The process of studying mathematics without formal instruction. _Don't_ use this tag just because you were self-studying when you came across the mathematical question you're asking; it is only for when fact that you're self-studying is what your question is _about_.

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39 views

Find a vector orthogonal to other two given and ends at a plane

I am reviewing Calculus III using Mahavier, W. Ted's material and get stuck on one question in chapter 1. Here is the problem: Assume $\vec{u},\vec{v}\in \mathbb{R}^3$. Find a vector ...
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2answers
45 views

Problem of continuous function

Define the function $g(x) = x^2\cos\frac1x$ for $x\ne 0$. What should be the value of $g(0)$ if $g(x)$ is a continuous function? Explain your work and justify your answer. Frankly, I have no ...
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2answers
198 views

Third and average price auction

Third price auction: the winner is the highst bidder but this time instead of paying the second highst bid, he would pay the third highst bid. -assume there are at least 3 bidders. - Average price ...
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1answer
55 views

Prove the assertion on the game theory.

if a dominant strategy for player1 is added to finite normal form game then the payoff to player1 at any equlibrium of the new game must be at least as great as any nash equlibrium payoff ...
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3answers
83 views

How do I find $\lim_{x\rightarrow \infty} x\sin \frac {c}{x}$?

How do I find the following limit for some real $ c $? $$\lim_{x\rightarrow \infty} x\sin \frac {c}{x} $$
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0answers
25 views

Polar coordinates: Slope of tangent

Would anyone mind telling me how to solve this problem? It seems strange as my answer is $-1$. Do I have to apply this formula, $(r'sinθ +rcosθ)/(r'cosθ -rsinθ)$ ?
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2answers
42 views

Need help with this absolute value equation

I need to solve the following equation involving absolute value: $$|x-1| = 1-x$$ Looking at the term $x-1$, I thought I'd divide the interval into parts: $x < 1$ and $x \geq 1$. Now, when ...
2
votes
1answer
73 views

Misuse of Tangent Vector

I am quite confused with the term tangent used in differential geometry books. It seems to me that people use this word quite loosely. For example, one definition about tangent space in my book is as ...
0
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1answer
40 views

Series representation of cosh or the hyperbolic cosine. [closed]

What is the Series representation of cosh or so-called the hyperbolic cosine? Can you help me guys?
2
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2answers
37 views

Question in relation to Fundamental Theorem of Calculus

I obtain two answers, one is $\dfrac1{\sqrt{1+x^6}}$ and another one is $\dfrac{2x}{\sqrt{1+x^{12}}}$ by using Fundamental Theorem of Calculus, but I am not so sure. Would anyone help me?
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vote
1answer
93 views

Washer method and shell method

(1) Sketch the region enclosed between the curve $y=sin^2x$ and the straight line $y =2x/π$ (2) Find the volume of the solid $S$ obtained by revolving the region in part (1) about the $y$-axis by ...
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2answers
243 views

Partitions of Natural Numbers [duplicate]

This is a question from Complex analysis by Stein. The question is Prove that it is not possible to partition $\mathbb N$ into finitely many infinite AP's with distinct common differences.(other ...
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3answers
56 views

Definite integrals: Calculating Volume

Suppose $D$ is the region in the $xy$-plane bounded by the parabola $y=x^2$ and the line $y=2x$. Find the volume of the solid generated by rotating $D$ about 1) $x$-axis 2) $y$-axis. Are the ...
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1answer
61 views

Calculus: Application of definite integrals

Suppose $a>0$ is a constant. Let $C$ be the curve $y=\cosh x$, for $-a \leq x\leq a$. Let $D$ be the region bounded by $C$, $|x| = a$ and the $x$-axis. 1) Find the length of $C$ 2) Find the area ...
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2answers
69 views

Proving this binomial identity

I'm required to prove the following binomial identity: $$\sum\limits_{k=0}^l {n \choose k} {m \choose l-k} = {n+m \choose l}$$ I tried various arrangements but reached nowhere. Finally I turned to ...
0
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1answer
65 views

Every metrizable space with a countable dense subset has a countable basis

I'm working on this problem from Munkres: Show that every metrizable space with a countable dense subset has a countable basis. Here's my attempt at a proof. Let $X$ be a metrizable space with ...
5
votes
3answers
260 views

Laplace transforms: Convolution

Find $$1*1*1*\cdots*1\quad n\,\,\text{ factors}$$ that is, a function $f(t)=1$ convolution with itself for a total of $n$ factors. Would anyone mind helping me? I have no idea what I should do. ...
3
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2answers
63 views

Two sequences are equivalent. Prove that one is Cauchy iff the other is Cauchy.

This question has already been asked and answered here Let $ϵ>0$ be given. With loss of generality, we may assume $ϵ$ is rational. Suppose $a_n$ is a Cauchy sequence and $b_n, a_n$ are ...
2
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0answers
87 views

Continuity of the right-hand derivative of a Convex function (help with the proof)

Hi everyone I have some trouble with one point in the following proof. Let $f$ be a convex function (strict convex function) on a real interval. If $f'_-(a)=f'_+(a)$ where $f'_-$ and $f'_+$ are ...
2
votes
1answer
2k views

Laplace transformation: second shifting theorem

I know the answer is $1/(s^2) +e^-6s (2/s^3 -14/s -1/s^2 )$, but can anyone tell me how to evaluate the solution? I really get stuck.
4
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0answers
260 views

Isolation and self-study

A little background: I am currently a sophomore (studying mathematics) at an unknown university in the Middle East. My mother is European so it does not make sense to study mathematics in the Middle ...
0
votes
1answer
33 views

Why is this a boolean algebra

Let $A = \{a,b\}$. The $\mathcal P(A) = \{\emptyset,\{a\},\{b\},A\}$. Let $+$ be $\cup$, $\cdot$ be $\cap$, complement be set complement, $1$ be $A$, and $0$ be $\emptyset$. I need to explain why ...
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2answers
66 views

strict midpoint convex $\Rightarrow$ strict convex (help with a proof)

Hi everyone I have trouble with the following I think is something very simple, but I cannot figure out yet the correct approach for the strict inequality If $f$ is continuous and $f$ is strict ...
2
votes
3answers
69 views

CDF of $max(X,X^2)$

$X$ is uniformly distributed on $[-1,1]$. And $Y=max(X,X^2)$. What is $F_{Y}(t)$ , the CDF of $Y$? My attempt: I tried to graph it, but I think I found wrong. I found the joint pdf $5/6$. Is this ...
2
votes
3answers
315 views

Any good introductory book/tutorial on Fourier Transform (up to FFT) with plenty of exercises and solutions?

I wonder what could be a good book to start learning in depth all aspects of the Fourier transform up to the FFT algorithm, and beyond. I am going to dedicate quite some time on the subject, so I ...
0
votes
1answer
21 views

Confused about a proposition about rational polynomial

I'm studing complex analysis.I saw that $R(z)$ is a rational function.Consider the function $R(1/z)$ which we can rewrite as a rational function $R_1(z)$,and set $R(\infty)=R_1(0).$ with the ...
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2answers
76 views

why $\infty$ +$\infty$, $\infty$ - $\infty$ and 0⋅$\infty$ are left undefined.

I'm reading http://en.wikipedia.org/wiki/Riemann_sphere, and having the following question. 1. What's the mean of symbol $\infty$?Is it a surreal number? 2. they write note that ∞ + ∞, ∞ - ...
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0answers
67 views

Create a finite-state machine

I need to create a finite-state machine which accepts strings whose characters are in {a,b,c} and produce output strings of T's and F's. The machine outputs a T once the characters ab is encountered ...
1
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1answer
32 views

A doubt regarding splitting field.

$\Bbb Q(\omega)= \Bbb Q(\sqrt3,\iota)$ This is written in my text book that i am following. But I think this is a typo. Since $\Bbb Q(\sqrt3,\iota)$ is a larger field. in which $\Bbb Q(\omega)$ is ...
0
votes
1answer
35 views

Deriving Stochastic Euler Equation

If a consumer has utility function \begin{equation*} u(c_t) = ac_t - \cfrac{b}{2}c_t^2 \end{equation*} and present value budget constraint \begin{equation*} \sum_{j=0}^\infty E_t[\beta^jc_{t+j}] = ...
0
votes
1answer
74 views

symmetric normalized Graph Laplacian and symmetric normalized Adjacency matrix eigenvalues

I am trying to show that the symmetric normalized Graph Laplacian and symmetric normalized Adjacency matrix have corresponding eigenvalues $\lambda_i$ and $1 - \lambda_i$ for i=1 to n. $\lambda$ is ...
3
votes
2answers
211 views

Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP

I am still struggling quite a lot with the Picard-Lindelöf-Theorem (also known as the Cauchy-Lipschitz-Theorem). Problem: Consider the following IVP with $\alpha \neq 1$ $$\begin{cases} y'&= ...
0
votes
1answer
129 views

Given a second countable space $X$, show $A \subset X$ has uncountable limit points

Here's my attempt at a solution and I'm wondering if it's correct. Let $X$ have a countable basis with $A \subset X$ an uncountable set. Show $A$ has uncountably many limit points. Let $A'$ be the ...
0
votes
1answer
55 views

Why is the set of positive definite matrices in $\mathbb R^{n\times n}$ a positive cone

The set of positive definite matrices in $\mathbb R^{n\times n}$ is geometrically a positive cone. This statement appears in almost every article on real positive definite matrices I read but without ...
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2answers
33 views

Finding a cycle with a specific property

I am reading the book Dummit and Foote - Abstract Algebra . One of the exercises is to find an $n$-cycle $(n \ge 5)$, $\sigma$ such that $\sigma^k = \tau$ for some positive integer $k$, where ...
0
votes
1answer
149 views

Question about $G_\delta$ set

I'm on summer break but I want to keep my math skills sharp so I'm self-studying a bit from Munkres. This question is from pg 194, chapter 4 about the Countability and Separation Axioms. I've ...
1
vote
1answer
53 views

Show that for $f,g: [0, + \infty) \to \mathbb{R}$ convex functions of Class $C^2$ their product $fg$ is convex

Problem: Let $f,g: [0, + \infty[ \to \mathbb{R}$ be two convex functions of Class $C^2$. Assume that $$ f(0) \geq 0, g(0) \geq 0 \text{ and } f'(0) \geq 0, g'(0) \geq 0 \tag{!}$$ Show that $fg$ ...
4
votes
4answers
189 views

Some aspects of inner products in $\mathbb R^3$

I read several descriptions about inner products recently due to an exercise (here it is no longer active so I like to ask again). I am still very confused about this concept. Any clarification will ...
0
votes
1answer
34 views

Differentiability of Norm $N: U \subset \mathbb{R}^n \to \mathbb{R}, \ x \mapsto \sum_{i=1}^n i|x_i|$

Problem: Let $U:= \lbrace x \in \mathbb{R}^n \mid x_i \neq 0 \text{ for } 1 \leq i \leq n \rbrace $ and show that the Norm given by $$ N: \begin{cases} U & \longrightarrow \mathbb{R} \\ x ...
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1answer
32 views

Question on closed sets using a convergent sequence

Intro: The following two questions are from my exam preparation sheet, it is not mandatory and will not be accredited (or improve marks and the like). There won't be a correction, merely an online ...
0
votes
1answer
47 views

Show a simple strategy.

Imagine that we have 49 cards with the values written on their faces, (they are all visible ) as follows; $$25, 24, 23, 22, ........3, 2, 1, 2, 3, .........23, 24, 25$$ suppose Paola and Victor are ...
0
votes
1answer
45 views

Help me writing Payoff matrix.

I guess, in order to answer this question, I need to write Payoff matrix. But I cannot write it. And then, I Will able to answer this question by myself. Thank you for helping. (These are just ...
2
votes
1answer
69 views

Check my answers: Dominant strategy.

I saw another question on Game theory. My answer for part a the nash equlibria (T, L) and (B,R). for part-b, Player-1's action T is strictly diominated. So Player1 never choose T. For part ...
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0answers
50 views

Is every curve measurable? [duplicate]

Does there exist a function $f:\mathbb R \rightarrow \mathbb R$ such that the set $E=\{(x,f(x)\mid x\in\mathbb R\}$ is non-measurable in $\mathbb R^2$?
0
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1answer
87 views

A question on Game theory

I'm studying Game theory, I saw the question: Consider two players; player A and player B playing the following estimation game. Each player chooses a number from {1, 2, 3}. If the difference ...
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0answers
80 views

Continuous transition kernel for Markov Chains

I am trying to show that the stationary distribution for a Markov Chain on a continuous state space can be obtained by building a transition density kernel, which obeys the detailed balance rule where ...
0
votes
1answer
25 views

Minumum value of integral and of partition are eventually the same?

I am trying to compare the minimum value of the integral of some function $f$ over the interval $[a,b]$ to the minimum value of the sum of the rectangles up to a certain point in a partition of ...
5
votes
2answers
164 views

Easy exercise (hint) Real Analysis

I've been stuck for a while with this problem. I suppose is something very easy, but I cannot figure out yet the correct approach. I'd really appreciated not a complete solution just some hints ...
0
votes
2answers
186 views

Measure Theory or Set Theory?

Having taken Real Analysis I before (the seven first chapters of baby Rudin) I have the option to take Measure Theory now. However I am torn between that and Set Theory. Which course would you go for ...
0
votes
1answer
50 views

Find $a,b,c\in \mathbb{R}$ that minimize the integral $\int_0^\infty |t^3 -at^2 -bt-c|^2e^{-t}$dt

Find $a,b,c\in \mathbb{R}$ that minimize the integral $\int_0^\infty |t^3 -at^2 -bt-c|^2e^{-t}$dt Hint:Use orthogonality of $(P_n)_{n=0}^\infty$ in $H=L_{2,\rho}(\mathbb{R}_+)$ with $\rho(t)=e^{-t}$ ...