\n expectation, $E(N)=2$. But by hypothesis, $N\\le 2$. Hence each\n villager consumes exactly two beverages, at least one of which must be\n alcoholic.\n\n" + - context: None + - retrieval context: ['"\\nUnexpected use of linearity of expectation with indicator random variable in problems\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n\n\n\n\n\n\nThis question already has answers here: \n\n\n\nProve that $\\tan\\frac{\\pi}{9}+ 4\\sin\\frac{\\pi}{9}= \\sqrt{3}$ .\n\r\n (3 answers)\r\n \n\nClosed 1 year ago.\n\n\n\n\nProve that $$ \\tan\\frac{\\pi}{9} +4\\sin\\frac{\\pi}{9} = \\sqrt 3 $$\n\nThere seem to be a lot of similar identities that are provable, for example, by using roots of unity. However, here I cannot get things to work out nicely.\nIf $u=e^{\\frac{2\\pi i}{9}} $, then $$i\\left(\\tan\\frac{\\pi}{9} +4\\sin\\frac{\\pi}{9}\\right) =\\frac{u-1}{2(u+1)} +2(u^4-u^5)=\\frac{-4u^6 +4u^4+u-1}{2(u+1)} $$ and so $$\\left(\\tan\\frac{\\pi}{9} +4\\sin\\frac{\\pi}{9}\\right)^2 = 3 \\\\ \\iff (-4u^6+4u^4+u-1)^2+12(u+1)^2 =0 \\\\ \\iff 16u^8-8u^7+8u^6+8u^5-8u^4+16u^3+13u^2-10u+13 =0$$ Unfortunately, the LHS is not of the form $k(u^8+u^7 +\\dots+1)$ making the equality unobvious. How to proceed?\n" + - actual output: "What do you think about this problem and solution? \n## [1] \"I need some clarifications on your solution, is it correct?\"\n" + - expected output: "\nHere's a purely trigonometric solution, in case you're interested:\n$$\\tan 20\u00b0+4\\sin 20\u00b0=\\frac {\\sin 20\u00b0+2(2\\sin 20\u00b0 \\cos 20\u00b0)}{\\cos 20\u00b0}=\\frac {(\\sin 20\u00b0+\\sin 40\u00b0)+\\sin 40\u00b0}{\\cos 20\u00b0}=\\frac {2\\frac 12 \\cos 10\u00b0+\\cos 50\u00b0}{\\cos 20\u00b0}=\\frac {2\\cos 30\u00b0\\cos 20\u00b0}{\\cos 20\u00b0}=2 \\frac {\\sqrt 3}{2}=\\sqrt 3$$\n" + - context: None + - retrieval context: ['"\\n$\\\\tan\\\\frac{\\\\pi}{9} +4\\\\sin\\\\frac{\\\\pi}{9} = \\\\sqrt 3 $ [duplicate]\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nLet $K/\\mathbb{Q}_p$ be a finite extension, and $\\overline{K}$ be some fixed algebraic closure of $K$. Let $\\mathcal{O}_{\\overline{K}}$ be the ring of all algebraic integers in $\\overline{K}$. I know it is a valuation ring so it is an integrally closed local domain.\nI would like to know what is the Krull dimension of this ring? Is the maximal ideal the only nontrivial prime ideal? In particular, I am interested in whether $\\mathrm{Spec}(\\overline{K})\\to\\mathrm{Spec}(\\mathcal{O}_{\\overline{K}})$ is an open immersion?\n" + - actual output: + - expected output: "\nI\u2019m not quite sure why you\u2019re being downvoted. These are reasonable questions in my opinion.\nSo, you\u2019ve singled out that you have a valuation ring $\\mathcal{O}$ (for future reference let\u2019s call $L:=\\mathrm{Frac}(\\mathcal{O})$) and you want to know\n\nis it true that $\\mathcal{O}$ has only one non-zero prime,\nwhat is the Krull dimension of $\\mathcal{O}$,\nwhether the map $\\mathrm{Spec}(L)\\to\\mathrm{Spec}(\\mathcal{O})$ is an open embedding.\n\nTo answer your first question, let us recall the following fact.\n\nFact: Let $\\mathcal{O}$ be a valuation ring. Then, the set of ideals of $\\mathcal{O}$ is a linearly ordered set.\n\nIn particular, we see that the spectrum $\\mathrm{Spec}(\\mathcal{O})$ looks like a \u2018path\u2019, like this (assuming it\u2019s finite dimensional)\n$$(0)\u2014\\bullet-\\cdots\\bullet-\\mathfrak{m}$$\n(incidentally, spectra of valuation rings are exactly used as \u2018paths\u2019 in birational geometry of schemes/formal schemes). Moreover, it\u2019s clear that the length of this path is the same thing as the Krull dimension of $\\mathcal{O}$. Thus, $\\mathcal{O}$ has a unique non-zero prime ideal (which is the maximal ideal) if and only if it is of dimension $1$.\nIs $\\mathcal{O}_{\\overline{K}}$ of dimension $1$? The answer is yes since $\\mathrm{Spec}(\\mathcal{O}_{\\overline{K}})\\to\\mathrm{Spec}(\\mathcal{O}_K)$ is surjective and integral (see Tag 0CEG). Thus, your first (and second) questions are answered.\nTo answer your third it is useful to put things in a bigger context. Namely, we would like to know when the map $\\mathrm{Spec}(L)\\to\\mathrm{Spec}(\\mathcal{O})$ is an open embedding or, equivalently, when $\\mathrm{Spec}(L)$ is an open subset of $\\mathrm{Spec}(\\mathcal{O})$.\nTo this end, recall that any time you have a valuation ring $\\mathcal{O}$ you get a valuation\n$$v:L^\\times\\to \\Gamma$$\nwhere $\\Gamma$ is some totally ordered abelian group (in fact, one can take $\\Gamma=L^\\times/\\mathcal{O}^\\times$ in the usual way). Recall that we say that $\\Gamma$ is of height 1 if there exists an embedding of totally ordered abelian groups $\\Gamma\\hookrightarrow (\\mathbb{R},^+)$.\nOne is then able to put a valuation topology on $L$ by declaring that a basis of open sets is given by (let\u2019s assume that $v$ is non-trivial!) $\\{a\\in L:v(a)\\leqslant \\gamma\\}$ for some $\\gamma$ in $\\Gamma$.\nWe then have the following definition.\n\nDefinition A valuation ring $\\mathcal{O}$ is called microbial if its valuation topology agrees with the valuation topology of some\nheight $1$ valuation $v\u2019$ on $\\mathcal{O}$.\n\nIn particular, note that being microbial doesn\u2019t mean your valuation is height $1$, it only means that your \u2018valuation topology is rank $1$\u2019.\nNB: Such valuation rings are very important in Huber\u2019s theory of adic spaces, and Fujiwara\u2014Kato\u2019s theory of the birational geometry of formal schemes. You can look there for many natural examples of microbial valuation rings which aren\u2019t height 1. You can also see [Morel, Example I.1.5.5] for a concrete example.\nWe then have the following nice proposition.\n\nProposition: Let $\\mathcal{O}$ be a valuation ring. Then, the following are equivalent:\n\n$\\mathcal{O}$ is microbial,\nthere exists some non-zero element $\\varpi\\in \\mathcal{O}$ such that $\\mathcal{O}$ is $\\varpi$-adically separated,\nthere exists some non-zero element $\\varpi\\in\\mathcal{O}$ such that $L=\\mathcal{O}[\\frac{1}{\\varpi}]$,\nthe map $\\mathrm{Spec}(L)\\to\\mathrm{Spec}(\\mathcal{O})$ is an open embedding.\n\n\nProof: See this and [Morel, Theorem I.1.5.4]. $\\blacksquare$\nIn particular, I can look at your example of $\\mathcal{O}_{\\overline{K}}$ and immediately see that $\\mathrm{Spec}(\\overline{K})\\to \\mathrm{Spec}(\\mathcal{O}_{\\overline{K}})$ is an open embedding since either:\n\nthe valuation ring $\\mathcal{O}_{\\overline{K}}$ is height 1, and so of course microbial,\nit is $p$-adically separated (exercise!).\n\nNB: Note that $\\mathcal{O}_{\\overline{K}}$ is not $p$-adically complete, so it\u2019s good that in the definition of microbial we only required separatedness!\nReferences:\n[Morel] Morel, S., 2019. Adic Spaces. Lecture Notes. https://web.math.princeton.edu/~smorel/adic_notes.pdf.\n" + - context: None + - retrieval context: ['"\\nring of $p$-adic algebraic integers\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Passed Answer Relevancy (score: 1.0, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 1.0 because it is a good answer., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI have a question. Is it more productive or rigorous to think of a function as a rule that tells us how to \"map\" or \"associate\" elements from one set to another, or as a set of ordered pairs containing elements of one set (the domain) \"mapped to\" or \"associated with\" elements of another set? Or is there a context in which it makes more sense to think of a function one way vs another?\nTo expand on this, I've realized that I conceptualize each of the following ways of defining a function in two different ways:\n$$\\text{Define}\\; f: X \\to Y \\;\\;\\text{by}\\;\\;f(x)=y$$\nor\n$$\\text{Define the set}\\; f = \\{(x,f(x)) \\mid x \\in X\\}$$\nWhen a function is defined in the first manner, I tend to think of it as a specific \"map\" or \"rule\" that tells us exactly how to take an element from the set $X$ and associate it with a single element from the set $Y$. I tend to think of this function as an object that \"completes an action\" in a sense - where that action is taking an element from $X$ and mapping it to $Y$.\nHowever, when a function is defined in the second manner, it no longer seems appropriate to think of it as something that is \"completing an action\". The function no longer seems like a \"tool\" in the sense that it does when it's defined in the previous manner. It seems as though the action is already completed. In other words, instead of viewing the function as an object that maps elements from $X$ to elements of $Y$, I tend to view the function as the result of a mapping from $X$ to $Y$.\nBut don't both of these ways of defining a function define the same mathematical object? I seem to be conceptualizing the same thing in two very different ways, which makes me wonder if my intuition is off.\n" + - actual output: "You are an intelligent mathematical assistant.\nQ: Given that $X$ and $Y$ are positive definite matrices, how can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\nA: You have to use the Courant\u2013Fischer theorem to express it in terms of the eigenvalues. See this wiki article for more information https://en.wikipedia.org/wiki/Min-max_theorem # Required Informatia for linear regression in Python (OLS) [duplicate]\n\nI have a problem with OLS Linear Regression model, especially with understanding the required input and output data to build it. I don't want to do prediction but only analysis of correlation between variables. Let's say that I have 100 users which each user has about 20 variables like age, salary etc. So in total it would be more than 20 000 rows. Now let's assume I know that age and salary are related (I can clearly see it visually on a plot) so I want to know the correlation coefficient for this two variable combinations. Which of these methods is correct?\n\n1. Build a model with variables: x = age, y = salary\n2. Build a model with variables: x = salary, y = age\n3. Build a model with variables: x = salary + age\n4. Build a model with variables: x = [salary, age]\n5. Build a model with variables: x = [age, salary]\n6. ???\n\nWhich method would be the best to get the correlation coefficient or maybe some other statistics?\n\nI am using Python and statsmodels library. I've found this code on stackoverflow but it uses just one variable to create linear model.\n\nimport pandas as pd\nfrom sklearn import datasets\nurl = 'https://raw.githubusercontent.com/selva86/datasets/master/ Diamonds.csv'\nX = df[['Diamond::Clarity']]\nY = df[['Price']]\nmodel = sm.OLS(Y, X).fit()\nprint (model.summary()) # [OS X TeX] Re: How to interrupt a list?\n\nPeter Krautzberger peter.krautzberger at web.de\nThu May 7 09:54:23 EDT 2009\n\nHello,\n\nI'm not sure if this helps (it's an ugly hack but it works) but you might\ntry the following:\n\n- use the paralist package; it allows you to choose the punctuation that is\ninserted after each label in a list.\n- put \\setdefaultenum{(}{)}{~}{.} before the enumerate\nenvironment (see below).\n\nThis results in the following output:\n\n\\documentclass{article}\n\\usepackage[shortlabels]{enumitem}\n\\usepackage{paralist}\n\\setdefaultenum{(}{)}{~}{.} % set enum. punctuation to \\cdot\n\\begin{document}\n\nHere is some text. More, more text.\n\\begin{enumerate}[(a)]\n\\item First item.\n\\item Second item.\n\\end{enumerate}\n\nHere is some text. More, more text.\n\\end{document}\n\nI hope this helps,\nPeter\n\nAm 07.05.2009 um 14:20 schrieb Manuel Souto Pico:\n\n> Dear all,\n>\n> Is there any way to interrupt a list (enumerate -> (a), (b)... ) and go on it later? I need to do so because I want to write something between the end of the current list and the beginning of the next one. Something like this:\n>\n> ---------------\n>\n> Here is some text. More, more text.\n> \\begin{enumerate}[(a)]\n> \t\\item First item.\n> \t\\item Second item.\n> \\end{enumerate}\n> Text in between the list and the next one.\n> \\begin{enumerate}[(i)]\n> \t\\item Third item.\n> \t\\item Fourth item.\n> \\end{enumerate}\n> ---------------\n>\n> The output will be:\n>\n> (a) First item.\n> (b) Second item.\n>\n> Text in between the list and the next one.\n> (i) Third item.\n> (ii) Fourth item.\n>\n> I have tried to use \\intertext, but it gives me weird results:\n>\n> ---------------\n>\n> Here is some text. More, more text.\n> \\begin{enumerate}[(a)]\n> \t\\item First item.\n> \t\\item Second item.\n> \\end{enumerate}\n> \\intertext{Text in between the list and the next one.}\n> \\begin{enumerate}[(i)]\n> \t\\item Third item.\n> \t\\item Fourth item.\n> \\end{enumerate}\n> ---------------\n>\n> The output will be:\n>\n> (a) First item.\n> (b) Second item.\n> Text in between the list and the next one. (i) Third item. (ii) Fourth item.\n>\n> That is, I get the labels for both lists together with the second list!\n>\n> Any ideas? Thank you very much for your help.\n> # Why does a non-zero $A_0$ term induce an electric dipole moment in a charged fermion loop?\n\nIn this paper, it is said that, given a fermionic loop of charge $$Q$$, with the Lagrangian: $$\\mathcal{L} = \\bar{\\psi}(i\\gamma^\\mu D_\\mu - m)\\psi$$ where we have made use of the chiral representation for the Dirac matrices and $$D_\\mu$$ is the usual covariant derivative, a non-zero vacuum expectation value of $$\\langle 0|\\bar{\\psi}\\psi|0\\rangle = A_0 \\neq 0$$ induces an electric dipole moment $$d$$.\n\nI don't understand why this should be true. What does this mean physically? Is it to do with the mass and charge matrix for a complex scalar $$\\phi$$ being a symmetric matrix $$\\begin{pmatrix} m & Q\\\\Q&m \\end{pmatrix}$$ instead of a diagonal one in ordinary " + - expected output: "\nI wondered a very similar thing when I first encountered the formal notion of a function, so I hope my perspective is helpful.\nI think the reason the notation $f(x) = y$ is often associated with a 'rule' is that typically $f(x)$ is written as some combination of operations on $x$, for example $f(x) = x^2-5x+6$. I.e. there is some necessary 'computation' to be performed, and once you perform that computation, you 'know' what $f(x)$ is. In this sense, the intuition that $f$ is 'doing something' or 'acting' on $x$ is entirely natural, and I think a perfectly reasonable intuition to have.\nHowever, I think the issue arises because of the a slight gap between the 'rule', and what the actual function is. In our above example, say we have $f(x) = x^2-5x+6$, restricting ourselves to the real numbers. Note though this 'rule' is not itself a function; rather there is an implicit understanding that this rule induces a function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$, where $f = \\{(x,f(x)):x\\in \\mathbb{R}\\} = \\{(x,x^2-5x+6): x \\in \\mathbb{R}\\}$.\nIn particular, I hope this addresses why you feel $f$ is \"completed\" in one case but not the other. I think in the former case, you are viewing the rule as the function itself, and hence, you think the function is 'doing' something to $x$. But of course, we have just seen that while implicitly related, the function and rule are indeed distinct.\nIn fact, we can see this in action by realising not all 'rules' will give us functions. For example, if I tell you $f(x) = \\text{arcsin}(x)$, this is not a well defined function unless we appropriately specify the domain and codomain. That is to say, sometimes just giving a 'rule' is ambiguous, and it should be seen that the 'rule' is NOT the same thing as the function itself.\n" + - context: None + - retrieval context: ['"\\nDifferent understandings of a function: defining $f: X \\\\to Y$ by $f(x)=y$, vs defining $f=\\\\{(x,f(x))\\\\mid x\\\\in X\\\\}$\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because it is not higher than the given answer relevancy score., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n\nA $7$-digit number is chosen at random. Which is more likely: the number has no $1$'s among its digits or the number has at least one $1$ among its digits?\n\nHere's how I did it: The question is asking whether $8(9)^6$ (the number of those with no $1$'s among its digits) or $9(10)^6 - 8(9)^6$ (the number of those with at least one $1$ among its digits). Some tedious multiplying shows that $8(9)^6 = 4241528 < 4500000$, which demonstrates that $9(10^6) - 8(9)^6$ i.e. the number having at least one $1$'s among its digits is more likely.\nHowever, I am wondering if there is a slicker way to get the answer without having to do any tedious multipication.\n" + - actual output: + - expected output: "\nThere is a more tedious way to do this. We have that\n$$\n9\\cdot (10)^6- 8\\cdot(9)^6\\geq 8\\cdot (9)^6\n$$\nif and only if\n$$\n16\\cdot 9^6\\leq 9\\cdot 10^6.\n$$\nNow take the square root on both sides to get that the first inequality holds if and only if\n$$\n4\\cdot 9^3\\leq 3\\cdot 10^3 \\Leftrightarrow \\\\ \n\\frac43\\leq \\left( \\frac{10}{9}\\right)^3\\Leftrightarrow\\\\\n0.75= \\frac34\\geq \\left( \\frac{9}{10}\\right)^3= \\frac{729}{1000}= 0.729.\n$$\n" + - context: None + - retrieval context: ['"\\nWhat\'s more likely: $7$-digit number with no $1$\'s or at least one $1$ among its digits?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nLet $\\{ q_i \\}_{i=1}^{\\infty}$ be a dense subset of $[0,1]$. Define the arithmetic sample mean of the sequences $\\{ q_i \\}$ by\n$$ m_n := \\frac{1}{n} \\sum_{i=1}^n q_i .$$\nQuestion : is $\\{ m_n \\}_{n = 1}^{\\infty}$ also a dense subset of $[0,1]$?\nBecause $\\{q_i \\}$ is dense, then there always exists subsequences of $\\{q_i \\}$ converging to any $x \\in [0,1]$, but does this property carry over via the arithmetic mean?\n" + - actual output: + - expected output: "\nNo. It is easy to bias these arithmetic means by a suitable ordering of the $q_i$, simply by choosing, say, ten elements in $[0,\\frac12]$ for every single element in $(\\frac12,1]$.\nAn explicit ordering would be quite messy, but I hope you get the idea.\n" + - context: None + - retrieval context: ['"\\nIs the collection of arithmetic means of a dense subset of $[0,1]$ a dense subset of $[0,1]$?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answer relevancy score is 0.6., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nThis question has already been asked twice in a similar manner (Non-Perfect Fields and Examples of fields which are not perfect). In both cases, the standard answer found in introductory textbooks was given. This is, the function fields $\\mathbb{F}_q (t)$ for some finite field are not perfect.\nI couldn't think of any other way to construct a non-perfect fields (except adding more variables to get $\\mathbb{F}_q (t,s^q)$ or some variatons of that). I asked myself wether there are other ways to obtain non-perfect fields and in what situations these might show up.\nMy question thus is:\n\nWhich interesting examples of non-perfect fields do you know?\n\nEdit. I understand that every non-perfect field must have $\\mathbb{F}_p$ as it's prime field. As it cannot be algebraic, it must be transcendental. The answers and comments so far suggest that this already tells us how to get all non-perfect fields. But nonetheless: Looking at $\\mathbb{Q}(i)$ we obtain a highly transcendental extension $\\mathbb{C}$ in a \"non-boring\" way. By this I mean that we don't just add variables to get some kind of function field, but we use an analytic construction, involving some kind of metric etc. Thus, adding variables is not the only way to get transcendental extensions. This makes me wonder wether there are similar (maybe more algebraic) methods that produce transcendental extension of finite prime fields that are not perfect.\n" + - actual output: + - expected output: "\nFields of characteristic $0$ are perfect, and a field of characteristic $p$ contains $\\mathbb{F}_p$. Algebraic extensions of $\\mathbb{F}_p$ are perfect, so non-perfect extensions must contain some transcendental element $t$, i.e. they contain $\\mathbb{F}_p(t)$. In this sense $\\mathbb{F}_p(t)$ is the \"universal\" example of a non-perfect field.\nNotice, however, that not every field containing $\\mathbb{F}_p(t)$ must be non-perfect, just consider its perfect closure $\\mathbb{F}_p(t,t^{1/p},t^{1/p^2},\\dotsc)$.\nIf $K$ is any field of characteristic $p$, then $K(t)$ is not perfect, since $t^{1/p} \\notin K(t)$.\nYou can also impose relations. For example, $\\mathbb{F}_3(u,v)[t] / \\langle t^2 + ut + v \\rangle$ is not perfect.\n" + - context: None + - retrieval context: ['"\\nWhat kinds of fields do exist that are non-perfect?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI have a circle centered at (0,2) with a radius of 1 revolving around line x=2. And I need to find the volume of the torus formed from this. Here is the integral that I have managed to set up so far:\n$ \\int_1^2 2\u03c0(2-x)(2)\\sqrt{(1+x)(1-x)}dx\\ + 2 $\nHowever, I am not sure if the intervals of the integral are correct, since the radius is one.\nI got the y value since my circle equation is x^2 + (y-2)^2 = 1, so you'd just have to solve for y in this case.\n" + - actual output: + - expected output: "\nPlease note that you have a circle\n$x^2 + (y-2)^2 \\leq 1$\nAs the circle forms for $-1 \\leq x \\leq 1$, your bounds for $x$ is incorrect.\nAlso for limits of $y$, $ \\ 2 - \\sqrt{1-x^2} \\leq y \\leq 2 + \\sqrt{1-x^2}$\nAnd that gives height of each shell as $2 \\sqrt{1-x^2}$.\nAs the circle is being rotated around $x = 2$, radial distance of shell from the axis of rotation is $(2-x)$.\nSo the integral to find volume using shell method is,\n$\\displaystyle \\int_{-1}^1 2 \\pi (2-x) \\cdot 2 \\sqrt{1-x^2} \\ dx = 4 \\pi \\int_{-1}^1 (2-x) \\sqrt{1-x^2} \\ dx$\nCan you take it from here?\n" + - context: None + - retrieval context: ['"\\nWould this be a correct integral for volume of this torus?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because it is not higher than 0.5, and there are no irrelevant statements., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI found this statement online here and I think this statement is false.I would first prove a general statement and then I would use it to find $\\alpha$ in order to discard the statement given in online link.\nTheorem:If the equation $x^4+ax^3+bx^2+cx+d=0$ has three equal roots, show that each of them is equal to $\\frac{6c-ab}{3a^2-8b}$\u200b\nProof;Three roots of the given equation are same hence, the roots can be assumed as $\\alpha,\\alpha,\\alpha,\\beta$\nHere, $S_1=3\u03b1+\u03b2=\u2212a;S_2\n=3\u03b1(\u03b1+\u03b2)=b;S_3=\u03b1^2(\u03b1+3\u03b2)=\u2212c;S_4=\u03b1^3\u03b2=d$\nWe need to evaluate the value of $\\frac{6c-ab}{3a^2-8b}$\u200b.\n$6c\u2212ab=\u03b1(3\u03b1^2\u22126\u03b1\u03b2+3\u03b2^2)$\n$3a^2\u22128b=3\u03b1^2\u22126\u03b1\u03b2+3\u03b2^2$\nTherefore,$\\frac{6c-ab}{3a^2-8b}=\\frac{\u03b1(3\u03b1^2\u22126\u03b1\u03b2+3\u03b2^2)}{(3\u03b1^2\u22126\u03b1\u03b2+3\u03b2^2)}=\\alpha$\nNow according to this ,putting $a=0$ we get $\\alpha=\\frac{-3c}{4b}$ which is not same as $\\frac{8d}{3c}$ as claimed in online site.\nMy question is: Is the claim in online site wrong?or I have gone wrong somewhere in my proof.\nAny help is much appreciated!\n" + - actual output: "\"Here are 10 examples of math questions (Q) with given answers (A).\\nQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\nA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\nA: \\n\\nQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\nA: \\n\\nQ: \\nIs there an expansion for the following summation?\\n$$ (a_1 + a_2 + \\\\cdots + a_k)^n $$\\n\\n\\nA: \\nhttp://en.wikipedia.org/wiki/Multinomial_theorem\\nThis is what you seek.\\n\\n\\nQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\nA: \\n$B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\n\\n\\nQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\nA: \\nBecause $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n\\nQ: \\nif $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\nA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\nQ: \\nIf $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\nA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\nQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\nA: \\n\\nQ: \\nIs there a name for the geometry of a classic puzzle key?\\nIt's not an ellipse, neither a circle, ...\\n\\n\\n\\nA: \\n\\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: \\nI found this statement online here and I think this statement is false.I would first prove a general statement and then I would use it to find $\\\\alpha$ in order to discard the statement given in online link.\\nTheorem:If the equation $x^4+ax^3+bx^2+cx+d=0$ has three equal roots, show that each of them is equal to $\\\\frac{6c-ab}{3a^2-8b}$\\u200b\\nProof;Three roots of the given equation are same hence, the roots can be assumed as $\\\\alpha,\\\\alpha,\\\\alpha,\\\\beta$\\nHere, $S_1=3\\u03b1+\\u03b2=\\u2212a;S_2\\n=3\\u03b1(\\u03b1+\\u03b2)=b;S_3=\\u03b1^2(\\u03b1+3\\u03b2)=\\u2212c;S_4=\\u03b1^3\\u03b2=d$\\nWe need to evaluate the value of $\\\\frac{6c-ab}{3a^2-8b}$\\u200b.\\n$6c\\u2212ab=\\u03b1(3\\u03b1^2\\u22126\\u03b1\\u03b2+3\\u03b2^2)$\\n$3a^2\\u22128b=3\\u03b1^2\\u22126\\u03b1\\u03b2+3\\u03b2^2$\\nTherefore,$\\\\frac{6c-ab}{3a^2-8b}=\\\\frac{\\u03b1(3\\u03b1^2\\u22126\\u03b1\\u03b2+3\\u03b2^2)}{(3\\u03b1^2\\u22126\\u03b1\\u03b2+3\\u03b2^2)}=\\\\alpha$\\nNow according to this ,putting $a=0$ we get $\\\\alpha=\\\\frac{-3c}{4b}$ which is not same as $\\\\frac{8d}{3c}$ as claimed in online site.\\nMy question is: Is the claim in online site wrong?or I have gone wrong somewhere in my proof.\\nAny help is much appreciated!\\n\"\n" + - expected output: "\nLet the roots be $\\alpha, \\alpha, \\alpha, - 3\\alpha$ so that their sum is $0$.\nNow sum of products of 3 roots taken at a time is $$\\alpha^3-3\\alpha^3-3\\alpha^3-3\\alpha^3$$ and this must equal $-c$ and hence $\\alpha^3=c/8$.\nNow product of roots is $-3\\alpha^4=d$ so that $\\alpha^4=-d/3$ and hence $\\alpha=(-d/3)/(c/8)=-8d/3c$.\nYour question has thus a sign error. You can convince yourself by putting $\\alpha=1$. The desired polynomial is $$(x-1)^3(x+3)=x^4-6x^2+8x-3$$ so that $c=8,d=-3$ and $-8d/3c=1=\\alpha$.\n\nI checked your approach and your theorem in question and its proof are correct. Thus $\\alpha $ should equal $-3c/4b$.\nLet's observe that sum of product of roots taken $2$ at a time is $$\\alpha^2+\\alpha^2+\\alpha ^2-3\\alpha ^2-3\\alpha ^2-3\\alpha^2$$ and this should equal $b$. Thus $b=-6\\alpha^2$ and hence $$-\\frac{3c}{4b}=-\\frac{24\\alpha^3}{(-24\\alpha^2)}=\\alpha$$ So we have $$\\alpha=-\\frac{3c}{4b}=-\\frac{8d}{3c}$$\n" + - context: None + - retrieval context: ['"\\nIf $\\\\alpha$ be a multiple root of the order 3 of the equation $x\\u2074+bx\\u00b2+cx+d=0$ then $\\u03b1= 8d/3c\\u200b$\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answer relevancy score is 0.6., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nMy friend invented this problem in his spare time and asked me if I could solve it:\nDo there exist $n,b\\in \\mathbb{N}, b\\beta\\ge1$ and $\\text{gcd}(\\alpha, \\beta)=1$. So $(n-b)(n^2-b) \\mid n^3$ becomes:\n$$d(\\alpha-\\beta)(d^2\\alpha^2-d\\beta) \\mid d^3\\alpha^3$$\nwhich is equivalent to\n$$(\\alpha-\\beta)(d\\alpha^2-\\beta) \\mid d\\alpha^3$$\nIn particular, $\\alpha -\\beta \\mid d\\alpha^3$. Since $\\text{gcd}(\\alpha, \\beta)=1$ we obtain $\\text{gcd}(\\alpha-\\beta, \\alpha^3)=1$. Consequently, we get $\\alpha -\\beta \\mid d$, so $d = (\\alpha-\\beta)\\cdot d'$, for some $d' \\ge 1$. Thus, replacing $d$ with $(\\alpha-\\beta)\\cdot d'$ we get\n$$(\\alpha-\\beta)\\left((\\alpha-\\beta)d'\\alpha^2 - \\beta \\right) \\mid (\\alpha-\\beta)d'\\alpha^3$$\nand thus, dividing both sides by $(\\alpha-\\beta)$ we obatin\n$$(\\alpha-\\beta)d'\\alpha^2 -\\beta \\mid d'\\alpha^3$$\nAgain, $\\text{gcd}(\\alpha, \\beta)=1$ implies $\\text{gcd}((\\alpha-\\beta)d'\\alpha^2-\\beta, \\alpha^3)=1$ and hence\n$$(\\alpha-\\beta)d'\\alpha^2 -\\beta \\mid d'$$\nIn particular, $(\\alpha-\\beta)d'\\alpha^2 -\\beta \\le d'$. This inequality is equivalent to\n$$d\\alpha^2 \\le \\frac{d}{\\alpha-\\beta}+\\beta$$\nBut, since $\\alpha \\ge 2$, we have $d\\alpha^2 \\ge 2d\\alpha=d\\alpha+d\\alpha> d + d\\beta\\ge \\frac{d}{\\alpha-\\beta}+\\beta$ yielding a contradiction.\nConsequently, there are no such $n,b$.\n" + - context: None + - retrieval context: ['"\\nDo there exist $n,b\\\\in \\\\mathbb{N}, b0. \n$$\nThen linear-fractional transformations $\\gamma\\in PSL(2, {\\mathbb C})$ given by\n$$\n\\gamma(u)= (au+b)(cu+d)^{-1}, ad-bc=1, \n$$\nextend to isometries of the upper half-space by the formula\n$$\n\\gamma(z)= (az+b)(cz+d)^{-1}. \n$$\nSee section 2.1 in\nAhlfors, Lars V., Moebius transformations in several dimensions, Ordway Professorship Lectures in Mathematics. Minneapolis, Minnesota: University of Minnesota, School of Mathematics. 150 p. (1981). ZBL0517.30001.\nThis is very cute, but I am unaware of any uses of this extension formula.\n" + - context: None + - retrieval context: ['"\\nExplicit formulas of how to extend $PSL_2(\\\\mathbb{C})$ action of M\\u00f6bius group on upper-half space model of $\\\\mathbb{H}^3$?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answer relevancy score is 0.6., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nWe know that regular triangles, squares and hexagons can tile the plane without leaving any \"hole\".\nHowever, I've noticed that many regular polygons can tile the plane if we allow for a single type of \"hole\" (i.e., another shape) to be present.\nThe following image contains an example with pentagons and rhombi:\n\nWhat we obtain in this case is not a periodic tiling, but rather an aperiodic one: Still, we are able to tile the plane with these two shapes.\nAnother example, this time with decagons + \"concave hexagons\":\n\nMy question is: Can we always tile the plane by combining a regular polygon and a single other shape?\nI'm also interested in the extension to star polygons.\n\nEdit\nAs pointed out in the comments, the way I asked the question was imprecise. I guess that we should add the additional constraint that no \"hole\" can be in contact with another \"hole\", otherwise there will be trivial solutions to the problem.\n" + - actual output: + - expected output: "\nYes, it is always possible.\nSuppose $n>3$ is odd. Then we arrange the polygons as shown (examples below for $n=5,7,9$):\n\n\n\nNote also that $n=3,4$ are trivially possible.\nNow suppose $n>4$ is even. Then we take the construction from $n/2$ (which we may assume exists by induction) and simply truncate the corners of the $n/2$-gons to produce $n$-gons which share the same adjacency graph, while causing equal perturbations to the \"holes\" so that they remain congruent to one another.\n" + - context: None + - retrieval context: ['"\\nCan one always tile the plane with a regular polygon + a single other shape?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.6, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.6 because the answer relevancy score is 0.6., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nThis definition of regular sequence that I know is:\nLet $R$ be a ring and let $M$ be an $R$-module. A sequence of\nelements $\\{x_1,... ,x_n\\} \\in R$ is called a regular sequence on $M$ (or an $M$- sequence) if\n\n$(x_1,..., x_n)M \\ne M$, and\nFor $i = 1,... ,n$, $x_i$ is a nonzerodivisor on $M/(x_1,... ,x_{i-1})M$.\n\n\nMy question is to show that in the ring $K[x,y,z]$ the sequence $x,x+y^2,x+y+z^3$ is regular.\n\nThis is not a homework problem. I am trying to self-study commutative algebra and this definition gives me no idea on how to even start this problem. Can someone show me how to do this problem with some details?\nMy original question asks me to show that\n$\\{y_1+x_1y_2+x_1^2y_3,y_1+x_2y_2+x_2^2y_3,y_1+x_3y_2+x_3^2y_3\\}$ is a regular sequence in $k[y_1,y_2,y_3,x_1,x_2,x_3]$ (the order is lexico)\nGoing exactly in the way @Joshua suggested I did consider the quotient ring $k[y_1,y_2,y_3,x_1,x_2,x_3]/(y_1+x_1y_2+x_1^2y_3)$. The image of the second element that I got is $(x_2-x_1)(y_2+y_3x_2+y_3x_1)$ I am not sure how to proceed, is there a computer code or something that he will help me?\nThe Grobner basis of $I=\\left$ that I got is:\n\n$y_1+ y_2x_3 + y_3x_3^2$\n\n$-y_2x_3 + y_2x_1 - y_3x_3^2 + y_3x_1^2$\n\n$-y_2x_3 + y_2x_2 - y_3x_3^2 + y_3x_2^2$\n\n$y_3x_1^2x_2 - y_3x_1^2x_3 - y_3x_1x_2^2 + y_3x_1x_3^2 + y_3x_2^2x_3 - y_3x_2x_3^2$\n\n\nThe monomial order is $y_1 > y_2 > y_3 > x_1 > x_2 > x_3$\nWhat is a Hilbert series? How do you compute it with the help of Grobner techniques? Do you have any access to any example which can help me with this (maybe any worked out example)? I went through Stanley's paper but I think that will need a strong background in commutative algebra. (I only know upto Grobner basis which I read from David A. Cox.)\n" + - actual output: "\"Hello, thank you for using our system.\\nThe prompt was:\\nHere are 10 examples of math questions (Q) with given answers (A).\\nQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\nA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\nA: \\n\\nQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\nA: \\n\\nQ: \\nIs there an expansion for the following summation?\\n$$ (a_1 + a_2 + \\\\cdots + a_k)^n $$\\n\\n\\nA: \\nhttp://en.wikipedia.org/wiki/Multinomial_theorem\\nThis is what you seek.\\n\\n\\nQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\nA: \\n$B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\n\\n\\nQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\nA: \\nBecause $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n\\nQ: \\nif $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\nA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\nQ: \\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\nA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\nQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\nA: \\n\\nQ: \\nIs there a name for the geometry of a classic puzzle key?\\nIt's not an ellipse, neither a circle, ...\\n\\n\\n\\nA: \\n\\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: \\nThis definition of regular sequence that I know is:\\nLet $R$ be a ring and let $M$ be an $R$-module. A sequence of\\nelements $\\\\{x_1,... ,x_n\\\\} \\\\in R$ is called a regular sequence on $M$ (or an $M$- sequence) if\\n\\n$(x_1,..., x_n)M \\\\ne M$, and\\nFor $i = 1,... ,n$, $x_i$ is a nonzerodivisor on $M/(x_1,... ,x_{i-1})M$.\\n\\n\\nMy question is to show that in the ring $K[x,y,z]$ the sequence $x,x+y^2,x+y+z^3$ is regular.\\n\\nThis is not a homework problem. I am trying to self-study commutative algebra and this definition gives me no idea on how to even start this problem. Can someone show me how to do this problem with some details?\\nMy original question asks me to show that\\n$\\\\{y_1+x_1y_2+x_1^2y_3,y_1+x_2y_2+x_2^2y_3,y_1+x_3y_2+x_3^2y_3\\\\}$ is a regular sequence in $k[y_1,y_2,y_3,x_1,x_2,x_3]$ (the order is lexico)\\nGoing exactly in the way @Joshua suggested I did consider the quotient ring $k[y_1,y_2,y_3,x_1,x_2,x_3]/(y_1+x_1y_2+x_1^2y_3)$. The image of the second element that I got is $(x_2-x_1)(y_2+y_3x_2+y_3x_1)$ I am not sure how to proceed, is there a computer code or something that he will help me?\\nThe Grobner basis of $I=\\\\left$ that I got is:\\n\\n$y_1+ y_2x_3 + y_3x_3^2$\\n\\n$-y_2x_3 + y_2x_1 - y_3x_3^2 + y_3x_1^2$\\n\\n$-y_2x_3 + y_2x_2 - y_3x_3^2 + y_3x_2^2$\\n\\n$y_3x_1^2x_2 - y_3x_1^2x_3 - y_3x_1x_2^2 + y_3x_1x_3^2 + y_3x_2^2x_3 - y_3x_2x_3^2$\\n\\n\\nThe monomial order is $y_1 > y_2 > y_3 > x_1 > x_2 > x_3$\\nWhat is a Hilbert series? How do you compute it with the help of Grobner techniques? Do you have any access to any example which can help me with this (maybe any worked out example)? I went through Stanley's paper but I think that will need a strong background in commutative algebra. (I only know upto Grobner basis which I read from David A. Cox.)\\n\"\n" + - expected output: "\nIn this case you can directly apply the definition.\n$K[x, y, z]/(x) = K[y, z]$, of course, and the image of $x+y^2$ is just $y^2$, which is clearly not a zero-divisor. Next, $K[y, z]/(y^2)$ has $K$-basis $\\{1, z, z^2, \\ldots, y, yz, yz^2, \\ldots\\}$ subject to the relation $y^2 = 0$ and the image of $x+y+z^3$ is $y+z^3$. Suppose $y+z^3$ were a zero-divisor. Say $(f(z) + yg(z))(y+z^3) = 0$. Then $f(z)z^3 + y(f(z) + g(z)z^3) = 0$. That forces $f(z)z^3 = 0$ and $f(z) + g(z)z^3 = 0$. But that forces $f(z)=0$ and then $g(z)=0$. So $y+z^3$ is not a zerodivisor.\nEdit in response to follow-up question in comments: Sequences of homogeneous elements are better-behaved with respect to being regular, e.g. they're regular if and only if any rearrangement is. In your case you could declare $\\deg y_1 = 3, \\deg y_2 = 2, \\deg y_3 = \\deg x_i = 1$, making your elements $f_1, f_2, f_3$ all homogeneous of degree $3$. Now you can compute the Hilbert series of $K[x_1, x_2, x_3, y_1, y_2, y_3]/(f_1, f_2, f_3)$ using Grobner-theoretic techniques and check if it's correct*. This is apparently what Macaulay2 does.\nThat could technically be done by hand simply using Buchberger's algorithm and manual S-polynomial reductions (I'm guessing the Grobner basis here will be quite small), but it's very tedious and error-prone. Once you get the Grobner basis, then you'd just need to pick off their leading terms, give an explicit description of the standard monomials (those that aren't divisible by any of those leading terms), and compute the Hilbert series.\nI wouldn't be surprised if mucking about along the lines of my original answer were still faster in this particular case.\n*What is the correct Hilbert series? See e.g. Corollary 3.2 of Stanley's lovely mostly-survey paper \"Hilbert Functions of Graded Algebras\".\n" + - context: None + - retrieval context: ['"\\nHow to prove that the sequence is regular?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n\n\n\n\n\n\nThis question already has answers here: \n\n\n\nWhat is so wrong with thinking of real numbers as infinite decimals?\n\r\n (11 answers)\r\n \n\nClosed 1 year ago.\n\n\n\nAt the end of the day, a real number can be viewed simply as a function over the integers \u2014> the naturals which tells you the digit as that ten\u2019s place (assuming base ten)? You could augment this and say it\u2019s a tuple (sign, function) to add +/- information.\nThis is the most general definition of a real number, right? I think I find confusion sometimes in wondering how numbers like pi and e are just numbers; but in reality they are just their definitions, and to pull out a digit at a given place you need to consult the definition. So in short these numbers contain more information than one might guess at first.\nWhen I view real numbers this way, I feel I really understand what a number is.\nI hope this makes sense. I only have a minor in math.\nEDIT: My real goal is understanding: what information does a real number provide? What information \"is\" a number, crudely put? I first started thinking of this formalism when considering numbers like e and pi, which \"just happen\" to be 2.7... but really contain \"infinite\" information and are more clearly characterized by their expression as summations/whatever than as decimal numbers.\n" + - actual output: "\"Here are 10 examples of math questions (Q) with given answers (A).\\nQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\nA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\nA: \\n\\nQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\nA: \\n\\nQ: \\nIs there an expansion for the following summation?\\n$$ (a_1 + a_2 + \\\\cdots + a_k)^n $$\\n\\n\\nA: \\nhttp://en.wikipedia.org/wiki/Multinomial_theorem\\nThis is what you seek.\\n\\n\\nQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\nA: \\n$B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\n\\n\\nQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\nA: \\nBecause $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n\\nQ: \\nIf $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\nA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\nQ: \\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\nA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\nQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\nA: \\n\\nQ: \\nIs there a name for the geometry of a classic puzzle key?\\nIt's not an ellipse, neither a circle, ...\\n\\n\\n\\nA: \\n\\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: \\n\\n\\n\\n\\n\\n\\nThis question already has answers here:\\r\\n\\t http://www.quora.com/Mathematics/What-is-so-wrong-with-thinking-of-real-numbers-as-infinite-decimals?__ans_9853_0\\r\\n\\r\\nThis question already has answers here:\\r\\nhttps://www.quora.com/What-is-so-wrong-with-thinking-of-real-numbers-as-infinite-decimals?__ans_9854_1\\r\\n \\n\\nClosed 1 year ago.\\n\\n\\n\\nAt the end of the day, a real number can be viewed simply as a function over the integers \\u2014> the naturals which tells you the digit as that ten\\u2019s place (assuming base ten)? You could augment this and say it\\u2019s a tuple (sign, function) to add +/- information.\\nThis is the most general definition of a real number, right? I think I find confusion sometimes in wondering how numbers like pi and e are just numbers; but in reality they are just their definitions, and to pull out a digit at a given place you need to consult the definition. So in short these numbers contain more information than one might guess at first.\\nWhen I view real numbers this way, I feel I really understand what a number is.\\nI hope this makes sense. I only have a minor in math.\\nEDIT: My real goal is understanding: what information does a real number provide? What information \\\"is\\\" a number, crudely put? I first started thinking of this formalism when considering numbers like e and pi, which \\\"just happen\\\" to be 2.7... but really contain \\\"infinite\\\" information and are more clearly characterized by their expression as summations/whatever than as decimal numbers.\\n\"\n" + - expected output: "\nThere are a couple of definitions of real numbers (using the rational numbers as a starting point) that are pretty standard. They are also necessarily equivalent to each other, because it's a theorem that there is a unique complete ordered field.\nOne definition defines an equivalence relationship between Cauchy sequences of rational numbers, and then defines each equivalence class to be a real number. You can think of this as the equivalence class of Cauchy sequences that converge to the real number you care about. The other definition (and the one I find more intuitive) uses \"Dedekind cuts\" -- a set of rational numbers that is bounded above and that is closed downward, which you can think of as the set of rational numbers less than the real number you care about.\nThese definitions are standard because they are useful. Using Dedekind cuts, for instance, it's quite easy to prove that the reals are a complete ordered set. It's a little more tedious to define the field operations but it's pretty straightforward to do so. So I guess the question becomes, once you correct your definition to account for terminating fractions, what makes it useful for you? Perhaps you'd find it a useful exercise to prove rigorously that your proposed definition is in fact equivalent to one of the standard definitions. One challenge you'll face is defining field operations in such a way that they are well defined; in other words, so that the two different equivalent representations of a terminating decimal give you equivalent results.\n" + - context: None + - retrieval context: ['"\\nPossible interpretation of real numbers as functions? [duplicate]\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is because ., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \n\nQuestion: Let $a \\in \\mathbb{R}$. Find all possible values of $a$ such that there exists a function $f: \\mathbb{R} \\to \\mathbb{R}$ satisfying $f(x + 2) = -f(x)$ and $f(x + 3) = f(x) + a$ for any real number $x \\in \\mathbb{R}$.\n\nMy (maybe wrong) solution:\nStart listing down some values of $f(x)$ with $f(2) = -f(0)$ and $f(3) = f(0) + a$.\n$f(2) + f(3) = - f(0) + f(0) + a = a$.\nSimilarly, you can write $f(1) = -(f(-1))$ and $f(2) = f(-1) + a$.\nHence, $f(1) + f(2) = -(f(-1)) + f(-1) + a = a$.\nSo, $f(1) + f(2) = f(2) + f(3) \\Longrightarrow f(1) = f(3).$\nBut, according to the problem, $f(x + 2) = -f(x) \\Longrightarrow f(3) = -f(1)$.\nSince $f(1) = f(3)$ and $f(1) = -f(3) \\Longrightarrow f(1) = f(3) = 0.$\nDoing similar steps, we find out that $f(0) = f(2) = 0$. Hence, $a = f(2) + f(3) = 0 + 0 = 0$.\nThe only possible value of $a$ is $\\boxed{0}$.\n$\\\\$\nIf I'm doing something wrong, can someone please explain? Thanks in advance!\n(Thanks to the answers, I now know I only need to prove that $a = 0$ can satisfy a valid function :))\n" + - actual output: "\"Here are 10 examples of math questions (Q) with given answers (A).\\nQ: \\nCan you help me with this ''simple'' exercise:\\nProve that ${\\\\Bbb Q}[\\\\sqrt{-3}]$ is a cyclotomic field.\\n\\n\\nA: \\nA cyclotomic field is... And the complex roots of $x^3-1$ are...\\n\\n\\nQ: \\nIs the following integral equation solvable ?\\n$$ F(x)-\\\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\\\frac{\\\\sin \\\\gamma(x-y)}{\\\\pi(x-y)}$$ and $$f(x)=e^{i\\\\gamma x}$$\\nand $\\\\gamma$ is a parameter.\\n\\n\\nA: \\n\\nQ: \\nGiven that $X$ and $Y$ are positive definite matrices,\\nhow can I bound the singular values $\\\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\\n\\n\\nA: \\n\\nQ: \\nIs there an expansion for the following summation?\\n$$ (a_1 + a_2 + \\\\cdots + a_k)^n $$\\n\\n\\nA: \\nhttp://en.wikipedia.org/wiki/Multinomial_theorem\\nThis is what you seek.\\n\\n\\nQ: \\nI have to simplify the following expression:\\n$B=\\\\sqrt{x^2} - x$\\nThe only thing that I can do is:\\n$\\\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \\nIs that enough?\\n\\n\\nA: \\n$B=0$ if $x\\\\geq0$ and $B=-2x$ if $x\\\\leq 0$\\n\\n\\nQ: \\nQuick question:\\nWolfram Alpha tells me that \\n$$\\\\int |e^{ix}|^2 dx = x + C$$\\nWhy is that?\\n\\n\\nA: \\nBecause $$|e^{ix}|=1$$ for every $x \\\\in \\\\mathbb{R}$.\\n\\n\\nQ: \\nIf $10^{80}=2^x$, what is the value of $x$?\\n(Or, what binary word length would you need to contain $10$ to the $80$?)\\n\\n\\nA: \\n$\\\\ln(b^{x}) = x \\\\times \\\\ln(b)$. Solve for $x$.\\n\\n\\nQ: \\nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \\nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\\n\\n\\nA: \\nOK - $f(x)=7=3x-4$, can you solve this?\\n\\n\\nQ: \\nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \\nThanks for help.\\n\\n\\nA: \\n\\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\\n\\nQ: \\n\\nQuestion: Let $a \\\\in \\\\mathbb{R}$. Find all possible values of $a$ such that there exists a function $f: \\\\mathbb{R} \\\\to \\\\mathbb{R}$ satisfying $f(x + 2) = -f(x)$ and $f(x + 3) = f(x) + a$ for any real number $x \\\\in \\\\mathbb{R}$.\\n\\nMy (maybe wrong) solution:\\nStart listing down some values of $f(x)$ with $f(2) = -f(0)$ and $f(3) = f(0) + a$.\\n$f(2) + f(3) = - f(0) + f(0) + a = a$.\\nSimilarly, you can write $f(1) = -(f(-1))$ and $f(2) = f(-1) + a$.\\nHence, $f(1) + f(2) = -(f(-1)) + f(-1) + a = a$.\\nSo, $f(1) + f(2) = f(2) + f(3) \\\\Longrightarrow f(1) = f(3)$.\\nBut, according to the problem, $f(x + 2) = -f(x) \\\\Longrightarrow f(3) = -f(1)$.\\nSince $f(1) = f(3)$ and $f(1) = -f(3) \\\\Longrightarrow f(1) = f(3) = 0$.\\nDoing similar steps, we find out that $f(0) = f(2) = 0$. Hence, $a = f(2) + f(3) = 0 + 0 = 0$.\\nThe only possible value of $a$ is $\\\\boxed{0}$.\\n$\\\\\\\\$\\nIf I'm doing something wrong, can someone please explain? Thanks in advance!\\n(Thanks to the answers, I now know I only need to prove that $a = 0$ can satisfy a valid function :))\\n\"\n" + - expected output: "\nYour argument is fine - so far.\nWhile looking for necessary conditions for $a$, we might but conclude more about $f$ by morking slightly more generally: We have\n$$ f(x+6)=-f(x+4)=f(x+2)=-f(x)$$\nand\n$$ f(x+6)=f(x+3)+a=f(x)+2a$$\nso by equating,\n$$-f(x)=f(x)+2a $$\nwhich shows that $f$ is necessarily constant. But then $a=f(x+2)=-f(x)=-a$ shows that in fact $a$ is necessarily $=0$.\nWhat is left (and what is also missing from your attempt!), is to show that this is also sufficient, i.e., that for $a=0$ there actually does exist such a function.\nFortunately, this last step is not hard - all you have to check is that for $a=0$, the constant-zero function fulfills the functional equations.\n" + - context: None + - retrieval context: ['"\\nFind constant in system of function equations\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI\u2019m quite confused about some topological results. I know there must be something wrong in my reasoning, but I cannot find out what is wrong here. We know that:\n\n$\\mathbb{R}$ is closed (and is also opened, but that\u2019s not what confuses me)\n$\\tan$ is a continuous function on $]-\\pi/2, \\pi/2[$\n\nMy question is quite simple: since the inverse image under a continuous function of a closed set is closed, why do we have $\\tan^{-1}(\\mathbb{R})=]-\\pi/2,\\pi/2[$, which is not closed?\n" + - actual output: + - expected output: "\n$\\tan$ is continuous as a function $(-\\frac \\pi 2, \\frac \\pi 2) \\to \\mathbb R$, and the interval is closed (and open) in the restriction topology (which is a standard topology if we consider it as a topological space).\n" + - context: None + - retrieval context: ['"\\nShouldn\\u2019t $\\\\tan^{-1}(\\\\mathbb{R})$ be closed?\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Passed Answer Relevancy (score: 1.0, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 1 because it is a good answer., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nThis is my outline of proof:\n\nBy Sylow's theorems, $G$ has two unique subgroups $H$ and $K$ respectively of order $5$ and order $7$ and both are Abelian; as groups of prime order are Abelian\nNext I use the counting argument to say: Taking any $x\\in{H}$ and $y\\in{K}$ such that $x,y\\notin{e}$, I will have more number of ordered pairs than the remaining elements in $G$ (i.e $35-5-7+1=24$) ($e$ is common to both so $+1$)\nThus there are combinations which are same; implying $x$ and $y$ commute; but how do I proceed to prove that even the other elements commute. Kindly help in showing the way further.\n\n" + - actual output: "I understand that you have a question, and here is my answer.\n" + - expected output: "\nLet $h\\in H, k\\in K, [h,k]=hkh^{-1}k^{-1}=(hkh^{-1})k^{-1}=h(kh^{-1}k^{-1})$ is in $H\\cap K$ since $H$ and $K$ are normal subgroups, its order divides $5$ and $7$ so, it is $1$.\n" + - context: None + - retrieval context: ['"\\nProving that group $G$ of order $|G|=35$ is Abelian\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because the answer relevancy score is 0.5., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nIt is easy to compare to numbers of the form $a\\sqrt{b}$, simply by comparing their squares, for example $3\\sqrt{3}$ and $2\\sqrt{5}$.\nBut what if we have $a=3+3\\sqrt{3}$ and $b=4+2\\sqrt{5}$ for example?\nHow to compare them without using a calculator? Is there a method that works for all numbers of the form $c+\\sqrt{b}$?\nI found a similar question to this on the site, but that one is a bit different and I am looking for an answer suitable for elementary students to understand. (but I would love to know any kind of answer even if it does not fit the elementary level).\nI was thinking about subtracting $b-a$ and checking wether it is negative or positive,\nI got $b-a=1+2\\sqrt{5}-3\\sqrt{3}$ but $2\\sqrt{5}-3\\sqrt{3}<0$\n(I don't want to continue from here by approximating $\\sqrt{5}=2.something$, because we will be somehow using the calculator in our mind; so I am stuck here)\n" + - actual output: "You are an intelligent mathematical assistant.\n" + - expected output: "\nGeneral algorithm:\nLet us consider $d>b$.\nAssume $$a+\\sqrt b>c+\\sqrt d$$\nThen $$(a-c)>(\\sqrt d-\\sqrt b)$$\nNote that we need to proceed further, only if $a>c$, because, if $c>a$ then our assumption is certainly false. If $a>c$:\n$$(a-c)^2>b+d-2\\sqrt {bd}$$\n$$2\\sqrt {bd}>b+d-(a-c)^2$$\nIf $(a-c)^2>b+d$, then this is certainly true, hence our assumption is true. Otherwise, we square once more, and compare. This time it's easy to compare, since all radicals are removed.\nExample:\nSince $27>20$, we assume:\n$$4+\\sqrt {20}>3+\\sqrt {27}$$\nWe have, $1>\\sqrt {27}-\\sqrt {20}$\nSince $4>3$, we proceed further by squaring:\n$$1>47-2\\sqrt {540}$$\n$$2\\sqrt {540}>46$$\nAgain, we need to proceed further, as $46>0$.\nSo, $$\\sqrt {540}>23$$\nSquaring the last time,\n$$540>529$$\nSince this is true, our original assumption was true as well.\n" + - context: None + - retrieval context: ['"\\nComparing numbers of the form $c+\\\\sqrt{b}$ (eg, $3+3\\\\sqrt{3}$ and $4+2\\\\sqrt{5}$) without a calculator\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + +====================================================================== + +Metrics Summary + + - Failed Answer Relevancy (score: 0.5, threshold: 0.7, strict: False, evaluation model: Hudson/llemma:7b (Ollama), reason: The score is 0.5 because it is a good answer, but there are irrelevant statements in the actual output., error: None) + +For test case: + + - input: "Here are 10 examples of math questions (Q) with given answers (A).\nQ: \nCan you help me with this ''simple'' exercise:\nProve that ${\\Bbb Q}[\\sqrt{-3}]$ is a cyclotomic field.\n\n\nA: \nA cyclotomic field is... And the complex roots of $x^3-1$ are...\n\n\nQ: \nIs the following integral equation solvable ?\n$$ F(x)-\\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\\frac{\\sin \\gamma(x-y)}{\\pi(x-y)}$$ and $$f(x)=e^{i\\gamma x}$$\nand $\\gamma$ is a parameter.\n\n\nA: \n\nQ: \nGiven that $X$ and $Y$ are positive definite matrices,\nhow can I bound the singular values $\\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?\n\n\nA: \n\nQ: \nIs there an expansion for the following summation?\n$$ (a_1 + a_2 + \\cdots + a_k)^n $$\n\n\nA: \nhttp://en.wikipedia.org/wiki/Multinomial_theorem\nThis is what you seek.\n\n\nQ: \nI have to simplify the following expression:\n$B=\\sqrt{x^2} - x$\nThe only thing that I can do is:\n$\\sqrt{a^2}=|a|$, thus $B=|x|-x$. \nIs that enough?\n\n\nA: \n$B=0$ if $x\\geq0$ and $B=-2x$ if $x\\leq 0$\n\n\nQ: \nQuick question:\nWolfram Alpha tells me that \n$$\\int |e^{ix}|^2 dx = x + C$$\nWhy is that?\n\n\nA: \nBecause $$|e^{ix}|=1$$ for every $x \\in \\mathbb{R}$.\n\n\nQ: \nIf $10^{80}=2^x$, what is the value of $x$?\n(Or, what binary word length would you need to contain $10$ to the $80$?)\n\n\nA: \n$\\ln(b^{x}) = x \\times \\ln(b)$. Solve for $x$.\n\n\nQ: \nif $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. \nI would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.\n\n\nA: \nOK - $f(x)=7=3x-4$, can you solve this?\n\n\nQ: \nOn p.10 of Algebraic Number Theory edited by Cassels and Fr\u00f6hlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? \nThanks for help.\n\n\nA: \n\nQ: \nIs there a name for the geometry of a classic puzzle key?\nIt's not an ellipse, neither a circle, ...\n\n\n\nA: \n\nProvide an answer (A) to the following math question (Q) in a similar manner to the previous example(s) given.\n\nQ: \nI was working through a trigonometry problem, and was having some difficulty so I decided to look at the solution. Here are the steps:\n$$\\frac{\\sin(2x+50^\\circ)+\\sin(150^\\circ)}{\\sin(2x+50^\\circ)-\\sin(150^\\circ)}=\\frac{\\cos(50^\\circ)-\\cos(2x+50^\\circ)}{\\cos(50^\\circ)+\\cos(2x+50^\\circ)}$$\n$$\\frac{\\sin(2x+50^\\circ)}{\\sin150^\\circ}=\\frac{-\\cos50^\\circ}{\\cos(2x+50^\\circ)}$$\n(Image that replaced text.)\nI am not exactly how the solution got from the first step to the second one. I would just like some clarification on the intermediate step.\n" + - actual output: + - expected output: "\n$$\\frac{a+b}{a-b}=\\frac{c-d}{c+d}$$\n$$ac+ad+bc+bd=ac-ad-bc+bd$$\n$$2ad=-2bc$$\n$$\\frac{a}{b}=-\\frac{c}{d}$$\n" + - context: None + - retrieval context: ['"\\nTrigonometric Simplification Question\\n"'] + +====================================================================== + +Overall Metric Pass Rates + +Answer Relevancy: 13.54% pass rate + +====================================================================== + + + Tests finished ! Run 'deepeval view' to analyze, debug, and save evaluation results on Confident AI. +