1
vote
1answer
18 views

Local maxima when multiplying two functions

I have two functions, $f(x)$ and $g(x)$, where: $f(x) = \frac{1}{x^2 + a}, a>0$ and the only things I know about $g(x)$ are that: $g(x) > 0, \forall x \in \mathbb R$ $g(x)$ is a polynomial ...
0
votes
1answer
220 views

Coming up with an alternative proof by induction

Kindly refer to Q4 of this handout. "$2n$ dots are placed around the outside of the circle. $n$ of them are colored red and the remaining $n$ are colored blue. Going around the circle clockwise, you ...
0
votes
1answer
316 views

complex numbers find greatest value of z

I've to sketch the complex number $z$ such that it satisfy both the inequality $|(z-2i)|\le2$ and $ 0\le \arg(z+2)\le 45\deg $ I was able to sketch and shade the region that satisfies both ...
1
vote
1answer
96 views

Proving that if $\sum\|f_n-e_n\|^2< 1$, $\{f_n\}$ is a complete sequence

Let $\{e_n\}$ be a complete orthonormal sequence in an Hilbert space $H$ and let $\{f_n\}$ be an arbitrary sequence of elements in $H$ s.t $$\sum_{n=1}^\infty\|f_n-e_n\|^2<1$$Show that ...
0
votes
0answers
52 views

Computational cost of solving $Ax_i = b_i$ for $i=1,…,m$

$A$ is an invertible matrix square $n$ matrix. The exercise is about 3 different ways you can solve this and I have to determine its efficiency. It's always the same matrix $A$ but a different right ...
5
votes
0answers
32 views

(Topological Groups) Show that $g_{\alpha}(x) = x *\alpha$ is a homeomorphism of G.

Note: This is not homework. Can someone please verify my proof or offer suggestions for improvement? Let $\alpha$ be an element of a topological group $G$. Show that the map $g_{\alpha}: G ...
0
votes
1answer
45 views

Third isomorphism Theorem and Free groups

Let $F(a,b)$ be the free group on the set $\{a,b\}$, and let $\left<\left<\{aba^{-1}b^{-1}\}\right>\right>$ and $\left<\left\{aba^{-1}b^{-1},a\}\right>\right>$ be the normal ...
1
vote
1answer
168 views

Primary decomposition of $(XY,(X-Y)Z)$ in $k[X,Y,Z]$

How to find the primary decomposition of $I=(XY,(X-Y)Z)$ in $R=k[X,Y,Z]$? It has minimal primes $(x,y),(y,z),(z,x)$. I tried to calculate $J=S^{-1}I\cap R$, where $S=R-(x,y)$, but it seems ...
1
vote
1answer
133 views

Metric topology induced by the sum of two metrics

I have to show the following: Let $X$ be a set with metrics $d_1$ and $d_2$ inducing metric topologies $\tau_1$ and $\tau_2$. Define a new metric on $X$ where $d(x,y) = d_1(x,y) + d_2(x,y)$ for ...
4
votes
4answers
137 views

What are non-obvious examples of measures obtained from linear functionals by the Riesz representation theorem?

In chapter two of Rudin's "Real and Complex Analysis" there is a "Riesz Representation Theorem" that dominates the chapter. My understanding of the statement of the thm. is that given a complex-valued ...
2
votes
2answers
163 views

Origin of the words arithmetic and geometric progression

Why are arithmetic progression and geometric progression called arithmetic and geometric respectively?
0
votes
2answers
53 views

In a metric space with a countable base, how does every open cover has a countable subcover?

Let $X$ be a mertic space, and let $\{V_{\alpha}\}$ be a collection open subsets of $X$ such that, for every $x \in X$ and for every open set $G \subset X$ with $x\in G$, there is some $V_\alpha$ such ...
4
votes
1answer
41 views

Find the largest value of x given the equation…

Find the largest value of $x$ for which $x^2 + y^2 + z^2 = x + y + z$. What I did was subtract the RHS, to get $$x^2 - x + y^2 - y + z^2 - z = 0$$ $$x^2 - x + \frac{1}{4} + y^2 - y + \frac{1}{4} + ...
0
votes
2answers
90 views

When does Mean Square Error increase?

As far i know, we want the model to include as few regressors as possible because the variance of the prediction $\hat y$ increases as the number of regressor increases. But from the hald cement ...
0
votes
1answer
65 views

How are the real numbers complete?

The completeness axiom for the real numbers states that any subset of the reals that is bounded above has a supremum. But if we take an example: Completeness of the real numbers - Least upper bound ...
1
vote
1answer
121 views

(compact, non-empty boundary )Surface Geodesics on Hyperbolic Geometry

I have a basic knowledge of hyperbolic geometry . I am trying to understand the meaning of "a compact surface S with non-empty boundary(which is neither a disk nor an annulus )with a complete ...
0
votes
1answer
31 views

Proving for each seperatble hilbert space exist complete sequence

Let $H$ be a separable Hilbert Space. Prove that exists orthonormal complete sequence and give example for one non-orthonormal sequence. I thought taking orthonormal basis for $H$ denoted by ...
1
vote
1answer
286 views

Proving an equivalence relation(specifically transitivity)

I'm currently learning about equivalence relations. I understand that an equivalence relation is a relation that is reflexive, symmetric, and transitive. But I'm having trouble proving the transitive ...
0
votes
2answers
226 views

Probability of picking balls out of bins

Question: You have two bins with four different balls in each bin. Bin A: 2 White Balls and 2 Black Balls Bin B: 3 Black Balls and 1 White ball You cannot tell which bin contains what balls. Given ...
1
vote
1answer
64 views

Integration by Residue

How do I evaluate $$\int_{C(0,e)} \frac{1-\cos z}{(e^z-1)\sin z}dz.$$ This looks simple but I have a hard time to find the residue at $z=0$. At $z=0$ the function say $f(z)$ is even undefined. I ...
1
vote
1answer
105 views

Finding the probability of exactly one event in a series of independent events, why used (-1)?

Learning programming and trying to understand this example. Given multiple independent events, each with a probability of occurring, what is the probability of just one event occurring? If we have ...
4
votes
1answer
88 views

When does pairwise independence imply independence?

We know if a collection of events are independent, then they are pairwise independent. In general, the converse is not true. However, I'm wondering if there's a condition under which the converse ...
0
votes
1answer
154 views

A continuous nonconstant function has uncountable range

How would I prove that if I have a function $f$ on $\mathbb R^1$, that is continuous and non constant, that its range is non countable? Here's my thought. Let $f$ be a continuous, non constant ...
4
votes
0answers
555 views

Mean Absolute Deviation for a Stable Distribution as a Function of the Tail Exponent

Consider the standard Lévy-Stable (or Alpha Stable) distribution $S(\alpha,\beta, \mu, \sigma)$ where $\alpha$ is the tail exponent, $1 \leq \alpha \leq 2 $. Picking the symmetric case with $0$ mean ...
4
votes
2answers
91 views

How to calculate the value of the special integral

I get $${\left. \frac{\partial ^2}{\partial n^2} \left( \frac{\partial ^2}{\partial m^2} B(m,n) \right) \right|_{m = \frac{1}{2},n = 0}} = \int_0^1 \frac{\ln^2 x \ln^2 (1 - x)}{\sqrt x (1 - x)} \, dx ...
3
votes
4answers
290 views

Simplification of geometric series.

Can someone please help simplify this series? $$\sum_{k=1}^\infty k\left(\frac 12\right)^k$$ In general, $$\sum_{k=1}^\infty\left(\frac 12\right)^k = \frac{1}{1-\frac{1}{2}} =2.$$ However, I am ...
2
votes
2answers
262 views

Inhomogeneous Wave Equation Derivation

This is an assignment question which I've been working on to solve the inhomogeneous wave equation $u_{tt} - c^{2}u_{xx} = f(x,t)$. I separated the equation out into a system of two equations: $u_{t} ...
5
votes
1answer
179 views

Determining a consistent estimator/asymptotic relative efficiency

Question: Let $X_1,\ldots,X_n$ be i.i.d. as $N(0,\sigma^2)$. a) Show that $\delta_1 = k \sum_{i=1}^n |X_i|/n$ is a consistent estimator of $\sigma$ if and only if $ k = \sqrt{\pi/2}$. b) Determine ...
13
votes
2answers
581 views

Extract real and imaginary parts of $\operatorname{Li}_2\left(i\left(2\pm\sqrt3\right)\right)$

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g. ...
0
votes
3answers
109 views

How do you solve this equation?

$$\sqrt{x + \sqrt{x + \sqrt{x + \cdots}}}= 5$$
1
vote
0answers
70 views

Limit definition question

Just curious, the definition of a limit is: For every $\epsilon\gt0$, there exists a $\delta\gt0$, such that for every $x$, the expression $0\lt|x-c|\lt\delta$ implies $|f(x)-L|\lt\epsilon$. Is ...
4
votes
0answers
156 views

Exact values of error function

The error function is defined as $$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$ We know that the Gaussian integral is $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
0
votes
1answer
59 views

If $P$ is a projection operator, is $1-P$ also a projection operator?

Show that if $P$ is a (hermitian) projection operator, so are (a) $1-P$ and (b) $$ U^{+}PU $$ for any operator $U$
1
vote
1answer
86 views

The solution of $\Delta u=u^3$ with zero boundary values is identically zero

My question: My attempt: I tried to use the Representation using Green's formula: Since $u=0$ on the boundary and $f(x)=x^3$, then the formula becomes: $$u(x)=\int_\Omega y^3G(x,y)dy \quad ...
2
votes
2answers
226 views

If $AB = I$, the identity matrix prove $\mathrm{rank}(B)$

Let $A$ be an $m \times n$ matrix and $B$ be an $n \times m$ matrix. Show that if $AB = I$, where $I$ is the identity matrix, then $\mathrm{rank}(B) = m$. I'm not exactly sure how to start this ...
2
votes
2answers
81 views

how to solve $\log{x}=cx^4$ for $x$

I was wondering if there is a general solution for this form of equations: $$\log{x}=cx^4$$ Tried: $$ x = e^{cx^4}\\ xe^{-cx^4}=1$$
1
vote
3answers
177 views

determine whether $f(x, y) = \frac{xy^3}{x^2 + y^4}$ is differentiable at $(0, 0)$.

I am new to multivariable calculus and my textbook doesn't give out solutions so I'm just wondering how you go about proving something like this? I know that a function is differential at a point $a$ ...
1
vote
1answer
59 views

proof of DCT with weak condition(almost everywhere)

I have a question about a proof of the dominating convergence theorem, with weak requirements. Before I show the proof from the book, note that in my book you are allowed to integrate functions that ...
0
votes
1answer
57 views

How do you determine where a polynomial evaluates to a perfect square?

How do you determine where a polynomial evaluates to a perfect square? One example would be $f(x)=x^2+148x-288$. $f(68) = 14400 = 120^2$. Another one would be $f(x)=x^2+204x-88$. $f(2) = 324 = ...
2
votes
1answer
32 views

Matching moments implies matching densities?

If $X$ and $Y$ are random variables with matching moments (ie: $\mu_X^i = \mu_Y^i (\forall i \in \mathbb{Z}^+)$ then are the density functions of $X$ and $Y$ identical (almost everywhere)? Idea: I'm ...
1
vote
1answer
40 views

Convexity problem

Let $S$ be a convex set. If $x\in$ int$S$ and $y\in$ cl$S$, show that relint[x,y] $\subset$ int$S$. I easily proved this for a case where y is in the interior of S, but am stuck if y is in the ...
2
votes
1answer
192 views

Closed-form of $\int_{a}^{b}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx$ for some $a<b$

In this question I asked to prove that $$\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}.$$ If we take a look at the plot of the integrand, then we could see some symmetry-property. ...
2
votes
1answer
143 views

Extremal set theory problem

What are good bounds(asymptotic bounds preferred) on the cardinality of the largest family $S$, of $m$-element subsets of an $n$-element set, if any pair of elements intersect in a set that has ...
0
votes
1answer
39 views

Solve the following derivative through its definition

We have a function $$f(x) = \frac{5}{\sqrt{x} + 1}$$ and its definition states that $$f'(x) = \lim_{x \to 0}\frac{f(x+h)-f(x)}{h}.$$ Therefore, I attempted it by computing the following $$\lim_{x ...
2
votes
0answers
137 views

Clarification: Rudin Theorem 3.7: Subsequential limits are closed

My question is this: Why does Rudin use $\delta$ in this proof? Would it not work just as well if $\forall i \ge1,$ $$x_{i}\in N_{2^{-(i+1)}}(q) \cap E^* $$ $$p_{n_i}\in N_{2^{-(i+1)}}(x_i) ...
0
votes
1answer
113 views

Calculate winner of soccer match

I am writing a program that simulates a soccer tournament between countries using their FIFA rankings. I am looking for a function that takes two country rankings and outputs a number between (about) ...
0
votes
2answers
105 views

A fair coin is flipped until the first tail appears, in general we win \$ $2^k $. St. Petersburg problem.

For the St.Peterburg problem (Example 3.5.5), find the expected payoff if (a) the amounts won are $c^k$ instead of $2^k$, where $0 < c < 2$. (b) the amounts won are $\log(2^k)$. The original ...
0
votes
1answer
25 views

Maxima and Minima, If s = 60, what should the side of the cut out be…

A square piece of steel, s cm on a side, is to made into an equipment chassis by cutting equal squares out of the corners, folding up the sides , and welding the seam to form a pan. $A)$ If $s=60$, ...
2
votes
2answers
212 views

Notation involving the Lebesgue integral.

I have a measurable function $f : \mathbb{R}^d \to \mathbb{R}$. Let $E$ be a measurable subset of $\mathbb{R}^d$. Then then $$\int_{E} f(x) \, dx = \int f(x) \chi_E (x) \, dx.$$ If we are taking an ...
0
votes
2answers
186 views

Correct notation for union of all elements in a set?

Say I have a set $H$ and I want to describe the union of all elements in $H$. How would I write that? I believe I've seen a big U with a subscript used before.

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