-4
votes
1answer
36 views

Volume of water [on hold]

Please Calculate volume of water in a sphere container with radius r that is filled with water up to the height h.
1
vote
2answers
29 views

Finding polynomial optimal in terms of least squares approximation

Find polynomial $w$ of degree at most $2$ optimal in terms least squares approximation for a function $f(x)=x^3$ in the norm $\|g\|=\sqrt{(g,g)}$, given that: $$ (f,g) = \int\limits^1_0 f(x)g(x)dx. ...
3
votes
0answers
29 views

Notion of a distribution as acting on tangent spaces

I'm reading a paper that uses distributions in a way I'm not entirely comfortable with. To be precise, I'm not sure what definitions the author is working with and can't find any natural way to fill ...
0
votes
1answer
11 views

A simple function with no tangential limits but with non-tangential limits

I am going to be teaching a course about the Hardy space and I would like to show the students that the non-tangential limit is a necessary concept BEFORE telling them about Blaschke products. Is ...
0
votes
2answers
20 views

Calculating righ/left sided limits

For example, limit of cos(x)/sin(x) as x approaches pi from right/left... how does one do it, rigourously, instead of just evaluating the function at x = something just to the left of Pi, and x = ...
2
votes
1answer
14 views

Question about Hilbert transform(applying plancherel theorem)

Let $f\in S(\mathbb{R})$(Schwartz function on real line). Then Hilbert transform $H$ of $f$ is defined by $\displaystyle Hf(x)=\lim\limits_{t\rightarrow0}\int_{|y|>t}\frac{1}{y}f(x-y)\,dy$ One ...
28
votes
1answer
552 views

Has SGA 4$\frac 1 2$ been typeset in TeX?

The title says it all. I've CW'd the question since I'm answering it - this seemed like the best way to get the news out.
0
votes
0answers
7 views

Prove $\int s d \mu = \sum^n_{j=1} a_j 1_{A_j}$ for $\sum^n_{j=1} a_j 1_{A_j}$ not a standard representation of $s$.

Let $(X, \mathcal E, \mu)$ be a measure space. Let $s \in \mathcal S\mathcal M(\mathcal E)^+$ be a simple function written as $s= \sum^n_{j=1} a_j 1_{A_j}$ , $a_j \ge 0, A_j \in \mathcal E$. I want ...
0
votes
0answers
28 views

remarkable relation between fibonacci numbers and its squares!

there is a remarkable relation between fibonacci numbers and its squares:$F^{2}_{n} +F^{2}_{n+1}=F_{2n+1}$. I know the proof using $F_{n}=\frac{\sqrt{5}}{5}((\frac{1+\sqrt{5}}{2})^n ...
1
vote
1answer
46 views

Exercise 31 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

The following problem shows that $\{e^{inx}\}_{n \in \mathbb{Z}}$ is an orthonormal basis of $L^2([-\pi, \pi])$. It is taken from [1] (exercise 31 of chapter 4: "Hilbert Spaces: An Introduction", pp. ...
1
vote
1answer
220 views

Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
0
votes
1answer
24 views

Inner Product in Hilbert Space

Let $H$ be a Hilbert space and $\phi_{1}, \dots, \phi_{n} \in H$ are linearly independent vectors. How can we construct the inner product on $H$ such that $\phi_{1}, \dots, \phi_{n}$ become orthogonal ...
0
votes
0answers
31 views

How to solve these equations?

How to solve these equations where $\gamma$ and $\mu$ are constants? $$ab^{\gamma}=de^{\gamma},\frac{(ab)}c=\frac{(de)}f=\frac{(dg)}h=\mu $$ in a form: $$f(a,c,d,h,\gamma,\mu)=0$$ I always end up ...
0
votes
0answers
20 views

Solving Black scholes PDE using Laplace transform

I'm trying to obtain the Laplace transform of Call option price with repect to time to maturity under the CEV process. The well known Black scholes PDE is given by $$ ...
2
votes
0answers
106 views
+50

Proof of Laplace expansion using minors

I've come across with the following proof of the Laplace expansion: Let $\Delta=\sum_{j=1}^n (-1)^{1+j} a_{1j}\bar M_j^1$ and $\tilde{\Delta}= \sum_{j=1}^n (-1)^{i+j} a_{ij}\bar ...
0
votes
0answers
8 views

Normal distribution around and extreme value

I'm trying to create an artificial dataset with users, items, and ratings given by the users to the items. Creating the dataset, I pick the average rating for every item randomly, and let the ratings ...
1
vote
1answer
27 views

Question concerning a sequence in GAP

I would like to know, what's the best (fastest) way to programm the following in GAP (perhaps using some functionality from the QPA package): Input: $n\geq 2$ Output: A list of all sequences ...
0
votes
4answers
23 views

Is functions of cauchy sequences is also Cauchy?

Recently i saw in some book that if a sequence is Cauchy then function of that sequence is also Cauchy.I have confusion about this. Please help me.
1
vote
1answer
35 views

A doubt from Milnor's “Topology from a Differentiable Viewpoint”.

This is a doubt from Milnor's "Topology from a Differentiable Viewpoint". For a smooth $f:M\to N$, with $M$ compact, and a regular value $y\in N$, we define $n(f^{-1}(y))$ to be the number of ...
0
votes
1answer
25 views

local dimension of irreducible varieties

If $X$ is an irreducible (quasi-)affine variety it is well known that each maximal sequence $C_0 \subsetneq C_1 \subsetneq \dots \subsetneq C_d$ of irreducible closed subsets has the same length $d = ...
0
votes
3answers
27 views

Modular calculus and square

I want to prove that $4m^2+1$ and $4m^2+5m+4$ are coprimes and also $4m^2+1$ and $4k^2+1$ when $k\neq{m}$ and $4m^2+5m+4$ and $4k^2+5k+4$ when $k\neq{m}$. Firstly : Let $d|4m^2+1$ and $d|4m^2+5m+4$ ...
2
votes
2answers
20 views

Force non-consecutive colours for pie chart

Background Calculating colours for pie chart wedges. Consider: $$ \begin{align} d(n)&=\frac{\theta}{t}\times n\\ \end{align} $$ Where: $\theta$ is the degrees in a circle (360) $t$ is the ...
1
vote
1answer
26 views

$S^1$ with length metric is not isometric to any subset of Euclidean plane (metric given by restriction)

Let $S^1$ denote point whose radius is 1 from the center. Metric is given by distance between two point is the shortest distance, that is the length metric. Prove that $S^1$ with this metric is not ...
1
vote
1answer
26 views

Sequence, Monotone Convergence Theorem.

Suppose that $x_0 \geq 2$ and $x_n = 2 + \sqrt{x_{n-1} - 2}$ for all natural $n$. Use the Monotone Convergence Theorem to prove that either $x_n \rightarrow 2$ or $x_n \rightarrow 3$ as $n$ grows. ...
1
vote
1answer
133 views

True or False. Convergent subsequence

Is the statement true or false? If $(x_n)$ has a convergent subsequence,then $(x_n)$ is bounded. The statement is False. However, can someone please show me an example of a sequence with ...
0
votes
1answer
42 views

I'm new to proving inequalities. How does one prove this?

If a, b, and c are non-negative real numbers and $a + b + c = 2 $, prove that $ 2 \ge a^2 b^2 + b^2 c^2 + c^2 a^2 $
1
vote
2answers
25 views

Generating function for number of integer solutions, no computer

Use a generating function to solve the number of integer solutions for $$x_1+x_2+x_3=17$$ Where $2\leq x_1 \leq 5, 3\leq x_2 \leq 6, 4\leq x_3 \leq 7$ Now all this takes is doing: ...
1
vote
1answer
26 views

How to get the ratio from a function of N?

The exercise gave us a chart which showed the running time as a $N$ increases: \begin{array}{c|c} N & \text{seconds}\\\hline 256 & 0.000\\ 512 & 0.000\\ 1024 ...
0
votes
1answer
12 views

Max no. of piece in k cut

Suppose I have large piece of rectangular sheet. Cutting is allowed only vertically and horizontally. My approach is if no. of cut is even then max. no of piece is (n/2)*(n/2) if no of cut is odd ...
0
votes
1answer
19 views

Unique extrema of sum of monotonically increasing and decreasing functions on an interval

If I have two functions, f and g, defined on the interval [0, 1] with both f and g non negative (i.e. f(x), g(x) >= 0) f(x) is monotonically increasing, while g(x) is monotonically decreasing. and ...
0
votes
3answers
76 views

Sum of the infinite series $\frac16+\frac{5}{6\cdot 12} + \frac{5\cdot8}{6\cdot12\cdot18} + \dots$

We can find the sum of infinite geometric series but I am stuck on this problem. Find the sum of the following infinite series: $$\frac16+\frac{5}{6\cdot 12} + \frac{5\cdot8}{6\cdot12\cdot18} + ...
1
vote
1answer
24 views

Operator induced by continuous function and measures

If $X$ is a compact metric space, and $T:X \rightarrow X$ is continuous map, what would be meant by $T_\ast$ is the operator on measures induced by $T$? Allow $\mu$ to be some Borel regular normed ...
-2
votes
0answers
19 views

Gauss curvature K in polar coordinates

EDIT: A surface is given in Monge's form: $z=f (x,y)$ the partial derivatives of $z$ are.. $$ p = \frac{\partial z}{\partial x}, \; q = \frac{\partial z}{\partial y}, \; ...
1
vote
0answers
24 views

The number of Balanced Boolean functions

Suppose we have n-variable Boolean function (BF) and we know that the weight of a Balanced BF is $2^{n-1}$. The total number of BFs are $2^{2^n}$, Affine BFs are $2^{n+1}$ and Linear BFs are $2^n$. In ...
0
votes
2answers
35 views

How to determine the orbits of points under the tripling map $f(x)=3x\bmod 1$?

Let $f$ be the tripling map $f(x) = 3x \mod(1)$. Determine the complete orbit of the points $\frac{1}{8}$ and $\frac{1}{72}$. Indicate whether each of these points is periodic, eventually periodic, or ...
19
votes
4answers
1k views

Evaluating $\int_0^{\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx$

I need to solve $$ \int_0^{\Large\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx $$ I tried to use symmetric properties of the trigonometric functions as is commonly used to compute $$ ...
1
vote
1answer
17 views

Vector Analysis (Parametized curve)

The question is find a familiar parameterized curve that has the property $r(t) \times\dfrac{dr}{dt}=0$. The only curve that I can see that works is the line through the origin. I was just wondering ...
1
vote
1answer
35 views

Pulsating waves of zeta function

Below is an animation of the partial sums of $\operatorname{li}x-2\Re\sum_{k=1}^{N}\operatorname{Ei}(\rho_k \log x)-\log2+\int_{x}^{\infty}\dfrac{\text{d}t}{t(t^2-1)\log t},$ for $1\leq N\leq100$ ...
0
votes
1answer
28 views

Probability: student passing an exam by randomly guessing (no calculator)

Assuming you can't use a calculator, how do you estimate the answer to the following problem? Suppose an exam has 40 questions, all multiple choice. Each question has 5 choices and you need 20 ...
0
votes
1answer
20 views

Probability of Fair Die

Hi everyone I have a question about probability: Fair die thrown two times, final score is calculated as follows. If the number on the second throw is a 5 he multiplies the two numbers together, and ...
4
votes
1answer
275 views

Rank of matrix as a difference of ranks

If $X$ is an $n \times p$ matrix of rank $r$ and $C = AX$ for some $q \times n$ matrix $A$ with rank$(A) = q$, how do I show that rank $(X(I-C^{-}C))=$ rank$(X)-$ rank$(C)$? I can show that rank ...
5
votes
5answers
47 views

Is $\{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\}$ open or closed?

I am trying to figure out if the set $\{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\}$ is open or closed or maybe none of that. I hope someone could provide a hint to solve this. Can this set be ...
0
votes
2answers
26 views

every irreducible polynomial has a root in some field extension

We know the following fact from field theory. Let $F$ be a field and $p(X)$ an irreducible polynomial in $F[X]$. Then we can find a field extension $L$ of $F$ such that $p(X)$ has a root in $L$. ...
2
votes
0answers
81 views
+50

What is the smallest possible angle of this polygon?

A convex polygon contains a square with side-length 1 and is contained in a parallel square with side-length 2 (which is its smallest containing square). What is the smallest possible angle of the ...
1
vote
0answers
10 views

How to visualize probability distributions in terms of sets - joint and marginal?

Let there be two sets, $\mathcal{X},\mathcal{Y}$, both finite, and they represent the set of values that the discrete random variables, $X,Y$ can take. $\mathcal{P}_{Y|X}$ be all possible ...
-1
votes
3answers
37 views

Logic problem?? [on hold]

When Paul's age in years and Evan's age in years are written down one after the other, they form a four digit number. Each of them are over 10 years old. The resulting number is a perfect square. In ...
2
votes
1answer
27 views

If any two norms on a vector space are equivalent then the space is finite-dimensional [duplicate]

I need to prove: If any two norms on a vector space are equivalent then the space is finite-dimensional. I am aware of the converse of this result that on a finite dimensional vector space any two ...
0
votes
0answers
43 views

numerical method (Implicit) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$ I need a numerical method (implicit , backward difference or forward difference) for estimate $A$ in this nonlinear PDE: $$ A_t + \mu(\lambda -\lbar ) ...
0
votes
1answer
25 views

Lipschitz functions and its closure

I came across the following statements in a math book without proof. Denote $M_k$ as the set of functions from $C[a,b]$ that is K-Lipschitz continous. i.e $\forall x,y,|f(x)-f(y)|\le K|x-y|$ 1) The ...
6
votes
1answer
131 views

invariant subspace of a Hardy space

Let $T$ be the unit circle and $H^1=\{f\in L^1(T): \int_0^{2\pi} f(e^{it})\chi_n(e^{it})dt=0 \text{ for } n>0\}$ where $\chi_n(e^{it})=e^{int}$. Let $M$ be a closed subspace of $H^1$. Then ...

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