Win At Mathematics

One of our user asked: I will be taking the American Math Contest. Various topics are covered in this test including advanced geometry and algebra. It would be great if any of you could provide books/references which contain challenging or hard problems related to advanced geometry and algebra.

One of our user asked: I am a competition math student, and I was doing a practice test when I found this problem: "The fourth degree polynomial equation $x^4 - 7x^3 + 4x^2 + 7x - 4 = 0$ has four real roots a, b, c and d. What is the value of the sum $1/a + 1/b + 1/c + 1/d$ ? Express your answer as a common fraction." First of all, I don't think this polynomial can be factored, so I don't know how to find the reciprocals of the roots. I need help and any help would be appreciated! Note: When they say "Express your answer as a common fraction.", it means that the answer IS a fraction. I tried asking this question already, but I worded it differently, and they didn't get a fraction so I knew it was wrong. This is the link to my other question: Link

One of our user asked: I could do this on a calculator but I thought there would be a more efficient way to do this than punching on my calculator the following: $3.5 \cdot 1.03 + 3.5 \cdot 1.03^2 + 3.5 \cdot 1.03^3 + \ldots + 3.5 \cdot 1.03^{10}$ So basically, $xy + xy^2 + xy^3 + ... + xy^{10}$ How can I simplify this to make it easier to calculate on a calculator?

One of our user asked: Write down the cubic equation given that $\alpha + \beta + \gamma = 4$ , $\alpha^2 + \beta^2 + \gamma^2 = 66$ , and $\alpha^3 + \beta^3 + \gamma^3 = 280$ Ok so, the sum of roots is given and I'm able to use the sum of the roots and the sum of the roots squared to get the sum of the combination of roots, but I'm unable to get the product of roots, because I can't seem to manipulate the sum of the cubes of roots to resemble the sum of roots and sum of squares of roots.

One of our user asked: When I type this equation ( $6x + 25 = 7y$ ) into WolframAlpha, it is able to tell me that the integer solution for this equation is: $x = 7n + 4$ , $y = 6n + 7$ , where n in the set of all integers How can I arrive at this solution on my own?

One of our user asked: A man want 762 by a number, but he trade it and its quotient result 13 and its rest was 13. What the quotient and rest of the original number? $$ q = ?; r=? $$ $$ 762 = q \cdot \overline{ ab } + r $$ $$ 762 = 13 \cdot \overline {ba} + 21 $$ Another way to write this is: $$ \frac{762-r}{ \overline{ba}}=r $$ $$ \frac{762-21}{ \overline{ba}}=13 $$

One of our user asked: We can consider $\mathbb{R}^+$ with two operation: +: $\mathbb{R}^+ \times \mathbb{R}^+\to \mathbb{R}^+$ that maps $(a,b)$ to $a+b:=ab$ and $ \cdot : \mathbb{R} \times \mathbb{R}^+ \to \mathbb{R}^+$ that maps $(\lambda,a)$ to $a^\lambda$ . With respect to these operation we have that $\mathbb{R}^+$ is a vector space. We want find an example of linear application from $\mathbb{R}$ to $\mathbb{R}^+$ . We can fix $a\in \mathbb{R}^+$ and we can define $f_a:\mathbb{R}\to \mathbb{R}^+$ such that maps every $\lambda$ to $ f_a(\lambda):=a^\lambda$ . From the rules of powers, the map $f_a$ is a linear map for every $a\in \mathbb{R}^+$ . A natural question can be if all linear maps from $\mathbb{R}$ to $\mathbb{R}^+$ are an exponential maps. The answer is positive because for each linear map $F: \mathbb{R}\to \mathbb{R}^+$ we have that $F(\lambda)=F(\lambda \cdot 1)=(F(1))^\lambda=f_{F(1)}(\lambda)$ so $F=f_{F(1)}$ We can also observe that the m...

One of our user asked: I quite sure about in but didn't found a prove. Is it true?

One of our user asked: Find the remainder of the division of $x^n+5$ with $x^3+10x^2+25x$ over $\mathbb{Q}$ What I tried to do is to write $x^n+5=p(x)(x^3+10x^2+25x)+Ax^2+Bx+C$ , where $p(x)$ is a polynomial of degree $n-3$ . If I set $x=0$ I obtain that $C=0$ . Now $x^3+10x^2+25x=x(x+5)^2$ and by setting $x=-5$ , I get that $5A-B=(-1)^n5^{n-1}$ But I need one more equation to be able to find the coefficients of the remainder, and I can't get one. What should I do?

One of our user asked: I wonder if someone can help me to find the differentiation with respect to the upper limit of a summation as shown below !! $\frac{d}{dx}\sum\limits_{n=1}^{f(x)} g(n)$ .

One of our user asked: How to prove that if n is composite then there are integers a and b such that n divides ab but n doesn't divide either a or b.

One of our user asked: Here are few of the questions from the previous years' exams. I've chosen the ones I'm not sure about. It's a simple TRUE/FALSE task. Would anyone be able to verify my solution? Some of my answers are good, some are just random guess according to my intuition. I don't really need a detailed explanation... Thanks! Domain of $f'$ is contained within domain of $f$ . - TRUE Boundary point of set A is also a cluster point of that set. - TRUE Every increasing sequence and bounded above is convergent. - TRUE Every increasing sequence and bounded below is convergent. - FALSE Every increasing sequence is always bounded below. - TRUE Every sequence is discontinuous function. - FALSE Every sequence is continuous function. - TRUE Every function integrable on $<a, b>$ is continuous on $<a, b>$ . - FALSE Function $f(x) = \ln{|x|}$ is discontinuous at $0$ . - TRUE The continuity is necessary for differentiability. - TRUE Func...

One of our user asked: In my various courses, for instance, linear algebra and vector calculus, I am somewhat confused with what precisely $\mathbb{R}^n$ is. From the definition of the Cartesian product, I would conceptualise $\mathbb{R}^n$ as the metric space with some distance operator, where all the points are just $n$ -tuples. This is surely a distinct notion from vectors as isn't the point $A = (1,2,3)$ , for instance, different from the vector $\vec{a} =\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}$ ? But if we were to consider the points in $\mathbb{R}^n$ as vectors then clearly it is a vector space. However I don't know whether these two conceptions of $\mathbb{R}^n$ are actually equivalent. Surely the vectors do not correspond to a specific point in space, unlike the points in $\mathbb{R}^n$ . Forgive me if this is a silly question, or if my question seems garbled. Also please help me with tags if they are inappropriate.

One of our user asked: FInd linear map $A: \Bbb{R^3} \rightarrow \Bbb{R^3}$ for given kernel and image. $$Ker(A)=L(\begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 1 \\ \end{pmatrix})\space ; \space Im(A)=L(\begin{pmatrix} 1 \\ 0 \\ 1 \\ \end{pmatrix}) \\$$ I've been reading some explonations about this kind of a problem but I didn't understand anything about expanding kernel base to the dimmension of $\Bbb{R^3}$ . But, according to this solution example , if I form matrix $A$ like $$\begin{bmatrix} 1 & a&b \\ 0 &c&d\\ 1 &e&f \\ \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix}= \begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix} \\$$ and $$ \begin{bmatrix} 1 & a&b \\ 0 &c&d\\ 1 &e&f \\ \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix}= \begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix} \\ $$ I don't get anything here for first equation, and for second I get $$\begin{bmatrix} 1+a+...

One of our user asked: Here are few of the questions from the previous years' exams. I've chosen the ones I'm not sure about. It's a simple TRUE/FALSE task. Would anyone be able to verify my solution? Some of my answers are good, some are just random guess according to my intuition. I don't really need a detailed explanation... Thanks! Domain of $f'$ is contained within domain of $f$ . - TRUE Boundary point of set A is also a cluster point of that set. - TRUE Every increasing sequence and bounded above is convergent. - TRUE Every increasing sequence and bounded below is convergent. - FALSE Every increasing sequence is always bounded below. - TRUE Every sequence is discontinuous function. - FALSE Every sequence is continuous function. - TRUE Every function integrable on $<a, b>$ is continuous on $<a, b>$ . - FALSE Function $f(x) = \ln{|x|}$ is discontinuous at $0$ . - TRUE The continuity is necessary for differentiability. - TRUE Func...

One of our user asked: One of my professor's lecture notes on Vector Spaces start by the following lines:- We have seen that if $det(A)$ = 0, then system $AX=O$ has infinite number of solutions. We shall now see that in this case, the set of solutions has a structure called vector space. My doubt is in what sense do the set of an infinite number of the solution of equation $AX = O$ (given |A|=0) is actually a structure of Vector Space ? How does the term Vector Space come into picture?

One of our user asked: Consider an irreducible smooth representation $\pi$ of the group $G=GL_n(\mathbb{Q}_p)$ with center $Z$ . Does there exist a unitary central character for $\pi$ ? More precisely, is there a (quasi-)character $\omega: G \to \mathbb{C}^{\times}$ such that $\pi \otimes \omega$ when restricted to the center $Z$ is a unitary character for $Z$ ? I find this result casually stated in many references, where they say it follows from Schur's lemma. But I am unable to see it directly from Schur's lemma.

One of our user asked: Can anyone please help me with the computation of following series: $$\sum_{n=1}^\infty \sum_{m=1}^\infty \frac{(m+n-1)!}{m!(n-1)!n!(m-1)!}a^m b^n.$$ My thoughts: Since $$\displaystyle \frac{(m+n-1)!}{m!(n-1)!n!(m-1)!} = \frac{\binom{m+n-1}{m}\binom{m+n-1}{m-1}}{(m+n-1)!},$$ by some arrangement this may be the probability of a hypergeometric distribution. Any help would be appreciated, thanks!

One of our user asked: i know how to find $g(x)$ if $f(x)$ and $f \omicron g (x)$ is known let say $$f \circ g (x) = \frac{x}{x-2}$$ and $$g(x) = x- 2$$ find $f(x)$

One of our user asked: Let $X_{i}\,(i=1,2,3,4,5,6)$ be i.i.d random variables. What will be the distribution of $\min(X_1+X_2+X_3,X_2+X_3+X_4,X_3+X_4+X_5,X_4+X_5+X_6)$ ? I just need to know the approach, so any help will be appreciated.

One of our user asked: The objective is to find all solutions to $$ x^3 - y^3 = 3(x^2 - y^2) $$ where $x,y \in \mathbb{Z}$ . So far I've got one pair of solution. Try $(x, y)=(0,0)$ : $$ 0^3-0^3=3(0^2-0^2) \\ 0=0 \qquad \text{equation satisfied}$$ Another try $ (x, y) = (x, x)$ then $$ x^3 - x^3 = 3(x^2 -x^2) \\ 0 = 0 \qquad \text{equation satisfied}$$ But the above are just particular solutions. To find all $(x, y)$ pairs, I tried: $$\begin{aligned} (x-y)(x^2+xy+y^2) &= 3(x+y)(x-y) \\ x^2+xy+y^2 &= 3(x+y) \end{aligned}\\ \begin{aligned} x^2+xy+y^2-3x-3y &= 0 \\ x^2+x(y-3)+y(y-3) &=0 \\ x^2 + (y-3)(x+y) &= 0 \end{aligned}$$ What now? Am I on the right track?

One of our user asked: I am having difficulty understanding why it is false. As I understand, the range is the output of everything from the domain into the transformation function, or the result of the domain being outputted to the co-domain via the transformation. Essentially, I cannot think of a counterexample where the range of a linear transformation is not in the domain. I would appreciate if anyone could shed some light on this basic concept that I am struggling with.

One of our user asked: So I'm given an assignment in probability and statistics that states the following: The download time for one document through the internet on a lab computer has an average of 25 seconds and that time has an exponential distribution. Find the expected time to download 3 documents. So judging by the assignment if the download time for a single document has an exponential distribution then: $\frac{1}{λ} = 25 => λ = \frac{1}{25}$ Now since we are looking for 3 documents the λ for 3 documents is equaled to: $λ = \frac{3}{25}$ Hence the expected value should be equaled to: $\frac{25}{3}$ seconds However here is my confusion: Since $\frac{25}{3} < 25$ and we're talking about the expected time to download 3 files then souldn't the expected value be grater than 25 ?

One of our user asked: Let $d\geq 3$ . Consider the $(2d-2) \times (2d-2)$ matrix with $a_{ii} = 1$ , $a_{ij} = -\frac{1}{3}$ if $|i-j|=1$ and $a_{ij} = \frac{1}{3}$ otherwise. Prove that its smallest eigenvalue is $0$ and its multiplicity is $d-2$ . It might be helpful to decompose the matrix as $AA^T$ (as far as I know, this has to be true) but I don't know how to do this effectively either - not much experience in Cholesky factorization. Any help appreciated!

One of our user asked: Let say $x \in M$ and $x \neq 0$ then is it true that for finitely generated projective module $M$ there is $g \in M^*$ such that $gx \neq 0$ . If yes how to prove it. For vector space dual this result is true what about projective module. Also if $f \in M^*$ $f \neq 0$ then $fy \neq 0$ is also true or not ?

One of our user asked: Let $A\in M_n(\mathbb C) $ . Denote its rows by $L_1,L_2,...,L_n$ . We know that $L_1,L_2,...,L_n$ are linearly independent. Construct the matrices $B\in M_n(\mathbb C)$ with the rows $O,L_2,...,L_n$ ( $O$ denotes a row of zeroes) and $C$ with the rows $L_2,...,L_n,O$ . Let $D=A^{-1}B$ and $E=A^{-1}C$ . a)Prove that $rank(D) =rank(D ^2)=...=rank(D^n)$ . b) Prove that $rank(E) >rank(E^2)>...>rank(E^n) $ . My approach : $rank(A) =n$ since all its rows are linearly independent. Also $rank D=rank B=n-1$ and $rank E=rank C=n-1$ by the same reasoning and using the fact that if a matrix is multiplied by an invertible matrix its rank doesn't change. I don't know how to do the same thing for $D^2,...,D^n$ and $E^2,...,E^n$

One of our user asked: I don't understand why the last step in the following equations is true. Could someone explain this to me please? Don't think context is important here, but just in case it's from a proof of a bound on the error of the explicit Euler method.

One of our user asked: Suppose $y'' + f(x)y = 0$ where $M \geq f(x) \geq m > 0$ on some interval $[a,b]$ , then the number zeros $N$ of a non trivial solution is $\lfloor\frac{(b-a)\sqrt{m}}{\pi}\rfloor \leq N \leq \lceil\frac{(b-a)\sqrt{M}}{\pi}\rceil$ Simple. Now suppose I have an equation $y''+4y'+\frac{8x+\sin(x)}{x+1}y = 0$ and I want to estimate the number of zeros of a non trivial solution. I can't use the theorem as is, because the ODE is not in the correct form, to fix this, we can multiply by $e^{2x}$ and get $y''e^{2x}+4y'e^{2x}+\frac{8x+\sin(x)}{x+1}ye^{2x} = 0$ Now if we let $ye^{2x} = z$ we have an ODE $z'' + (\frac{8x+\sin(x)}{x+1}-4)z = 0$ which is in the correct form How did the professor know to multiply by $e^{2x}$ ? Is there a method to this or was this just a lucky guess

One of our user asked: I have been all over the internet, but I just can't make sense of this stuff. I have done my best to learn from my textbook and different websites, but this is confusing for me. I haven't taken any calculus in years, and I'm jumping in headfirst. If anyone can help me understand how to prove these using Cartesian Tensor Notation, I would really appreciate it! First identity: ∇ x ( ∇ x a ) = ∇(∇ . a ) - ( ∇^2 ) a Second identity: ∇ . ( a b ) = a . ∇ b + b ( ∇ . a ) Third identity: ∇ . ( f δ ) = ∇ f Fourth identity: δ : ∇ a = ∇ . a Thanks everyone

One of our user asked: this seems like it should be simple, but i've run out of leads with a similar-triangles approach, and the algebraic approaches seem pretty daunting, so i'm asking for help. i'm working on a simulation where a ray of light reaches a point by bouncing once off a planar mirror. i know the position of the source, the destination point, and a point & normal of the mirror. (or any other convenient means of describing the mirror) For simplicity i'm happy to assume the mirror is infinite in extent. as i mentioned, a geometric solution via similar-triangles and so on has not revealed itself to me, but i'm good at missing those. for the algebraic approach i've set up a system of equations constraining the on-mirror point to 1) lie on the line of the mirror and 2) the angle between the incoming ray and normal and the angle between the outgoing ray and the normal are equal, by asserting that those Dot-products are equal. I haven't carried ...

One of our user asked: I am attempting to prove that $$J=\int_0^\infty\frac{\log^2(x)}{x^2+1}\mathrm dx=\frac{\pi^3}8$$ With real methods because I do not know complex analysis. I have started with the substitution $x=\tan u$ : $$J=\int_0^{\pi/2}\log^2(\tan x)\mathrm dx$$ $$J=\int_0^{\pi/2}\log^2(\cos x)\mathrm dx-2\int_{0}^{\pi/2}\log(\cos x)\log(\sin x)\mathrm dx+\int_0^{\pi/2}\log^2(\sin x)\mathrm dx$$ But frankly, this is basically worse. Could I have some help? Thanks.

One of our user asked: It's easy to check that in orthogonal matrix dimension $2 \times 2$ if there is entry $0$ in the matrix necessary one additional zero must be present. Then the total number of zeros is $2$ . In an orthogonal matrix dim. $3 \times 3$ number of zeros can be (if they are present) , I suppose from observations, only $4$ or $6$ - once again we obtain an even number of possible zeros. Can this observation be extended for other orthogonal matrices of greater dimensions? The number of zeros is always even? How to prove this? Maybe, it is known the explicit formula for the number of possible zeros in orthogonal matrices of any dimension?

One of our user asked: I know that the Fisher matrix is easily obtained from the Hessian matrix $I\left(\hat{\beta}\right)=-H\left(\hat{\beta}\right)$ Why is the covariance variance matrix the inverse of the Fisher information matrix?

One of our user asked: $$M=\{f\in C[0,2\pi],\int_{0}^{2\pi}f(x)sinxdx=\pi,\int_{0}^{2\pi}f(x)sin2xdx=2\pi\} $$ $a,b\in \mathbb R, g\in M, g(x)=asinx+bsin2x,x\in [0,2\pi]$ I've read on the answers that $a=1,b=2$ and I don't know how to calculate them. Can somebody explain me,please? By the way, the problem is to determine $ \int_{0}^{2\pi}(g(x))^2 dx$ so you have to first get the constants $a$ and $b$ .

One of our user asked: Let, the point $K$ is the center of inscribed circle of a triangle. The circle touching the edges of the triangle is mentioned. $\angle ABC=90°$ $AE=4$ and $CF=12$ Find the radius of inscribed circle of the triangle: A)3 B)4 C)5 D)6 E)7 My attempts: The radius of inscribed circle, we have, $r=\frac{a+b-c}{2}$ , where, $c=\sqrt{a^2+b^2}$ . Then, I need $AB$ and $BC$ . Or, I must know what are $BE$ and $BF$ . I 'm stuck.

One of our user asked: A real matrix $B$ is called non-negative if every entry is non-negative. We will denote this by $B\ge 0$ . I want to find a non-negative matrix $B$ satisfying the following two conditions: (1) $(I-B)^{-1}$ exists but not non-negative. Here $I$ is the identity matrix. (2) There is a non-zero and non-negative vector $\vec{d}$ such that $(I-B)^{-1}\vec{d}\ge 0$ . I tried all the $2\times 2$ matrices, but it did not work. I conjecture that such a $B$ does not exist, but don't know how to prove it.

One of our user asked: I found a theorem mentioned in a couple of places, but could not find a proof. The theorem states the following: Let $A, B \in \mathbb{F^{m,n}}$ , $p=min(m,n)$ with singular values $\sigma_1(A) \geqslant...\geqslant \sigma_p(A)$ and $\sigma_i(B) \geqslant...\geqslant \sigma_p(B)$ respectively, then $\sigma_{i+j-1}(A+B) \leqslant \sigma_i(A) + \sigma_j(B)$ . I am looking for a proof of the above. Thanks in advance.

One of our user asked: Why is Euler's number $\mathtt 2.71828$ and not for example $\mathtt 3.7589$ ??? I know that e is the base of natural logarithms, I know about areas on hyperbola xy=1 and I know it's formula : $$e =\sum_{n=0}^\infty \frac{1}{n!}$$ And I also know it has many other characterizations. But, why is e equal to that formula (which sum is approximately $\mathtt 2.71828$ ) ??? I googled that many times and every time it ends in having "e is the base of natural logarithms", I don't want to work out any equations using e without understanding it perfectly. Thanks in advance.

One of our user asked: I want to calculate and evaluate the following integral: $$\frac{B}{2 D}\int_{0}^{\infty} xe^{\frac{-Bx}{D}} erfc(\frac{x+x_{0}-Bt}{2\sqrt{Dt}})$$ My idea was to integrate by parts by setting: $$u= erfc(\frac{x+x_{0}-Bt}{2\sqrt{Dt}}), du=-\frac{1}{\sqrt{Dt}} e^{-(\frac{x+x_{0}-Bt}{2\sqrt{Dt}})^2}$$ $$dv=xe^{-\frac{Bx}{D}},v=e^{-\frac{Bx}{D}}(\frac{Bx}{D}x+1)*\frac{D^2}{B^2}$$ I have calculated further and got some results, but I am not sure if my thinking is right, or even if there is some easier or more efficient way to do this analytically or numerically,(for example by expanding the error function as infinite series). Any tips would be appreciated. And of course, erfc is the complementary error function with: $$erfc(\frac{x+x_{0}-Bt}{2\sqrt{Dt}})=\frac{2}{\sqrt{\pi}}\int_{\frac{x+x_{0}-Bt}{2\sqrt{Dt}}}^{\infty} e^{-z^2} dz $$

One of our user asked: . Find the first 4 terms of the Taylor series expansion of the error function with respect to x=0. The error function is defined as: enter image description here `

One of our user asked: Define $B = \{\frac{1}{\sqrt{2}}I, \frac{1}{\sqrt{2}}X, \frac{1}{\sqrt{2}}Y, \frac{1}{\sqrt{2}}Z \}$ , where: $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}$ , $X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \\\end{bmatrix}$ , $Y = \begin{bmatrix} 0 & i \\ -i & 0 \\ \end{bmatrix}$ , and $Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}$ I am trying to show that B (modified from the Pauli matrices) span the set of $2 \times 2$ matrices with entries in $\mathbb{C}$ (denote $\mathcal{M}_2(\mathbb{C})$ ). If $A \in \mathcal{M}_2(\mathbb{C})$ , then we can write: $A = \frac{1}{\sqrt{2}} \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$ for some $a, b, c, d \in \mathbb{C}$ . I think that I want to show that $A = \frac{1}{\sqrt{2}}(\lambda_0I + \lambda_1X + \lambda_2Y + \lambda_3Z)$ ? This means I would need to show that the system given by: $\{a = \lambda_0 + \lambda_3, \ b = \lambda_1 + \lambda_2i, \ c = \lambda_1 - \lam...

One of our user asked: In class we saw that every linear transformation is differentiable (since there's always a linear approximation for them) and we also saw that a differentiable function must be continuous, so it must be true that all linear operators are continuous, however, I just read that between infinite dimensional vector spaces this is not necessarily true. I would like to know where's the flaw in my reasoning (I suspect that linear transformations between infinite dimensional vector spaces are not always differentiable).

One of our user asked: I include a bit of an introduction, even though my main question is more mathematical. I was tasked with finding the Maximum Likelihood Estimate for $\theta$ in $$\mathrm P(X>x) = \left(\frac ax \right)^\theta $$ where $X$ is a variable, and $x$ represents a value that variable can take on. The Probability Density Function is $\newcommand\diff[2]{\frac{\mathrm d#1}{\mathrm d#2}}\diff Fx=\frac{-\theta a^\theta}{x^{\theta + 1}}$ , where $F = \mathrm P(X>x)$ . I maximise the loglikelihood function $l = \ln(-\theta) + \theta \ln a - (\theta + 1)\ln x\ $ to get $\hat\theta(x_i) = \frac 1{\ln x_i - \ln a}$ , where the $\hat.$ indicates that $\hat\theta$ is an estimate of $\theta$ , based on the data sample. Now, the answer is supposed to be $$\hat\theta = \frac 1{\overline {\ln x} - \ln a}$$ where $\overline {\phantom{x}}$ indicates the average: $\overline{\ln x} = \frac 1n \sum_i \ln x_i$ . I am stumped as to how to get this answer directly from $\hat\...

One of our user asked: I noticed during calculating variance : $$S^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\overline{X})^2$$ that the sum $\sum_{i=1}^n(X_i-\overline{X})^2$ is equal to $\sum_{i=1}^n(X_i-\overline{X})*X_i$ . However I wasn't able to prove why this is true, or if there are cases in which it won't be equal.

One of our user asked: A 3,4,5 triangle resting on its side of length 3 is tilted so that it makes a 60 degree angle with the floor and rests on the wall. What are the dimensions of the projection of the triangle onto the wall. The height is $2\sqrt{3}$ I can't seem to figure out how to find the length of the base. The answer given is $\frac{3}{4}2\sqrt{3}$ . I'd appreciate any explanations in deriving this. Thank you.

One of our user asked: Is there an explicit example of an additive map $\mathbb{R}^n \rightarrow \mathbb{R}^m$ which is not linear? (I have mostly thought about the question when $m = n = 1$ , and I don't think the general case is any easier.) I know that something like $f: \mathbb{C} \rightarrow \mathbb{C}$ which sends $f: z \mapsto \text{Real}(z)$ would be additive but not $\mathbb{C}$ -linear. I also know that since $\mathbb{R}$ is a $\mathbb{Q}$ -vector space, I can find some example where $1 \mapsto 1$ and $\sqrt{2} \mapsto 0$ . Is there an explicit example? By explicit, I mean, given an element in the domain, there would be some procedure to decide where it maps. Thank you!

One of our user asked: Consider the system of equations $$x_1 + x_2 + x_5 + x_6 = 26 \\x_2 + x_3 + x_7 + x_8 = 26 \\x_3 + x_1 + x_9 + x_4 = 26 \\x_4 + x_5 + x_{10} + x_{12} = 26 \\x_6 + x_7 + x_{10} + x_{11} = 26 \\x_8 + x_9 + x_{11} + x_{12} = 26$$ We need to find a solution such that $x_i = \sigma(i)$ for some permutation $\sigma$ of the set $\{k : k \le 12, k \in \Bbb N \}$ . The need to solve this system arose for an exercise from Pearls in Graph Theory by Hartfield and Ringel . I know that using a system like this may not be an efficient way for the graph problem, but for now I would like some insight on the equations by themselves without explicitly relying on the graph problem. I only know that such solution(s) exist because the question asked for one and I would not be able to justify its existence without finding a solution. Here's what I have tried so far: Some things can be obtained from the symmetry (I'm not sure if this is the right word) like $$x_1 + x...

One of our user asked: I'm doing a split-test and need to calculate the confidence-level of the result. I need to implement the calculations on my own, not using any online-tools or excel. I managed to do this for binomial values like the conversion rate, but couldn't get any further with non-binomial values like the revenue. I have a lot of "old" data which I can use to calculate mean, variance, standard deviation and standard error specific for my data. I found some answers here, but (to be honsest) wasn't able to understand how to translate them into code. So any additional help is welcome.

One of our user asked: Consider the system of equations $$x_1 + x_2 + x_5 + x_6 = 26 \\x_2 + x_3 + x_7 + x_8 = 26 \\x_3 + x_1 + x_9 + x_4 = 26 \\x_4 + x_5 + x_{10} + x_{12} = 26 \\x_6 + x_7 + x_{10} + x_{11} = 26 \\x_8 + x_9 + x_{11} + x_{12} = 26$$ We need to find a solution such that $x_i = \sigma(i)$ for some permutation $\sigma$ of the set $\{k : k \le 12, k \in \Bbb N \}$ . The need to solve this system arose for an exercise from Pearls in Graph Theory by Hartfield and Ringel . I know that using a system like this may not be an efficient way for the graph problem, but for now I would like some insight on the equations by themselves without explicitly relying on the graph problem. I only know that such solution(s) exist because the question asked for one and I would not be able to justify its existence without finding a solution. Here's what I have tried so far: Some things can be obtained from the symmetry (I'm not sure if this is the right word) like $$x_1 + x...

One of our user asked: Prove that ${1\over x_1}+{1\over x_2}+\dots+{1\over x_n}\lt3$ if no $x_j=10^kx_i+n$ where $k,n\in\mathbb{Z^+}$ I have attempted this question multiple times and have barely reached anything. I tried to assume WLOG that $x_1\le x_2\le\dots\le x_n$ however I could not continue. I am still new to such inequality questions so any help would be appreciated. Thank you anyways.

One of our user asked: i have a problem with a quite complicated question regarding guests who visit a hotel and hopefuly getting a correct key for their room: problem: 2n people, n pairs, arrive to an hotel. each pair is assured to obtain a shared room at the hotel. however, the receptionist is a bit confused and hands out, randomly, 2n keys. for each of the n rooms there are 2 identical keys(out of 2n) that open its door. let X - number of guests who obtained a correct key for their room. Y - number of guests that can enter to that shared room, meaning number of pairs in which at least one of the people got the correct key to the room. calculate mean and variance of X and the mean of Y. what i tried to do: i declared an indicator $x_i$ that is 1 if the i guest got the correct key, and 0 if he doesn't, for each of i=1,...,2n keys available. next, i calculated the mean of $E[X_i]=P(x_i=1)=\frac{\binom21}{\binom{2n}1}=\frac{1}{n}$ . the variance is $var(X_i)=P(x_i=1)P(X_i=0)$...

One of our user asked: The series of functions $$ \sum_{n=0}^{+\infty} \frac{(-x)^n}{1+n!} $$ is pointwise convergent for any $x\in \mathbb{R}$ , thus it defines a function $f:\mathbb{R}\rightarrow \mathbb{R}$ . Is there a way to evaluate the limit $\lim_{x\rightarrow +\infty} f(x)$ ?

One of our user asked: I'm looking for an easy way to factor singular conics into linear forms in order the following exercise. Which of the following quadratic forms define a singular > conic? Write those as a product of two linear forms. (a) $x_0^2-2x_0x_1+4x_0x_2-8x_1^2+2x_1x_2+3x_2^2$ (b) $x_0^2-2x_0x_1+x_1^2-2x_0x_2$ (c) $3x_0^2-2x_0x_1$ The matrices of (b) and (c) have full rank and so the quadratic forms are non-degenerate. For (a), we have the matrix $$ M= \begin{bmatrix} 1 & -1 & 2 \\ -1 & -8 & 1 \\ 2 & 1 & 3 \end{bmatrix} $$ where $det(M)=0$ . One way to solve the exercise would be to orthogonally diagonalize M to get rid of the mixed terms. However this involves computing the eigenvalues of M and finding the corresponding eigenvectors. So my question is: Is there a quicker way to do this or do I have to go through the calculations?

One of our user asked: $1/n < x < 1/(n+1) \leftrightarrow n+1 < 1/x < n $ ? could someone explain me why this is true ?

One of our user asked: I am wondering why finding a function that is harmonic on the sphere and that respect some conditions on the frontiere of the sphere is important ? This is called the Dirichlet problem, and I don't understand why we are interesting in this problem ? Why are we want the function to be harmonic ? I know what it means for a function to be harmonic, but I don't understand what is the "advantage" of being harmonic. Moreover does the solution of the Dircihlet helps solving real-life/physic problems ? Thank you !

One of our user asked: The following that $$\int_0^1 f(x) \,\mathrm dx=\lim_{n\rightarrow\infty}\sum_{k=1}^{n-1}f(\frac{k}{n})\frac{1}{n} \, $$ is well-known fact! But if $$\lim_{n\rightarrow\infty}\frac{r_n}{n}=\alpha,$$ then is it true that $$\int_0^{\alpha} f(x) \,\mathrm dx=\lim_{n\rightarrow\infty}\sum_{k=1}^{r_n-1}f(\frac{k}{n})\frac{1}{n} \, $$ ? If this is true, why? Help me with big mercy!!

One of our user asked: Let $ x_1, ..., x_9 $ and $ y_1, ..., y_8 $ be two random samples of two populations. $ \bar x = 7 $ is the mean of the first and $ \bar y = 11 $ the mean of the second sample. The sample standard deviations are $ s_x = 2 $ and $ s_y = 3.5 $ . Now I want to calculate a 95% confidence interval for the difference of the mean of the two populations. I know how to calculate a 95% confidence interval for the mean for the populations: $$ \bar x -2*\left(\frac{s_x}{\sqrt 9}\right) $$ and $$ \bar y -2*\left(\frac{s_y}{\sqrt 8}\right) $$ But don't know how to proceed from here.

One of our user asked: Find a step function s such that $$\int_{0}^{2} s(x) dx=5 \quad \int_{0}^{5} s(x) dx=2$$ The given answer is $$s(x)=\dfrac{5}{2} \quad \text{if} \quad 0 \leq x < 2$$ $$s(x)=-1 \quad \text{if} \quad -2 \leq x \leq 5$$ I don't understand how does one arrive to this solution.. even graphically trying to understand it I didn't come to a solution. Can someone please help me figure out how?

One of our user asked: I have a vector function $f: \mathbb{R}^n \to \mathbb{R}^n$ defined with components $$ f_i(a) = \sum_{j=1}^n \sin(a_i - a_j) $$ which I want to integrate from ${\bf{\alpha}}^0$ to ${\bf{\alpha}}^1$ where ${\bf{\alpha}}^k = [\alpha^k_1, \ldots, \alpha^k_n]$ for $k \in \{1,2\}$ . So the problem looks like $$ \int_^0}^^1} f(a)^{\top} {\rm d}\, a. $$ I thought that I could integrate as below $$ \int_{\alpha^0}^{\alpha^1} \sum_{i=1}^n \left\{ \sum_{j=1}^n \sin(a_i - a_j)\right\} {\rm d}a_i $$ by expanding the inner product in the integrand. I think that I can then write the integral as $$ \int_{(\alpha_1^0, \ldots, \alpha_n^0)}^{(\alpha_1^1, \ldots, \alpha_n^1)} \sum_{i=1}^n \left\{ \sum_{j=1}^n \sin(a_i - a_j)\right\} {\rm d}a_i = \sum_{i=1}^n \int_{\hat{\alpha}_i^0}^{\hat{\alpha}_i^1} \left\{ \sum_{j=1}^n \sin(a_i - a_j)\right\} {\rm d}a_i $$ where $\hat{\alpha}_i^0$ treats every component of $a$ as fixed $\alpha_j^0$ for $j \neq i$ , which I think would...

One of our user asked: So I understand the intuition of taking the determinant of a 2x2 matrix, but what is the intuition for taking the determinant of 3x3, matrix? It makes zero intuitive sense just looking at it. Also, when finding the inverse of a matrix, why do we need to find the cofactor matrix and the adjugate matrix and transpose the cofactor matrix, and what is the point of the checkerboard matrix with + and - signs?

One of our user asked: I am reading "Calculus" by Takeshi Saito. In this book, Saito adopts the following axiom of $\mathbb{R}$ . I like this axiom. I think it is easy to understand what this axiom is saying. But I cannot find a book in which this axiom of $\mathbb{R}$ is adopted. Are there any other books which adopt this axiom of $\mathbb{R}$ ? Axiom 1.1.1: 1. If $a$ is a real number, then there exists an integer $n$ such that $n \leq a \leq n+1$ . 2. If $\{a_n\}$ is a sequence such that $a_i \in \{0, 1\}$ for all $i \in \{1, 2, \cdots\}$ , then there exists a real number $b$ such that $$\sum_{n=1}^m \frac{a_n}{2^n} \leq b \leq \sum_{n=1}^m \frac{a_n}{2^n}+\frac{1}{2^m}$$ for all $m \in \{0, 1, 2, \cdots\}$ . By Axiom 1.1.1.1, if $a$ is a real number, there exists a unique integer such that $m \leq a < m+1$ and we define this $m$ as $[a]$ . $[a]+1$ is the smallest integer which is greater than $a$ . Proposition 1.1.2: Let $a, b$ be real numbers...

One of our user asked: The following set : $$\{(x,y) \in \mathbb{R}^2 \mid x+y \leq 1, x\geq 0, y \geq 0 \}$$ is a triangle. One way to see it is simply that we draw all points under the line of equation $y = 1-x$ with positive coordinates. My question is : Is it possible with inequalities (just as the one that describe a triangle) to draw some other nice shapes like parallelogram or more generally regular polygons ? Moreover I suspect that there is some linear algebra behind these inequalities. So maybe for example linear algebra can help proving that the above inequality makes a triangle. Thank you !

One of our user asked: This follow my previous post here , where Song has proven that $\forall b>1,\lim\limits_{x\to 1^{-}}\frac{1}{\ln(1-x)}\sum\limits_{n=0}^{\infty}x^{b^n}=-\frac{1}{\ln(b)}$ , that is to say : $$\forall b>1,\sum\limits_{n=0}^{\infty}x^{b^n}=-\log_b(1-x)+o_{x\to1^-}\left(\log_b(1-x)\right)$$ (The $o_{x\to1^-}\left(\log_b(1-x)\right)$ representing a function that is asymptotically smaller than $\log_b(1-x)$ when $x\to1^{-}$ , that is to say whose quotient by $\log_b(1-x)$ converges to $0$ as $x\to1^{-}$ , see small o notation ) So we have here a first asymptotical approximation of $\sum\limits_{n=0}^{\infty}x^{b^n}$ . I now want to take it one step further and refine the asymptotical behaviour, by proving a stronger result which I conjecture to be true (backed by numerical simulations) : $$\sum\limits_{n=0}^{\infty}x^{b^n}=-\log_b(1-x)+O_{x\to1^-}\left(1\right)$$ (The $O_{x\to1^-}\left(1\right)$ representing a function that is asymptotically bounded wh...

One of our user asked: The following set : $$\{(x,y) \in \mathbb{R}^2 \mid x+y \leq 1, x\geq 0, y \geq 0 \}$$ is a triangle. One way to see it is simply that we draw all points under the line of equation $y = 1-x$ with positive coordinates. My question is : Is it possible with inequalities (just as the one that describe a triangle) to draw some other nice shapes like parallelogram or more generally regular polygons ? Moreover I suspect that there is some linear algebra behind these inequalities. So maybe for example linear algebra can help proving that the above inequality makes a triangle. Thank you !

One of our user asked: Which complex $k$ -dimensional subspaces of $\mathbb C^n$ are spanned by real vectors? Can we characterise them? (here $1<k<n$ ). By "complex", I mean that I am interested in subspaces $W \le \mathbb C^n$ , which admit $k$ vectors $v_1,\ldots,v_k \in \mathbb{R}^n$ , such that $W=\text{span}_{\mathbb C}(v_1,\ldots,v_k)$ In the case $n=2,k=1$ , we ask when $(z_1,z_2)$ can be expressed as $z_0\cdot(x_1,x_2)$ for some $z_0 \in \mathbb C$ and $x_1,x_2 \in \mathbb R$ . This is equivalent to $z_1$ being a real multiple of $z_2$ or vice versa.

One of our user asked: I'm studying for my calculus 1 exam and came across this sample question from the professor's collection: Calculate: $\lim\limits_{n\ \rightarrow\ \infty} \frac{1}{2\log(2)}+\frac{1}{3\log(3)} + \dots + \frac{1}{n\log n}$ (hint: separate into blocks) Unfortunately the sample questions don't include answers and I'm at a loss as to how to proceed; I'd really appreciate some help. Thanks!

One of our user asked: Rectangle - 15CM , 10Cm Circle Diameter - 0.7 MM So my question is To Fill Rectangle, How many Circles needed. Please let me know techniques to solve. Thank in advance

One of our user asked: I am following a proof in my vector calculus book but I am getting stuck. Let T(s) be the the unit tangent vector at s. and let k(s) = norm( T’(s)) here T’(s) is orthogonal to T(s) and let N(s) be the Unit vector such that T’(s)=k(s)*N(s) See photo Now, in the proof it says: Differentiate N(s) • T(s) =0 Gives N’(s)•T(s)+N(s)•T’(s) =0 Hence N’(s)•T(s)=-k(s) But I don’t see how this Is derived? Also , written on mobile as this is the only access I have at the moment, please go easy on the syntax.

One of our user asked: Assume that F is an infinite field, k is an infinite cardinal, and V = F ^ k is a vector space. Can it be prove proved that | V |= dim V ? My thought was that we know that| V |= max {dim V ,| F |}, so all I need is to prove is that if | V |=| F |, then also | F |=dim V . Now, from Konig theorem, if | V |=| F |, then k < cf(| F |), so I tried to prove that k < cf(| F |) is imposible, but i didn't know how to continue this line of thougt. Has sombodey know how to prove it?

One of our user asked: i'm trying to do a question for which I was given the following line equations: $\underline r = \underline a + \lambda \underline u$ $\underline r' = \underline a' + \lambda' \underline u'$ They then gave me this relationship without any justification, i've been trying to get my head around it but have not had much luck. $\lvert \underline r-\underline r'\rvert^2\lvert \underline u \times \underline u'\rvert^2=\lvert (\underline a - \underline a') \cdot (\underline u \times \underline u')\lvert^2+\lvert (\underline r - \underline r') \times (\underline u \times \underline u')\lvert^2$ I know that $\lvert (\underline r - \underline r') \times (\underline u \times \underline u')\lvert = \lvert \underline r-\underline r'\rvert\lvert \underline u \times \underline u'\rvert\sin \theta $ but cant get any further.

One of our user asked: This question is inspired by the Escalation Battles in Pokémon Shuffle. There's a couple of other Pokémon-related questions on here, but they don't address this specific problem. The way an Escalation Battle works is, the Nth time you beat it, you have an N% chance of catching the Pokemon. If you've already caught the Pokémon, you get items instead. When N=100, you're guaranteed to catch the Pokémon, but the chance of having not caught it by then must be vanishingly small. I've competed in a few Escalation Battles, and I always seem to catch the Pokémon when 15 ≤ N ≤ 25. It's been years since I studied statistical probability at school, but this doesn't seem very intuitive to me. So I started wondering about the cumulative probability - how likely you are to have caught the Pokémon after N levels. Is there a general formula to calculate the cumulative probability of having caught the Pokémon after N attempts? How many attempts wil...

One of our user asked: I have the manifold with boundary $M:= \lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1\geq 0, x_1^2+x_2^2+x_3^2=1\rbrace \cup\lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1= 0, x_1^2+x_2^2+x_3^2\leq1\rbrace$ and I need to find the boundary of this manifold. I think it is $\lbrace (x_1,x_2,x_3) \in \mathbb R^n : x_1= 0, x_2^2+x_3^2=1\rbrace$ , the other option is that the boundary is the empty set? I think the first is right? Am I wrong?

One of our user asked: i'm trying to do a question for which I was given the following line equations: $\underline r = \underline a + \lambda \underline u$ $\underline r' = \underline a' + \lambda' \underline u'$ They then gave me this relationship without any justification, i've been trying to get my head around it but have not had much luck. $\lvert \underline r-\underline r'\rvert^2\lvert \underline u \times \underline u'\rvert^2=\lvert (\underline a - \underline a') \cdot (\underline u \times \underline u')\lvert^2+\lvert (\underline r - \underline r') \times (\underline u \times \underline u')\lvert^2$ I know that $\lvert (\underline r - \underline r') \times (\underline u \times \underline u')\lvert = \lvert \underline r-\underline r'\rvert\lvert \underline u \times \underline u'\rvert\sin \theta $ but cant get any further.

One of our user asked: Consider plane $\beta: -x+y=1$ and the point $A=(0;0;2)$ . I'm asked to write a vectorial equation for the line perpendicular to plane $\beta$ which also goes through point $A$ . I start by identifying a normal vector to $\beta$ , which is $\vec{r}=(-1;1;0)$ . This vector must be the "director vector" (not sure of the correct term in english) to the line I want. Hence the equation for the line is $\left(x;\:y;\:z\right)=\left(0;0;2\right)+k\left(-1;1;0\right),\:k\:\in \mathbb{R}$ . However, when I plot it, with various values for $k$ , I get this: My questions are: is this right? And if so - where exactly is the perpendicular line?

One of our user asked: Let there exists a pure composite system of 2 subsystems; namely, "1" and "2". Suppose $\hat{\rho} = \frac{|00\rangle \langle 00|+|00\rangle\langle 11| + |11\rangle\langle 00| + |11\rangle \langle 11|}{2}$ is a pure state operator for the this pure composite system. The reduce state operator for subsystem "1" is $\hat{\rho}^{1} = Tr_{2}\left(\hat{\rho}\right)$ = $\frac{1}{2} Tr_{2} \left(|0\rangle\langle 0|\otimes|0\rangle \langle 0| + |1\rangle\langle0|+|0\rangle\langle1| \otimes|0\rangle\langle 1| \otimes |0\rangle\langle1|+|1\rangle\langle1|\otimes|1\rangle\langle1| \right)$ = $\frac{1}{2}\left(|0\rangle \langle0|\cdot Tr\left(|0\rangle \langle 0|\right) + \cdot \cdot \cdot \right)$ I am unable to understand how the third inequality comes about. Any help is appreciated.

One of our user asked: https://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&page=show_problem&problem=1228 can't find the 2nd and 4th case... 1st case is six equlatiral triangle and the area is sqrt(3)/2*side = 1/2*side*height so height =sqrt(3)/2*side and 3rd case is 2*height =sqrt(3)/2*side so height = (sqrt(3)/2*side)/2 here height is the radius.

One of our user asked: How does one get the following expansion : $$\lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x = \lim_{x \to \infty} \left( 1 + x\frac{1}{x} + \frac{x(x-1)}{2}\cdot \frac{1}{x^2} + \dots \right)$$

One of our user asked: I am writing a code on Matlab to calculate the matrix $B$ , given $C$ and $A$ , following the equation $$C=A B A^\mathrm{T}$$ $A^\mathrm{T}$ is non-invertible, so I can't just multiply $C$ by the inverse of the matrices. I can't solve this by hand either because $C$ is a 3144x3144 matrix...any help will be greatly appreciated!

One of our user asked: Lets say we have a set of numbers { 5, 7, 1, 2, 5, 100 }, I want to find a number x such that the sum of distances of every number from the set to x is minimal. My first thought was that x is the average of the sum of all element of the set: (sum / n), but it's not true it fail the above example. Any help or hint will be appriciated, thanks.

One of our user asked: There is a fixed circle $C_1$ with equation $(x - 1)^2 + y^2 = 1$ and a shrinking circle $C_2$ with radius $r$ and center the origin. $P$ is the point $(0, r)$ , $Q$ is the upper point of intersection of the two circles, and $R$ is the point of intersection of the line $PQ$ and the $x$ -axis. What happens to $R$ as $C_2$ shrinks, that is, as $r \to 0^+$ ? (The figure is made with GeoGebra ) In order to solve this problem, I made a script using GeoGebra in which the circle $C_2$ is a dynamic one whose radius $r$ can be adjusted with a slider. As I set $r \to 0^+$ , the figure seems to suggest that $R \to (4,0)$ . In particular, this is the state with $r = 0.001$ , in which $R$ is reported to be $(3.9999997523053,0)$ : However, I would like to find out a way to prove (or disprove, though unlikely) my guess that $$\lim_{r \to 0} R = (4,0).$$ But I have little idea. Any help would be appreciated.

One of our user asked: I everybody, I have to solve this exercise. I have to find two topological spaces $Y_1$ , $Y_2$ such that $\pi (Y_1)=\pi (Y_2)= (0)$ but $\pi (Y_1 \cup Y_2)\ne (0)$ . Can you help me?

One of our user asked: I am working on a project but need to find a skewed standard distribution, and we can't figure out how. We have two variables: $p$ , which is a probability and $PV$ which is an integer variable between 0 and 10 There are two demands that need to be meeted: The mean must lay at $PV \cdot p$ The values with non-zero probability must lay between 0 and PV. Is it possible to accomplish this? And how? (If somebody has a tip how to implement it in Python, let me know)

One of our user asked: There is a fixed circle $C_1$ with equation $(x - 1)^2 + y^2 = 1$ and a shrinking circle $C_2$ with radius $r$ and center the origin. $P$ is the point $(0, r)$ , $Q$ is the upper point of intersection of the two circles, and $R$ is the point of intersection of the line $PQ$ and the $x$ -axis. What happens to $R$ as $C_2$ shrinks, that is, as $r \to 0^+$ ? (The figure is made with GeoGebra ) In order to solve this problem, I made a script using GeoGebra in which the circle $C_2$ is a dynamic one whose radius $r$ can be adjusted with a slider. As I set $r \to 0^+$ , the figure seems to suggest that $R \to (4,0)$ . In particular, this is the state with $r = 0.001$ , in which $R$ is reported to be $(3.9999997523053,0)$ : However, I would like to find out a way to prove (or disprove, though unlikely) my guess that $$\lim_{r \to 0} R = (4,0).$$ But I have little idea. Any help would be appreciated.

One of our user asked: There is a fixed circle $C_1$ with equation $(x - 1)^2 + y^2 = 1$ and a shrinking circle $C_2$ with radius $r$ and center the origin. $P$ is the point $(0, r)$ , $Q$ is the upper point of intersection of the two circles, and $R$ is the point of intersection of the line $PQ$ and the $x$ -axis. What happens to $R$ as $C_2$ shrinks, that is, as $r \to 0^+$ ? (The figure is made with GeoGebra ) In order to solve this problem, I made a script using GeoGebra in which the circle $C_2$ is a dynamic one whose radius $r$ can be adjusted with a slider. As I set $r \to 0^+$ , the figure seems to suggest that $R \to (4,0)$ . In particular, this is the state with $r = 0.001$ , in which $R$ is reported to be $(3.9999997523053,0)$ : However, I would like to find out a way to prove (or disprove, though unlikely) my guess that $$\lim_{r \to 0} R = (4,0).$$ But I have little idea. Any help would be appreciated.

One of our user asked: How is $$\left(\frac{n!}{(2n +1)!!}\right)^2 4^n = \left(\frac{(2n)!!}{(2n +1)!!}\right)^2 ?$$ Could anyone explain this for me please?

One of our user asked: A rectangular, $A \in \mathbb{R}^{m \times n}$ matrix, where $m \ge n$ , can be decomposed (QR factorization): $$A = \begin{bmatrix}Q_1 | Q_2 \end{bmatrix}\begin{bmatrix}R\\0\end{bmatrix}$$ where $Q_1$ and $Q_2$ has orthonormal columns, and $R$ is upper triangular. I'm implementing a routine (based on Householder reflections) which calculates $Q_1$ and $R$ (so called thin/reduced QR decomposition). My question is: is it possible to calculate $Q_1$ without calculating $Q_2$ ? The problem is that a Householder matrix is $\mathbb{R}^{m \times m}$ , and $Q_1 \in \mathbb{R}^{m \times n}$ , so I cannot multiply them. My routine currently calculates $Q=[Q_1|Q_2]$ , and then throws away the $Q_2$ part.

One of our user asked: Firstly,if I evaluate behavior of function at $+\infty$ : $$f(x)\sim\frac{\ln(x^{3\over2})|\ln^{\alpha-{2\over3}}(x^2)|}{x^{\alpha+{2\over3}}}$$ And I know that it is convergent when $\alpha+{2\over3}\gt1\implies\alpha\gt{1\over3}$ . As for $x\to0^+$ $f(x)\sim\frac{|\ln^{\alpha-{2\over3}}(x)|}{x^\alpha}$ and I know that when $\alpha={2\over3}$ we have ${1\over x^{2\over3}}$ ,which is convergent because ${2\over3}\lt1$ .And I observe that $\forall\alpha\leq0$ limit of the quotient equals $0$ and $\forall\alpha\gt0$ to $\infty$ .But I am not sure what it gives me and I can't think of any "standart conditions" I can use,like I did before in this problem.

One of our user asked: Let, the point $K$ is the radius of the circle drawn inside the triangle. $\angle ABC=90°$ $AE=4$ and $CF=12$ The problem is, to find the radius of the circle drawn inside the triangle. My attempt. The formula for $r$ , we have $r=\frac{2A}{a+b+c}$ , where $A$ , is area of Triangle. So, I need, $a,b,c$ . It is obvious, $c=\sqrt{a^2+b^2}$ . Then, I need $AB$ and $BC$ . Or, I must know what are $BE$ and $BF$ . I 'm stuck..

One of our user asked: Let $$f(x)=\int_0^x\frac{dt}{\sqrt{1+t^3}}$$ Prove that $$2g''=3g^2$$ given $g(x)$ is inverse of $f(x)$ . I tried of applying Newton-Leibnitz both sides but could not succeed as the variable is $x$ on the left and $t$ on the right. How to do this?

One of our user asked: In a mathematical physical problem, one has to deal with a non-trivial two-dimensional recurrence problem involving the two sequences $D_{i,j}$ and $\psi_{i,j}$ , where $i,j = 1, 2, \dots, N$ . Specifically, \begin{align} 2a \left( \psi_{i+1,i+1}+\psi_{i+1,i-1}-\psi_{i-1,i+1}-\psi_{i-1,i-1} \right) + 16 \left( \psi_{i+1,i} - \psi_{i-1,i} \right) +3a \left( D_{i+1,i+1}+D_{i-1,i+1}+D_{i-1,i-1}+D_{i+1,i-1} \right) -16 \left( D_{i,i+1}+D_{i,i-1} \right) +64 \left( D_{i-1,i} + D_{i+1,i} \right) = 0 \, , \\ -----------------------------------\\ 4(4+a)\psi_{i,i} + 2a \left( \psi_{i+1,i+1}+\psi_{i-1,i+1}+\psi_{i-1,i-1}+\psi_{i+1,i-1} \right) +8 \left( \psi_{i,i+1}+\psi_{i-1,i}+\psi_{i,i-1}+\psi_{i+1,i} \right) +3a \left( D_{i+1,i+1}+D_{i+1,i-1}-D_{i-1,i+1}-D_{i-1,i-1} \right) +24 \left( D_{i+1,i} - D_{i-1,i} \right) = 0 \, . \end{align} Here, $a > 0$ . For a single unknown sequence the generating function method seems to be often a suitable approach for the recur...

One of our user asked: Let $A, B$ be two $3 \times 3$ (complex) symmetric matrices and suppose the equation $\det(xA - B) = 0$ has three distinct solutions. Prove that $A$ is invertible. Any help appreciated!

Picture of cone Hi, I know this is high school math but i feel kinda stupid right now so I am asking it here: Given is a truncated cone (upside down if relevant). Given is the volume as well as the upper and lower radius. Now the cone got filled with a given volume. How do i determine the height of the filled liquid? As you see in the picture, there is a cone (not a triangle). Given is r1w, r2w, and VW and VA With that at least I am able to calculate everything else, but not hA. Can someone of you please help me?

I have the following data about the surrounding vehicles and the main vehicle(let call it E-vehicle): X,Y,Z in earth co-ordinates Velocity Yaw angle Pitch angle I need to make use of these variables and write an algorithm to select the most important vehicles relative to ego vehicle. Example: To select the 10 most important vehicles from the 100 available in the traffic. I have an algorithm which uses the distance between these surrounding vehicles and the ego vehicle and selects the nearest ones. However, this is not an efficient algorithm since it can also take into account the parked vehicles which might not be that important when compared a vehicle which is approaching the E-vehicle but is a little far away. Any ideas on how I can use these variables to write a better algorithm would be appreciated.

How to approach a question like this: Norms

Consider the $4$ hyperplanes in $\mathbb{R}^5$ given by the equations $$x_1-x_2+x_3+x_4+2x_5=0$$ $$2x_1-x_2+6x_3+2x_4+6x_5=0$$ $$3x_1-3x_2+3x_3+4x_4+7x_5=0$$ $$x_1+x_2+9x_3+x_4+6x_5=0$$ Let $V \leq \mathbb{R}^5$ be the intersection of these hyperplanes. Find a basis for $V$ .

Consider the $4$ hyperplanes in $\mathbb{R}^5$ given by the equations $$x_1-x_2+x_3+x_4+2x_5=0$$ $$2x_1-x_2+6x_3+2x_4+6x_5=0$$ $$3x_1-3x_2+3x_3+4x_4+7x_5=0$$ $$x_1+x_2+9x_3+x_4+6x_5=0$$ Let $V \leq \mathbb{R}^5$ be the intersection of these hyperplanes. Find a basis for $V$ .

I'm reading Jane Eyre in class and we have to do questions. However, there are page numbers in the questions that relate to a different copy of the novel that I have. I was wondering if I could create a formula to translate the page number in the questions to the page number in my book. Here are the page numbers of the chapters from both books here is a table of the numbers i used I tried to do this with a simultaneous linear equation and ended up with the formula 5/6x + 31.5 this worked with the first 2 chapters as you can see with the screenshot but after that, the number got further away. Can someone help me to make a formula? thanks for all your help!