# All Questions

32 views

### More elegant way for solving $y(x) = y_{1}(x) + y_{2}(x)$ in $y'' - 10y' + 28y = 29xe^{-x}$

Is there a more elegant way for solving $y(x) = y_{1}(x) + y_{2}(x)$ in $y'' - 10y' + 28y = 29xe^{-x}$ than to use Euler's identity and get the general solution through brute computation?
22 views

### $F[t]$ has undecidable positive existential theory in the language $\{+, \cdot , 0, 1, t\}$

Consider the ring $F[t, t^{-1}]$ (the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$). Theorem 1. Assume that the characteristic of $F$ is zero. Then the existentia theory ...
27 views

### Continuity and Supremum

Attempt: I can't seem to gauge the points of continuity, for the attain supremum parts, I know I need to use the fact a continuous function on a closed, bounded interval is bounded and attains its ...
57 views

### What does it means to multiply a permutation by a cycle? $\pi(x_1\cdots x_n)\pi^{-1}=(\pi(x_1)\cdots\pi(x_n))$

I have to prove that $$\pi(x_1\cdots x_n)\pi^{-1}=(\pi(x_1)\cdots\pi(x_n))$$ but I can't understand what this means. My book doesn't defines what a permutation and cycle product would be. So, for ...
243 views

### New idea to solve this equation

I was teaching $\left \lfloor x \right \rfloor$ function properties and equation . I solved this equation in my class . My works are show below. Some students ask me for new Idea...,and now I am ...
16 views

### Understanding step in derivation of joint distribution

In a derivation I am trying to understand, there is the following argument: \begin{align} &=\int n!\prod_{i=1}^n f_X(x_i)\mathbb{I}_{x_1\le x_2\le\ldots\le ...
37 views

### General topology exercise (equivalent condition for simple connection)

Let $X$ be a pathwise-connected topological space. Prove that $X$ is simply connected iff every continuous $f:S^1\to X$ can be extended to a continuous function $g:D^2\to X$. How can I use the fact ...
17 views

### Lottery Combinations Question

Background: The Daily 3 game is a daily game. It consists of three sets of balls, each numbered from 0 through 8 (9 is omitted due to its visual similarity to 6). One ball is drawn from each set ...
33 views

### isomorphism between function space and complex matrices

How would you show that $\mathcal{L}(X) \cong \mathbb{C}^{n \times n}$, where $X= \mathbb{C}^{n}$. Note that $\mathcal{L}(X)$ denotes the space of linear bounded functions on $X$. Is this a specific ...
58 views

### Solving $\frac{9a^3-7ab^2+2b^3}{3a+2b}=3a^2-2ab-b^2+\frac{4b^3}{3a+2b}$

I have the following problem: $$\frac{9a^3-7ab^2+2b^3}{3a+2b}$$ The solution in the book is $$3a^2-2ab-b^2+\frac{4b^3}{3a+2b}$$ but I do not know how to get there. I could solve the other ...
52 views

### Is conditional probability always meaningful

Problem: A bag contains $4$ red and $5$ white balls. Balls are drawn from the bag without replacement. Let $A$ be the event that first ball drawn is white and let $B$ denote the event that the ...
14 views

32 views

### Is there a name for this property among variables?

I have a convex function of four variables, $f(w,x,y,z)$, which when solving for the symbolic arg min of one variable assuming the other three are known I got something similar to the following. ...
28 views

### Involutive distributions?

How do we check exactly that a distribution is involutive? I have the following definition in my book: A $k-$dimensional distribution $\Delta$ on a manifold $M$ is a smooth choice of a k-dimensional ...
25 views

### Iterative calculation of $\log x$

Suppose one is given an initial approximation of $\log x$, $y_0$, so that: $$y_0 = \log x + \epsilon \approx \log x$$ Here, all that is known about $x$ is that $x>1$. Is there a general method of ...
16 views

### Question about concyclic points on the coordinate axes

If the points where the lines $3x-2y-12=0$ and $x+ky+3=0$ intersect both the coordinate axes are concyclic,then the number of possible real values of k is (A)1 ...
37 views

### Proof of the integral operator in $L^2(\mathbb{R})$ being self-adjoint “by hand”

Suppose we have an integral operator $A$ such that $$Af(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-\frac{(x-y)^2}{2}}f(x) \, dy$$ This operator is bounded and $\|A\|=1$ (see Norm of the ...
33 views

### External bisectors of the angles of ABC triangle form a triangle $A_1B_1C_1$ and so on

If the external bisectors of the angles of the triangle ABC form a triangle $A_1B_1C_1$,if the external bisectors of the angles of the triangle $A_1B_1C_1$ form a triangle $A_2B_2C_2$,and so on,show ...
33 views

### Total derivative of $f(A,B)$ , where $f:M(n,\mathbb{R}) \times M(n,\mathbb{R}) \to M(n,\mathbb{R})$

Find the Total derivative of i)$f(A,B)=A+B$ , ii)$g(A,B)=AB$ iii)$h(A,B)=A^2$ where $f,g:M(n,\mathbb{R}) \times M(n,\mathbb{R}) \to M(n,\mathbb{R})$ and $h:M(n,\mathbb{R}) \to M(n,\mathbb{R})$ ...
18 views

### Trajectories that connect equilibrium points

Suppose I consider the autonomous system \begin{align*} x' &= F(x, y)\\ y' &= G(x, y) \end{align*} where $F$ and $G$ are nonlinear and my task is to draw the phase portrait of the above ...
46 views

### Condition on p for convergence of $\sum{\frac{1}{n(\log(n))^p}}$

For what values of $p$ is the series $\sum{\frac{1}{n(\log(n))^p}}$ divergent and for what values it is convergent?
52 views

### Complex numbers: Argument [on hold]

Can anyone help me with this? (a) Show that (zw)* = zw (b) Prove by induction that $\ (z^n)^*=(z^*)^n$ for all positive integers $n$.
51 views

### Trigonometry identity $\csc x\cot x=\frac{\cos ^3x}{\sin^2 x}+\cos x$

How to prove that $\csc x\cot x=\frac{\cos ^3x}{\sin^2 x}+\cos x$? I tried manupulating the left hand side but ended up in $\frac{\cos x}{\sin^2 x}$. Can someone show me? Thanks in advance.
48 views

Hello I have stucked with theese two questions: $\sqrt{a:\sqrt{a:\sqrt{a: \cdots}}} + \sqrt[3]{a\cdot\sqrt[3]{a\cdot\sqrt[3]{a\cdots}}} = 12$ $a=\text{ ?}$ ...
48 views

### Trigonometry problem

Okay..this one simple problem but I am really stuck and have no idea how to start.. $\cos(a-b)+\cos(b-c)+\cos(c-a)=-\frac32$ we need to prove $\cos(a)+\cos(b)+\cos(c)=\sin(a)+\sin(b)+\sin(c)=0$
75 views

DOUBT What i didnot understand is from where it is written our new goal means there exists a... I didnot understand how there exists word popped up here and why the new givens are written as they ...
34 views

### Probability of Two People Choosing the Same Number between 1 and 50 or choosing 2 numbers that add to 50 [on hold]

If two people are asked to each choose a number between 1 and 50, what is the probability that they choose the same number, or choose numbers that add to equal 50?
What steps should be taken in order to get a solution (that only depends on v) for the following?: $\dfrac{\partial ^2f}{\partial v^2}+\dfrac{1}{v}\dfrac{\partial f}{\partial ... 1answer 24 views ### Counter example:$X$and$Y$normal imply$(X,Y)$bivariate normal I vaguely remember this construction from one of my courses: Suppose that$X\sim N(0,1)$and$Z$is$\pm 1$with probability$\frac{1}{2}$each. If$X$and$Z$are independent, then$Y\equiv XZ$is ... 2answers 24 views ### If$UT=TU $, Why is the range of$U $invariant under$T $? My Linear Algebra book says the following: Let$V$be a vector space and$T$be a transformation, which commutes with another transformation$U$. Then the kernel and range of$U$are invariant ... 4answers 81 views ### How to solve$\lim_{x\to1}\left[\frac{x}{x-1}-\frac{1}{\ln(x)}\right]$without using L'Hospital's rule? $$\lim_{x\to1}\left[\frac{x}{x-1}-\frac{1}{\ln(x)}\right]$$ How can I solve this without using the L'Hopital's rule? Any tips or hints would be greatly appreciated. I tried using the substitution ... 0answers 73 views ### Why we wonder to know all derivations of an algebra? It is well-known that the space of all derivations of algebra of smooth functions on a manifold is its space of sections (vector fields on underlying manifold). But, I wonder to know why finding the ... 1answer 32 views ### integrating product of PDF and CDF I am trying to show that the following integral: $$\int_{-\infty}^a F(x)~f(x)~dx = \frac{F(a)}{2!}$$ Where$F$is the cumulative distribution function of some continuous random variable X, and$f$... 1answer 149 views ### Can we find this limit? Let$\{a_n\}_{n=1}^{\infty}$be a sequence such that$a_1=1$and$a_{n+1}=\sqrt{a_n^2+a_n}$for$n\geq 0$. Is it possible to find$\displaystyle\lim_{n\to\infty}\frac{a_n}{n}$? I have no any idea. ... 2answers 19 views ### Biased and fair coin in Hat flipped Two coins are in a hat. The coins look alike, but one coin is fair (with probability 1/2 of Heads), while the other coin is biased, with probability 1/4 of Heads. One of the coins is randomly pulled ... 3answers 69 views ### Series convergence$x+\frac{2^2x^2}{2!}+\frac{3^3x^3}{3!}+\frac{4^4x^4}{4!}+\cdots$[on hold] Choose the right option. The series$x+\dfrac{2^2x^2}{2!}+\dfrac{3^3x^3}{3!}+\dfrac{4^4x^4}{4!}+\cdots$is convergent if a.$0<x<1/e$b.$x>1/e$c.$2/e<x<3/e$d.$3/e<x<4/e$... 2answers 27 views ### How to choose the extent of approximation for this equation? Given the curve $$y={1 \over{x^2-1}}-{1\over {x^2}}$$ Find reasonable approximations for the intersections of this curve with the straight line$y=Ax$(a) when$A$is a very small positive number ... 2answers 56 views ### Is$\frac00=\infty$? And what is$\frac10$? Are they same? Does it hold true for any constant$a$in$\frac{a}0$I know that$\lim_{x\to0}\frac{x}{x}=$1. But in my text book, it is written that it is$\infty$and even$\frac10=\infty$. But how is it possible? And are they both same? What is the difference ... 4answers 30 views ### Manipulating equations question In the equation: $$T = 2\pi \sqrt {\frac lg}$$ it is for determining period of pendulum swing If I want to solve for$g$and I want to start by removing the root do I square everything in the ... 0answers 20 views ### Could someone give a detailed (yet elementary) proof for Jensen's inequality? I want to prove that Suppose there is a function$f:[a,b] \to \mathbb R$, and there are$x_i \in [a,b], w_i \gt 0 $for$i=1,\dots,n$such that$\sum_{i=1}^nw_i=1$, then if the function is convex, ... 1answer 18 views ### Finding Automorphisms of Irregular graph through Regular Sub-Graphs. Objective : To find a set of permutations for a irregular graph which is also a set of automorphism. This finding process uses permutations of 2 regular subgraphs of the given graph. Description and ... 1answer 21 views ### Is the closure of every bounded convex set , with non-empty interior , in$\mathbb R^n (n>1)$homeomorphic to a closed ball? Is the closure of every bounded convex set , with non-empty interior , in$\mathbb R^n (n>1)$homeomorphic to a closed ball (by closed ball I mean$B[a,r]:=\{x \in \mathbb R^n : d(x,a)\le r\}\$ , ...
I'm trying to solve problem 7 from the IMC 2015, Blagoevgrad, Bulgaria (Day 2, July 30). Here is the problem Compute $$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$ ...