Recent Questions - Mathematics Stack Exchange |
- Minimal distance between intersections of a regular grid parameterized by its change-of-basis matrix
- On partitioning triangles and pentagons
- Is $X$ a compact subspace of ${M}2(\mathbb{R})$?
- A Computable function for Partial Recursive Function
- Action of $S^1$ on a vector space
- Find a unique solution to the following Dirichlet problem.
- Referencing something: Proposition, Lemma, Corollary, etc.?
- How to find the Absolute Minimum of this Type of Function
- evaluate the improper integral from $0$ to $\infty$ [duplicate]
- A different look at Collatz conjecture
- Transformation Matrix with respect to a basis and the General Linear Group
- Pendulum with Dirac Comb excitation
- Simple combinations and commissions
- Values of f_m_l coefficients of function development in spherical harmonic series?
- What is $\int_{-\infty}^{\infty}\frac{1}{x^3+1}dx$?
- Max Volume out of a square with a open rectangular box(can you confirm my proof?)
- The probability of mistake of randomized program.
- Unable to solve the following question
- Meaning of $\frac{P(X\cap Y)}{P(X)P(Y)}$
- Show that $\int_0^\frac{\pi}{2}\sqrt{\sin x} dx \times \int_0^\frac{\pi}{2}\frac{1}{\sqrt{\sin x}} dx =\pi $
- Could this mapping $f$ be bijective?
- Find the area of the shaded region $CEOD$.
- How do points on a grid satisfy the grid equation?
- How do I integrate it?
- Solve the following equation for $x$ given $ \log_2(x)=1-(x-2)^2$
- Does a càdlàg martingale on a finite time horizon have an integrable maximum?
- Find the angles of given triangle ABC
- 6 cards are distributed between 3 people among other cards, what is the probability that someone will have at least 3 cards AND a specific card
- Finding angles plane geometry
| Minimal distance between intersections of a regular grid parameterized by its change-of-basis matrix Posted: 13 Jan 2022 08:40 AM PST Let a 2D grid basis $\mathcal{B}(\theta_1, \theta_2,r_1, r_2)$ whose change-of-basis matrix with respect to $\mathcal{B}_0$ the canonical basis of $\mathbb{R}^2$ writes : $$P_{\mathcal{B}_0}^{\mathcal{B}(\theta_1, \theta_2,r_1, r_2)} = \left(\begin{array}{cc} r_1\cos(\theta_1) & r_2 \cos(\theta_2) \\ r_1\sin(\theta_1) & r_2 \sin(\theta_2) \end{array}\right)$$ and the parameters of the grid satisfy $$ \begin{array}[ccc] &&\\ \theta_1 & \in & (-\frac{\pi}{2}, \frac{\pi}{2}]\\ \theta_2 & \in & [-\frac{\pi}{2}, \theta_1)\\ r_1,r_2 & > & 0 \end{array} $$ The intersections of the grid $\mathcal{B}(\theta_1, \theta_2,r_1, r_2)$ written in $\mathcal{B}_0$ is the following set $$ \left\lbrace P_{\mathcal{B}_0}^{\mathcal{B}(\theta_1, \theta_2,r_1, r_2)} z\;;\;\forall z\in \mathbb{Z}^2\right\rbrace$$ Given this setting, computing the minimal distance between two intersections of this grid is equivalent to solving the following problem $$\min_{z\in\left(\mathbb{Z}^*\right)^2}\|P_{\mathcal{B}_0}^{\mathcal{B}(\theta_1, \theta_2,r_1, r_2)} z\|_2$$ The solution of this problem can be tricky to compute in corner cases. For example, let the grid be as follows: $$r_1 = 2\;;\;r_2=5\;;\;\theta_1 = -10^° \;;\;\theta_2 = -11^°$$ With this configuration the minimal value for $\|P_{\mathcal{B}_0}^{\mathcal{B}(\theta_1, \theta_2,r_1, r_2)} z\|_2 $ is achieved for $z^*=(5,-2)$. Finally, my question is, is there a faster method than Integer Programming to solve this problem. Particularly I would like to know if there is a way to compute $z_i^-$, $z_i^+$ such that $$\textrm{arg}\min_{z\in\left(\mathbb{Z}^*\right)^2}\|P_{\mathcal{B}_0}^{\mathcal{B}(\theta_1, \theta_2,r_1, r_2)} z\|_2 \subset[z_1^-,z_1^+]\times[z_2^-,z_2^+] $$ |
| On partitioning triangles and pentagons Posted: 13 Jan 2022 08:38 AM PST
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| Is $X$ a compact subspace of ${M}2(\mathbb{R})$? Posted: 13 Jan 2022 08:38 AM PST I have been having problems to resolve this problem: In the vectorial space ${M}2(\mathbb{R})$ of order $2$ real matrixes, We define the topology such that the natural biyection from ${M}2(\mathbb{R})$ to $\mathbb{R}^4$ is an homeomorphism. We consider the ${M}2(\mathbb{R})$ subspace: $$X = \{A ∈ {M}2(\mathbb{R}); A^t A = I\}. $$ Is $X$ a compact subspace of ${M}2(\mathbb{R})$? |
| A Computable function for Partial Recursive Function Posted: 13 Jan 2022 08:33 AM PST We have a computable function $F = (x_0, x_1)$ for set $M = \{f(x_1) ; f $ is partial recursive$ \}$. That means $F$ is a partial recursive function as well. Suppose function $g \simeq F(x, x) + 1$. Hence $\exists n, \forall x $ $g(x) \simeq F(n, x)$ If $x = n$; $g(n) \simeq F(n, n) \simeq F(n, n) + 1$ How can this be true? |
| Action of $S^1$ on a vector space Posted: 13 Jan 2022 08:38 AM PST In page 21 of the article cohomologie équivariante et théorème de Stokes, there is a small paragraph which says :
What does it mean that $S^1$ on $V$ by a group of a parameter of linear transformations ? Why does the action of $S^1$ on $V$ is represented by the above matrix, is there a unique action of $S^1$ on a vector space ? |
| Find a unique solution to the following Dirichlet problem. Posted: 13 Jan 2022 08:26 AM PST I want to show that the following Dirichlet problem \begin{equation} \begin{cases} \Delta u = 0 & \text{in } \Omega,\\ u = g & \text{on }\partial \Omega, \end{cases} \end{equation} where $\Omega = (0,1)^2$ and $g(x, y) = 4xy - 7e^x$, has a unique solution. I clearly only have to show the existence as uniqueness is given by the maximum principle. In my functional analysis course, we've seen that such a problem admits a solution if and only if each point of the boundary admits a barrier, i.e. $$\forall x_0 \in \partial \Omega, \quad \exists w \in C(\bar\Omega) ~~\text{with }~ w(x_0) = 0~~ \text{and}~~ w > 0 ~~ \text{on }~\bar\Omega\backslash \{x_0\},$$ and $w$ is superharmonic. Therefore I have to find $w$ defined on the square $\Omega$ that satisfies the above conditions. The only functions I can think of have the form $$w(x, y) = |x - x_0| + |y - y_0|$$ but it doesn't seem to be superharmonic.. Any help ? |
| Referencing something: Proposition, Lemma, Corollary, etc.? Posted: 13 Jan 2022 08:34 AM PST I have question on how to refer to some mathematical results in order to highlight their hierarchical order. (i) I have a major result which is formulated like: "Assume Assumptions 1 holds. Then there exists B satisfying Assumption 2." (ii) Assumption 1 is high level. I have a collateral result which helps one to verify whether Assumption 1 holds: "Assumption 1 holds if and only if Assumption 3 holds." Assumption 3 is easier to verify. I thought about referring to (i) as a Proposition. However, I'm not sure how to refer to (ii). Following this, it does not seem to be either a Corollary or a Lemma. Also, should (ii) be introduced after or before (i)? |
| How to find the Absolute Minimum of this Type of Function Posted: 13 Jan 2022 08:23 AM PST I have been working on this question: Question: Given positive numbers $λ_{1}$,…, $λ_{n}$. Find the absolute minimum of $f(x)$ = $\max_{1≤i≤n}$$\frac{|x−λi|}{x+λi}$, $x$$≥$$0$. Justify your solution. I solve maximum and minimum problems by determining the critical points of the derivative of the function. However, the MAX function here throws me off balance. This is my attempt on the question so far: Attempt f(x) = $\max_{1≤i≤n}$$\frac{|x−λi|}{x+λi}$ Since $\frac{|x−λi|}{x+λi}$ is negative for $x$$<$ ${λ_i}$ for each $i$, then the absolute minimum would be $-1$ for ${λ_i}$$=$$0$ for some $i$. The problem here is that ${λ_i}$ is positive. Hence, there seems to be no lower bound. This means the absolute minimum does not exist. Does this line of reasoning make sense or is there something I am missing out? Thanks. |
| evaluate the improper integral from $0$ to $\infty$ [duplicate] Posted: 13 Jan 2022 08:23 AM PST Evaluate the following integral: $$\int_0^\infty \frac{1}{x^4+1}dx$$ Do I use Cauchy's Integral formula? |
| A different look at Collatz conjecture Posted: 13 Jan 2022 08:21 AM PST There can be a different look at Collatz conjecture. The conjecture in its original form is very hard to prove. The conjecture can be re-formulated to a more manageable form. Below is an analytical proof of the Collatz conjecture. The key to proving it is a lemma: a multiple divider is reduced to 1 or a single divider when Collatz division(s) are applied to it. The proof is below.
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| Transformation Matrix with respect to a basis and the General Linear Group Posted: 13 Jan 2022 08:25 AM PST I need some help going over this problem because I'm not entirely sure if my solution is sound.
My work: a) Let $B=\{B_1,B_2\}$ be a basis for $\mathbb{R^2}$ with $B_1= \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}, B_2= \begin{pmatrix} b_3 \\ b_4 \end{pmatrix}$. We find the image of $B_1$ and $B_2$ and then express them in terms of the basis with real coefficients $p_i,q_i$. $$ \phi(B_1)= \begin{pmatrix} \frac{3}{2}b_1-\frac{1}{2}b_2 \\ -\frac{1}{2}b_1+\frac{3}{2}b_2 \end{pmatrix}= p_1\begin{pmatrix} b_1 \\ b_2 \end{pmatrix}+ p_2\begin{pmatrix} b_3 \\ b_4 \end{pmatrix}$$ $$\phi(B_2)= \begin{pmatrix} \frac{3}{2}b_3-\frac{1}{2}b_4 \\ -\frac{1}{2}b_3+\frac{3}{2}b_4 \end{pmatrix}= p_1\begin{pmatrix} b_1 \\ b_2 \end{pmatrix}+ p_2\begin{pmatrix} b_3 \\ b_4 \end{pmatrix} $$ Since the transformation matrix has been given we find the coefficients as follows: $p_1=1,p_2=0,q_1=0,q_2=2$. Which simply means that $\phi(B_1)=B_1, \phi(B_2)=2B_2$. I then simply chose the vectors $\begin{pmatrix} 2\\2 \end{pmatrix}$, $\begin{pmatrix} 2\\-2 \end{pmatrix}$. b)This question was much easier and I guessed, correctly, that the matrix $\begin{pmatrix} 2 &2 \\2&-2 \end{pmatrix}$ was an element of the set. The last two parts of this problem is where I need help. I tried simply computing the results of $\lambda T \begin{pmatrix} \frac{3}{2} & -\frac{1}{2} \\ -\frac{1}{2} & \frac{3}{2} \end {pmatrix}T^{-1} $ but it got quickly very complicated and I was wondering if there was an easier way to show it? |
| Pendulum with Dirac Comb excitation Posted: 13 Jan 2022 08:35 AM PST There is a pendulum that is excited by a Dirac Comb. $l \ddot\theta+b\dot \theta+g\theta=G\,\sum_{-\infty}^\infty\delta(t-nT)$ where $l, b, g, G$ are constants and $T=\dfrac{2\pi}{\omega}$. Show that the resulting motion is given by $\theta(t)=\dfrac{G}{Tl\omega^2}+\dfrac{2G\cos(\omega t-\frac{\pi}{2})}{Tb\omega}$+[terms with frequencies $\ge$ 2$\omega$] and explain why the higher frequency terms are supressed. My first take was to rearrange to $ \ddot\theta+\dfrac{b}{l}\dot \theta+\omega^2\theta=\frac{G}{l}\,\sum_{-\infty}^\infty\delta(t-nT)$ where $\omega=\sqrt{\dfrac{g}{l}}$ Then, taking the Laplace transform of both sides I got $\Theta(s)=\dfrac{G}{l\,\sqrt{\left(\frac{b}{2l}\right)^2-\omega^2}}\, \dfrac{\sqrt{\left(\frac{b}{2l}\right)^2-\omega^2}}{\left(s+\frac{b}{2l} \right)^2-\left(\left(\frac{b}{2l}\right)^2-\omega^2 \right)}\, \dfrac{1}{1-e^{-sT}}$ which, as far as I'm concerned transforms to $\theta(t)=\dfrac{G}{l\,\sqrt{\left(\frac{b}{2l}\right)^2-\omega^2}}\sum_{n=0}^\infty\,H(t-nT)\,e^{-\frac{b}{2l}(t-nT)}\sin\left(\sqrt{\left(\frac{b}{2l}\right)^2-\omega^2}(t-nT) \right)$ And, assuming that this is a correct form of the solution, I can't see how that is equivalent with the function given in the question. I reckon it has something to do with using Fourier series/transform instead? If so, I'm not sure how to do that. Or, is there a way to convert my solution into the given one? I've been struggling with this for a good few days now, so any help would be much appreciated. |
| Simple combinations and commissions Posted: 13 Jan 2022 08:39 AM PST A staff committee by $3$ men and $3$ women to be chosen from a pool of $8$ men and $5$ women. What the number of committees in the event that H and M are not allowed to be on the committee simultaneously Attempt: Let's call $H$ the man referred to in the statement, and $M$ the woman he doesn't like. We can divide into cases: Case 1: $H$ participates, so $M$ will not participate $\displaystyle \binom{7}{2} \cdot \binom{4}{3}$ Case 2: $M$ participates, so $H$ does not participate $\displaystyle \binom{4}{2} \cdot \binom{7}{3}$ So, adding up the possible outcomes we get $126$. The answer is $434$, am I doing something wrong? |
| Values of f_m_l coefficients of function development in spherical harmonic series? Posted: 13 Jan 2022 08:23 AM PST Given the equation of the coefficients of the expansion of a function in spherical armonics, taken from here here, is there a page or a math calculator which can show me the values of this coefficients f_m_l for different values of m and l? Also f_m_l with function form is good! Thanks. |
| What is $\int_{-\infty}^{\infty}\frac{1}{x^3+1}dx$? Posted: 13 Jan 2022 08:35 AM PST I am told to compute the above integral in terms of its principal value. My method involved using a semi-circular contour on the upper half plane, with an indentation around $-1$, as this is a singularity. The maths reduces to the following; $$PV\int_{-\infty}^{\infty}\frac{1}{x^3+1}dx = 2{\pi} i \mbox{Res}\left(\frac{1}{z^3+1}, e^{\frac{{\pi}i}{3}} \right) +{\pi}i \mbox{Res}\left(\frac{1}{z^3+1},-1\right) = \cdots = \frac{\pi}{\sqrt{3}}$$ And after computing the residues, which are found in a straightforward manner as they are both simple poles, I get the answer of $\frac{\pi}{\sqrt{3}}$. However, Wolfram doesn't give an answer for it. It says that the integral doesn't converge. Thus, I can't be sure my answers correct. |
| Max Volume out of a square with a open rectangular box(can you confirm my proof?) Posted: 13 Jan 2022 08:32 AM PST I have a simply proof to do, but just want to get your thoughts on it if i haven't forgot something: I have a squared piece of metal with the side length of $ a $ and i need to cut out a open rectangular box out of it by cutting out the same quadratic edges. My drawing of the problem (at least how i understand it):
x is in my calculation the length of one side at one of the blue squares The definition of the volume for a rectangular box is: $$ V = a * b *c $$ since the shape we cut out the rectangular box is a square, $ a = b $ and there for: $$ V = (a-2x) * (a-2x) * x$$ By multiplication i get: $$ V = (a^2 - 4ax + 4x^2) * x$$ $$ V = (a^2x - 4ax^2 + 4x^3)$$ The first derivative of V is: $$ V' = 12x^2-8ax+a^2 $$ and i need to find the zero points: $$ V' = 12x^2-8ax+a^2 = 0 $$ therefor, using the quadratic formula i get: $$ \frac{-b \pm \sqrt{(b^2)+4ac}}{2*a}$$ and $ a = 12 $, $ b = -8a $ and $ c = a^2 $ $$ \frac{-(-8a) \pm \sqrt{(-8a)^2+4 * 12 * a^2}}{2 * 12} $$ $$ \frac{8a \pm \sqrt{64a^2+48a^2}}{24} $$ $$ \frac{8a \pm \sqrt{16a^2}}{24} $$ $$ \frac{8a \pm 4a}{24} $$ and therefor $$ x_1 = \frac{a}{2} $$ and $$ x_2 = \frac{a}{6} $$ End of proof. Is my proof correct or am i on the wrong path? Thanks in advance everyone! |
| The probability of mistake of randomized program. Posted: 13 Jan 2022 08:24 AM PST Consider this program to detect odd and even numbers:
How many times must the program run so that the probability of getting the correct answer is 0.99 (confidence). My solution If the program prints $odd$ then the number is certainly odd, because if it was even the program would print $even$. But if program prints $even$ there is a probability that the number is actually odd. So the probability of error is when it prints $even$ but the number is odd. So using Bayes rule we have: $$ P(odd | print\ even ) = \frac{P(odd \wedge print\ even) }{P (odd \wedge print\ even) + P(even \wedge print\ even)} $$ In this case we repeat the program until we either get an $odd$ or repeating $even$ for $K$ times. Is it correct? However I then don't know how to continue and what is the value for each of these components. You can assume the probability of a number is odd or even equal to 0.5 |
| Unable to solve the following question Posted: 13 Jan 2022 08:36 AM PST I am trying to solve the following question but I can not figure out a method to solve it. The question is: $$x+\sqrt{(y^2) - (xy) \frac{dy}{dx}} = y, y(1/2) = 1$$ Can somebody help me and guide me through? If you could solve it I would get the idea and solve the remaining questions. |
| Meaning of $\frac{P(X\cap Y)}{P(X)P(Y)}$ Posted: 13 Jan 2022 08:21 AM PST Imagine that we have a set $\Omega$ and $X$ and $Y$ are events that can happen, I mean, $P(X),P(Y)>0$. Then, what does it mean the ratio $\frac{P(X\cap Y)}{P(X)P(Y)}$? I know that $\frac{P(X\cap Y)}{P(X)P(Y)}=\frac{P(X|Y)}{P(X)}=\frac{P(Y|X)}{P(Y)}$ and if that ratio is equal to 1 then $X,Y$ are independent events, but I can't figure out what exactly it means... please give simple examples. I found this when reading about lift-data mining. |
| Posted: 13 Jan 2022 08:27 AM PST Show that $\int_0^\frac{\pi}{2}\sqrt{\sin x} dx \times \int_0^\frac{\pi}{2}\frac{1}{\sqrt{\sin x}} dx =\pi $ My teacher gave this question to solve but I was unable to solve it. I think there is surely any property of definite integral which I'm missing. I'm trying not to use exponential integral or any other special function. I tried the following method: $$\int_0^\frac{\pi}{2}\sqrt{\sin x} dx \times \int_0^\frac{\pi}{2}\frac{1}{\sqrt{\sin x}} dx$$ $$\int_0^\frac{\pi}{2}\sqrt{\sin x} \times \frac{1}{\sqrt{\sin x}} dx$$ $$\int_0^\frac{\pi}{2}dx = \frac{\pi}{2} $$ This is of course not true. What should I do with this? Kindly help me. |
| Could this mapping $f$ be bijective? Posted: 13 Jan 2022 08:25 AM PST Let us assume that we have some fields $T$, $(Y_i)_{i=1}^k$ and $X$, that are subsets of an arbitrary field $G$. Let the following mappling $f:T\times Y_1\times\cdots\times Y_k\to X$ defined as below $$f(t,y_1,\cdots,y_k)=x_1\oplus x_2\oplus\cdots\oplus x_k$$ where for every $Y_i$, $i=1,...,k$, holds that $|Y_i|\geq |T|$, and the $k$ pairs $(x_1,y_1)$, $(x_2,y_2)$, ... , $(x_k,y_k)$ are associated with only one $t$. Could this mapping be bijective? If yes could someone show how? |
| Find the area of the shaded region $CEOD$. Posted: 13 Jan 2022 08:34 AM PST For reference:
My progress: $\angle AOD = 90^\circ$ $[ECOD] = [AOD] - [AEC]$ $\triangle AEC \sim \triangle AOD \implies \dfrac{AE}{AO} = \dfrac{AC}{4\sqrt2} = \dfrac{EC}{OD}$ $\dfrac{[AEC]}{[AOD]}=\dfrac{AC\cdot AE}{4\sqrt2\cdot AO}$ $[ECOD] = \dfrac{OD+EC}{2}\cdot OE$ $[AOD] = \dfrac{AO\cdot OD}{2}$ $[AEC] =\dfrac{AE\cdot EC}{2} $ I couldn't see more...??? |
| How do points on a grid satisfy the grid equation? Posted: 13 Jan 2022 08:24 AM PST Currently, I'm working on a presentation regarding Penrose tilings. During my research, I've become interested in the Pentagrid method of construction, that was introduced by N.G. de Bruijn in 1981 (original paper). However, since I won't talk exclusively about this construction method, I kept searching for lower-level sources. A difficult task, it appeared, but I found a thesis of Laura Effinger-Dean called "The Empire Problem in Penrose Tilings". I read it through, but I'm only really interested in chapter 4 (Pentagrids) and I can't get a grasp of the grid equation. On page 33, we can read:
I guess we can choose whatever $k$ we want, but it must remain fixed for all points on the grid. Furthermore, I suppose $\vec{x}$ is a point on the grid (that must satisfy the equation). What I don't get is how we can construct a grid, with multiple (infinitely to be precise) parallell lines, using this equation. How should I interpret the $\vec{x}\cdot\vec{\epsilon}$? Since we're working with vectors, my initial thougt was a dot product, but I'm not sure. Then the first term would always be zero (every point must be orthogonal to the grid vector), but if $\gamma$ and $k$ are fixed, how can we get more than one line? Am I missing something or making wrong assumptions? Thanks in advance! |
| Posted: 13 Jan 2022 08:35 AM PST Here, how to integrate it step by step? $$I=\int\frac{x^2+x+1}{\sqrt{x^2+2x+3}} \ dx$$ First let's Substitute, we got $$x^2+2x+3=t^2$$ $$\implies {x}=\sqrt{t^2-2}-1$$ $$\implies dx=\frac{t}{\sqrt{t^2-2}}dt$$ Putting this back into the Integral followed by $$I=\int\frac{x^2+2x+3-x-2}{\sqrt{x^2+2x+3}}dx$$ $$=\int\frac{t^2-(\sqrt{t^2-2}-1)-2}{\sqrt{t^2}}dx$$ $$=\int\frac{t^2-\sqrt{t^2-2}+1-2}{t}\frac{t}{\sqrt{t^2-2}}dt$$ $$=\int\frac{t^2-2+1-\sqrt{t^2-2}}{\sqrt{t^2-2}}dt$$ $$=\int\frac{t^2-2}{\sqrt{t^2-2}}+\frac{1}{\sqrt{t^2-2}}-1dt$$ $$=\int\sqrt{t^{2}-2}+\frac{1}{\sqrt{t^{2}-2}}-1dt$$ Since we all know $$\int\sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\ln\left(\left|\sqrt{x^2-a^2}+x\right|\right)$$ and $$\int\frac{1}{\sqrt{x^2-a^2}}dx=\ln(x+\sqrt{x^2-a^2})$$ & $$a^2=(\sqrt{2})^2=2$$ Now $$I=\frac{t}{2}\sqrt{t^2-2}-\ln(t+\sqrt{t^2-2})+\ln(t+\sqrt{t^2-2}-t+C$$ $$\implies I=\frac{t}{2}\sqrt{t^2-2}-t+C$$ $$=\frac{t}{2}\sqrt{t^2}-\frac{t}{2}\sqrt{2}-t+C$$ $$=\frac{t.t}{2}-\frac{t}{2}\sqrt{2}-t+C$$ $$=\frac{t^2-t\sqrt{2}-2t}{2}+C$$ $$=\frac{t(t-\sqrt{2}-2)}{2}+C$$ Now Putting back x 's value , we get, |
| Solve the following equation for $x$ given $ \log_2(x)=1-(x-2)^2$ Posted: 13 Jan 2022 08:24 AM PST Solve for $x$ the following equation given : $\log_2(x)=1-(x-2)^2$ I tried to discuss the variation of the function $f(x)=\log_2(x)$ and $g(x)=1-(x-2)^2$ so I found that $x=1$ or $x=2$ but another solution is missing I tried a lot help please and Thank you |
| Does a càdlàg martingale on a finite time horizon have an integrable maximum? Posted: 13 Jan 2022 08:37 AM PST We work on a filtered probability space with finite time horizon $T$. Let $M$ be a non-negative càdlàg martingale. Define $M_T^* := \sup_{t\leq T}M_t$. Question: Is $M_T^*$ integrable, i.e. does $\mathbb{E}[M_T^*]<\infty$ hold? Any help is very much appreciated! |
| Find the angles of given triangle ABC Posted: 13 Jan 2022 08:30 AM PST
I've been stuck on this one for quite a long time. After denoting with $I$ the incenter of ABC and deriving that $\angle C = 96^\circ$ from $\angle AIB = 90^\circ + \frac12\angle C = 138^\circ$, I really don't know how to continue. I tried using Geogebra to see everything clearer or at least guess the answer, and I concluded that $\angle A$ and $\angle B$ should be $12^\circ$ and $72^\circ$ respectively, but I'm not sure how to prove it. Any help would be appreciated. If I come up with something, I will post it right away. Thanks in advance! |
| Posted: 13 Jan 2022 08:36 AM PST I play a card game called "Whist". In this game, all four players get 13 cards each, and you have to estimate how many tricks you will win with one trump color. Here is my scenario: In my hand, I have 7 hearts : Ace, King, Jack, 10, ... (the rest doesn't matter). My question is : what is the probability that I will win those 7 cards if the trump is Heart? The only way for me to lose the third trick is if somebody has 3 hearts or more AND the queen. This is because if the player who has the Queen has only one or two cards he will lose it when I play my Ace and my King (people have to follow your color). So my question is: I know there are 6 remaining hearts in the game, they are separated into 3 players, what is the probability that one player has 3 cards or more containing the Queen? I tried to solve it doing a probability tree but it quickly became complicated so I wonder which formula we can use. Thanks in advance for your response! (PS: I know that if for example I have Ace, King, Jack, 10, 8, etc, I could lose one trick if someone has 5 hearts and the 9 of hearts but I think if I am not mistaken that the probability of this is quite low (correct me if I am wrong)) |
| Posted: 13 Jan 2022 08:26 AM PST $\Delta ABC$ is obtuse on $B$ with $\angle ABC = 90 + \frac{\angle BAC}2$ and we have a point $D \in AC$ (in the segment, I mean D is in between A and C) such that $\angle BDA = \angle ABD + \frac{\angle BAC}2$ and $DC = BA$. Find $\angle BCA$. What I've done: I've drawn the bissector of $\angle BAC$ and let it meet $BD$ on $F$, then I've got the point $E \in AD$ such that $AE = DC = AB$ therefore we would have $\Delta ABF \equiv \Delta AEF $. The coolest thing I've got from this is that $AF$ is the perpendicular bissector of $BE$ and the triangles $ABC$ and $BEC$ are similar. Believe me or not I've been stuck here for some days ( ._.) I apreciate the trigonometric solution, if nobody comes with a plane geometry solution I will chose the first answer |
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