-2
votes
1answer
293 views

Another way of extending the Banach-Tarski paradox?

This question is kinda a follow-up on Extending Banach-Tarski paradox? On a sphere, we can do all kinds of translations. We will, as usual, look to the translations that are a string of two ...
0
votes
0answers
47 views

Relation between hyperbolic numbers and hyperbolic functions

Is there any relation between the hyperbolic (split-complex) numbers and hyperbolic trig functions? Or are they just named similarly by accident?
0
votes
1answer
14 views

Simple questions about a polynomial ring

Reading Pinter's algebra, I'm little bit confused. In ch.24, the author says that x which appears in a polynomial is to be considered as a 'placeholder' for a moment... All right, then i was trying ...
0
votes
1answer
17 views

Determine uniform convergence of series

I have problems with checking if series $\displaystyle S_n(x)= \sum _{n=1} ^{\infty} \frac{1}{n(1+(x-n)^2)}$ is uniform convergent at $(0 , +\infty)$ My try: consider $\displaystyle |S_n(x) - ...
0
votes
0answers
20 views

How can I prove this statement about square root?

Introduction In computer science there is a field called Formal Methods and Specifications. In this field software designers design softwares by specifying their functionalities in formal methods, ...
1
vote
1answer
16 views

A Banach space in between $L^{1}$ and $L^{2}$, does it make sense?

Let $L^{p} (A, B)$ be a collection of functions $f:A \mapsto B$ satisfying $$(\|f\|_{p})^{p} := \int_{A} |f(x)|^{p} dx <\infty.$$ Now we consider functions $f:[0,1]^{2} \mapsto [0,1]$. We say ...
0
votes
0answers
7 views

Functional equation $f(x)=f(x-1)+f(x-a)$

Please help to solve functional equation for real numbers, we have $a>1$ and $f(x)=1$ if $x<a$ $f(x)=f(x-1)+f(x-a)$ for $x>=a$
0
votes
0answers
4 views

Banach Tarski Notation

Okay, I think I have a full notation and the rules of it how to extend the Banach-Tarski Paradox to an abritary number of cutoffs, as I introduced in Another way of extending the Banach-Tarski ...
-1
votes
1answer
10 views

Singularities and Residue

For part (a) the singularity is 1/root2 + i/root2 ? And it is a pole of order 1? I am having trouble calculating the residue So far I have: residue = limit (as z tends to 1/root2 + i/root2) of ...
0
votes
1answer
16 views

Conditional expectation of random variable

I have this home assignment in Introduction to Probability, and I'm not comfortable with definitions and heuristics. I really need someone to check if I'm even in the right direction. The question: ...
0
votes
0answers
2 views

Weighted Decision Making on the basis of Two Significance Indexes

I'm trying to make the best decision for assignment of books(some of the books have multiple authors that belong to different organizations).Only one author can be represented by a single book. I have ...
1
vote
3answers
11 views

A Linear Operator of Rank 1

Let $T$ be a linear operator with rank $1$ on a finite dimensional vector space $V$.Then Which of the following are true? 1)either $T$ is diagonalizable or $T$ is nilpotent. 2)$T$ is both ...
1
vote
5answers
50 views

Help With SAT Maths Problem (Percentages and Numbers)

I usually solve SAT questions easily and fast, but this one got me thinking for several minutes and I cannot seem to find an answer. Here it is: In 1995, Diana read $10$ English and $7$ French ...
0
votes
1answer
22 views

uniqueness of identity of a group G

The theorem for uniqueness of identity of a group says there is one identity element $e$ in a group and this element $e$ is unique. My book states the proof as follows: $a.e=a$ for all $a \in G$ and ...
0
votes
0answers
12 views

Let $X\subset \mathbb{R}$ be Borel measurable. Can it be that $\aleph_0 <|X|<2^{\aleph_0}$?

I want to know if every Borel measurable set in the real line has cardinality either that of the naturals or of the reals. Of course the Continuum Hypothesis is not assumed. It is clear that every ...
2
votes
1answer
5 views

Equation for sinusoidal wave with fixed wavelength and amplitude

I am a programmer. I am writing a program in which I need to show a graph plotted to the user when the user adjusts two sliders, which are the amplitude and wavelength of the wave, say...
0
votes
0answers
4 views

Hydraulic Engineering

How do you calculate the water capacity of a dam, given the surface area of the dam, and the height? Also, the flow rate to the dam is given for 5 days. How do I calculate the water capacity for each ...
2
votes
1answer
31 views

Borel regular measure: Approximate any measureable set by compact sets

Let $(K,\mathcal{F},\mu)$ be a measure space. Let $K$ be a compact Hausdorff space and $\mu$ be a regular finite measure. We said that it is regular if $\mu(A) = \inf\{\mu(B): B \text{ open }, ...
0
votes
0answers
2 views

Finding the Optimal Function Composition to Maximize Goodness of Fit

Assume I have one function $f(x)$, which optimally models empirical data in the range $[x_0, x_n]$ and a second function $g(t)$, which is optimal for the range $[x_{n+1}, x_{max}]$. $^1$ The function ...
-1
votes
1answer
26 views

Applications of set theory

We know that science and specially mathematics are based on the set theory. But, I would like to know some direct applications of set theory for computer science and engineering. For example, is there ...
0
votes
1answer
18 views

Show the following including number of divisors d(n)

I know how to show that $(d ∗ \mu)(n) = 1$ for all n ≥ 1.But.. I have two solutions. Firstly... result is trivial, because $d = 1 ∗ 1$ Secondly We know that both sides are multiplicative. Thus it ...
2
votes
1answer
29 views

How to prove the sign test

Please correct me if I'm wrong, but a version of the sign test assumes under $H_0$ that there is some distribution $F$ where $X_i \sim F, Y_i \sim F$ and $X_i, Y_i$ are iid. Then it states that $T = ...
2
votes
1answer
596 views

expectation and sign of the Radon-Nikodym derivative

I am new here. I have some questions about the Radon-Nikodym derivative. I hope someone is willing to help me with these. The questions are stated below. Also I added my attempts to the problem to ...
3
votes
1answer
38 views

Non-compactness of the resolvent

Consider a complete non-compact Riemannian manifold $M$ and the resolvent of the Laplacian $(-\Delta + \lambda I)^{-1}$. It is known that the resolvent is in general not a compact operator. I am ...
0
votes
2answers
11 views

CDF of a Uniform probability density function

I want to find Cumulative distribution function (CDF) of the following density function: $ f(x)= \begin{cases} 3/20 & \text{for } 2 \leq x \leq 4 \\[8pt] 4/20 & \text{for }4 < x \leq ...
0
votes
0answers
8 views

Find Taylor-Maclurine expansion of function

Find Taylor-Maclurine expansion of function: $f(x)=\sin(x)\cdot \cos(x) \cdot \arctan x^2$ my try: $f(x)=\frac{1}{2}\sin{2x} \cdot \arctan x^2$ and we have $\displaystyle \sin{2x} = \sum _{n=0} ...
1
vote
4answers
42 views

What are some simple examples illustrating the definition of “cover”

In my class the word "cover" is used very informally such as this set covers another set (this is for a class in PDE not topology by the way). Can someone provide a trivial example of cover to get ...
0
votes
0answers
10 views

The Coproduct of two spaces is the same as the disjoint union and is homeomorphic to the union when the spaces are disjoint

Let $X$ and $Y$ be two topological spaces. Consider the set $$X \cup Y = \{ x\; |\; x \in X \text{ or } x \in Y\}$$ In all the following I suppose that $X$ and $X$ are disjoint. I want to ...
0
votes
0answers
12 views

asking about the meaning of a notation in a 1942 article

In this article http://projecteuclid.org/euclid.bams/1183505105 what is the meaning of the notation $YX\wedge$, that is what does it mean when you write the notations for two sets together side by ...
0
votes
0answers
6 views

Will step-size $\frac{1}{n}$ work in the proof of approximation of measurable function by simple function

What will be the problem if I divide the range of the real measurable function $f$ by a step size of $\frac{1}{n}$ instead of $\frac{1}{2^n}$ in the proof of its approximation by simple function. I am ...
-1
votes
0answers
7 views

Doesn't axiom of choice only apply to countable many groups?

The process of selecting an object from each group seems to be similar to counting. It doesn't make sense to apply it to uncountable many groups.
0
votes
0answers
2 views

Do you know how get differential equations of HSIR model of propagation malware?

I have differential equations but I don't know how get it? thank you for help me.
6
votes
6answers
182 views

Is the set of all pairs of natural numbers countable? [duplicate]

Say that $\Bbb N \times \Bbb N$ is the set of all pairs $(n_1, n_2)$ of natural numbers. Is it countable? My hypothesis is yes it is countable because sets are countable. But I am unable to come up ...
0
votes
0answers
6 views

Are these random variables on the same probability space?

In the following exercise in David Williams' Probability with Martingales, an argument is made that the random variables are under the same $\Omega$ and $\mathbb{P}$. It does not say so however in ...
1
vote
0answers
14 views

Continuously bounding a matrix valued function by another matrix valued function

Notation: The symbol $\mathbb{S}^{n \times n}$ denotes the set of symmetric matrices of order $n \times n$ and $A \succ 0$ denotes that $A$ is a positve definite matrix. Let $M:S \rightarrow ...
0
votes
0answers
3 views

If a homeomorphism between two metric spaces and its inverse are uniformly continuous, do we have “X is bounded implies Y is bounded”?

Let $(X,d_X)$, $(Y,d_Y)$ be metric spaces, $\phi:X\to Y$ be a homeomorphism. Assume $\phi$ and $\phi^{-1}$ are uniformly continuous. Prove or disprove: If $X$ is bounded, then $Y$ ...
0
votes
0answers
23 views

Laurent series $\frac{-1}{z}+\frac{1}{2(z-1)} +\frac{1}{2(z+1)}$ at |z-1| > 2

So I have come across this question in a past paper and I am a bit unsure what to do... I have split it up in to partial fractions and obtained this: $\frac{-1}{z}+\frac{1}{2(z-1)} ...
2
votes
2answers
85 views

Replacing $\sin(z)$ with $1 - e^{2iz}$

I have seen many integral evaluations within logs where they change the sine to: $$\sin(z) \rightarrow 1 - e^{2iz}$$ Such as here: Contour integral evaluation. I dont understand how those ...
0
votes
2answers
20 views

How to prove that $a^{|G|}=e$ if a $\in G $

How to prove that; $a^{|G|}=e$ if a $\in G $ if $G$ is a finite group and $e$ is its identity. I think this could be done through pigeonhole principle but I don't want to use the Lagrange ...
0
votes
0answers
14 views

Differentiating a sum involving logs

I was doing the problem provided in the picture but I do not understand how do they obtain the answer. I am not sure how to differentiate the sum. I end up getting: alpha - 1 - 1/K. I believe I need ...
0
votes
0answers
11 views

Examples of vector bundles

I know three examples of interesting vector bundles (not taking into consideration what one gets from them using algebraic operations): Tangent bundle of a smooth manifold Tautological bundle over ...
2
votes
0answers
26 views

Lebesgue Decomposition Theorem only true for Borel sets?

In Evan's book "Geometric Measure Theory and Fine Properties of Functions", we have the following two theorems: Differentiation Theorem for Radon measures. Let $\nu, \mu: \mathcal P(\Bbb R^n) \to ...
3
votes
1answer
22 views

If $\sum \|f_n -f \|_{L^1} < \infty$ then $f_n \rightarrow f$ almost uniformly

Consider $(X,m)$ a measure space, $f_n, f : X \rightarrow \mathbb R$ s.t. $\sum_{n=1}^{\infty} \|f_n -f \|_{L^1} < \infty.$ How to show that $f_n \rightarrow f$ almost uniformly? I will have ...
0
votes
0answers
8 views

Difference of the $2$ sums is $O(x\log(x))$

If $g(x)$ is real-valued on $\{x\in\mathbb R:x\ge1\}$ and satisfies the condition $|g(x)|\le Cx$, with a constant $C$ for all $x\ge1$ then show that; $$\sum\limits_{n\in\mathbb N\atop{n\le ...
1
vote
0answers
11 views

Random Variables and Moment Generating functions

Let $(X_i)_{i∈\Bbb{N^+}}$ be a sequence of i.i.d random variables and for $n ∈ \Bbb{N^+}$ set $S_n := \sum _{i=1}^{n} X_i$ and $Y_n := max(X_1, . . . , X_n)$. Assume that the moment generating ...
1
vote
2answers
24 views

Prove: $\gcd(n^a-1,n^b-1)=n^{\gcd(a,b)}-1$ [duplicate]

I'm trying to prove the following statement: $$\forall_{a,b\in\Bbb{N^{+}}}\gcd(n^a-1,n^b-1)=n^{\gcd(a,b)}-1$$ As for now I managed to prove that $n^{\gcd(a,b)}-1$ divdes $n^a-1$ and $n^b-1$: Without ...
1
vote
1answer
12 views

Convergence in probability

I need to prove that given the r.v. Xn with the same distribution functions, the sequence of r.v. Xn/n tends to 0 in probability. Following the definition i find: P(|Xn/n| > a) = P(|Xn| > na) for ...
0
votes
0answers
11 views

Primality radius and quadratic reciprocity law

Given an integer $n>1$, I say that $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are primes. Goldbach's conjecture asserts that every integer greater than $1$ admits a primality radius. ...
9
votes
3answers
98 views

$A^3-A+I=0$ then $A$ is invertible

Prove or disprove: If $A^3-A+I=0$ then $A$ is invertible. Is it possible to say the characteristic polynomial of $A$ is $t^3-t+1=0$ and $A$ is invertible since $0$ is not an eigenvalue of the ...
3
votes
2answers
45 views

Show that there exists a vector $v$ such that $Av\neq 0$ but $A^2v=0$

Let $A$ be a $4\times 4$ matrix over $\mathbb C$ such that $rank A=2$ and $A^3=A^2\neq 0$.Suppose that $A$ is not diagonalisable. Then Show that there exists a vector $v$ such that $Av\neq 0$ but ...

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