0
votes
3answers
45 views

Prove that if $A$ is independent of $B$, $A$ is independent of $C$, then $A$ is independent of $B\cup C$.

Prove that if $A$ is independent of $B$, $A$ is independent of $C$, then $A$ is independent of $B\cup C$. $\mathbb{P}(A)\mathbb{P}(B)=\mathbb{P}(AB)$ $\mathbb{P}(A)\mathbb{P}(C)=\mathbb{P}(AC)$ So ...
4
votes
2answers
241 views

Is $f(x)=x+\sin x$ bounded variation

Does $f(x)=x+\sin x$ has a bounded variation on $\Bbb R$? I dont know the concept of bounded variation.
2
votes
1answer
25 views

What is the min integer factor which makes $4x+2y$ integer?

Sorry for the english translation. $x$ and $y \in \Bbb R$. $15$ is the minumum integer which makes $x$ integer when you multiply. $18$ is the minumum integer which makes $y$ an integer when you ...
0
votes
0answers
34 views

What is the Cumulative Distribution Function of $a/x^b$? [on hold]

I was just wondering what the CDF of $$\frac{a}{x^b}$$ would be? $a$ and $b$ are positive constants and $b \gt 1$ ($1.22$ to be exact). $x \in [0, \infty)$ theoretically but in practice once $x$ has ...
1
vote
1answer
25 views

Probability that in bridge game the Players N,E,S,W have a,b,c,d spades respectively.

There are 52 cards in bridge and 13 cards of each suit. The formula for numerator is: $${13\choose a}{39 \choose 13-a}{13-a\choose b}{26+a\choose 13-b}{13-a-b\choose c}{13+a+b\choose 13-c}$$ But i ...
0
votes
1answer
48 views

Finite Modules Isomorphism

Two vector spaces are isomorphic if and only if they have the same dimension. In particular, two vector spaces over a finite field are isomorphic if and only if they have the same cardinality. For a ...
0
votes
2answers
324 views

Minimal polynomial of a diagonal matrix

How can I show that the minimal polynomial of a diagonal matrix is the product of the distinct linear factors $(A-\lambda_{j}I)$? In particular, if we have a repeated eigenvalue, why is it that we ...
0
votes
0answers
6 views

How low density in LDPC codes leads to best code

I'm want to use LDPC code as a channel coding mechanism. In LDPC we are using sparse matrix for parity check matrix (H). Why we are going for low-density. What are the advantages?.why it is works for ...
5
votes
0answers
53 views

What did Lagrange, Euler, Gauss etc. learn in order to know what they knew?

What did the great mathematician, like Lagrange, Euler and Gauss, learn in order to know what they knew? It seems that they were extremely good in the most basic rules/structures/issues of math: ...
10
votes
3answers
912 views

Intermediate digits of 34!

Problem: Given that $34!=295232799cd96041408476186096435ab000000$. Find $a, b, c, d$. $a, b, c, d$ are single digits. I am able to find $a$ and $b$ but cant find $c, d$. I did the prime factorisation ...
3
votes
2answers
2k views

Sample variance converge almost surely

Suppose $X_{1},X_{2},\ldots$ be i.i.d. random variables such that $E\left[X_{i}\right]=\mu$ and $Var(X_{i})=\sigma^{2}<\infty$. Let $\bar{X}=\left(X_{1}+\cdots+X_{n}\right)/n$. Show that ...
0
votes
1answer
23 views

Reflection from the upper half ball to the whole ball is harmonic

I have a question about problem 9(b) in Chapter 2 of Evans' PDE book. It says if we have $u$ is harmonic in the open upper half ball $U^+$ and $u\in C^2(U^+)\cap C(\bar{U^+})$, $u=0$ for $x\in ...
0
votes
1answer
29 views

Probability of Team A winning where a draw is not allowed

I have the probabilities for a range of final scores for a sports team A and also for a sports team B. I assume that these probabilities are fixed and not affected by outside factors including the ...
0
votes
2answers
27 views

Approximate solution to a matrix equation

Let $A$ and $B$ be $n \times m$ matrices. I am looking for a $m \times m$ matrix $X$ which would be an approximate solution to the equation $AX = B$ (an exact solution is very unlikely to exist). More ...
1
vote
0answers
12 views

Complexity of fast multipole method

I am trying to implement the Fast multipole Method on Matlab. But i have a question: if we call N the number of source point we need (in the FMM algorithm) to sort the data points. Such a procedure ...
0
votes
1answer
9 views

Name generator prefix+suffix

By googling "word permutation" and "word combinations" I found http://textmechanic.com/Permutation-Generator.html But I would like to send a bulk of prefix and suffix into an engine, and let it give ...
2
votes
1answer
21 views

How to understand the Mobius transform as a group action?

The group $SL(2,R)$ acts on the upper half-plane by the formula $$ \left(\begin{array}{cc} a & b \\ c & d \end{array} \right) z = \frac{az + b}{cz + d} .$$ It is indeed straightforward to ...
-1
votes
1answer
58 views

How to simplify the conditional expectation $E[v_3\mid v_1 < \max\{v_2,v_3\}, v_3=\max\{v_2,v_3\}]$ [on hold]

Suppose $v_1,v_2,v_3$ are three random variables drawn independently from the same distribution $\mathrm{uniform}(0,1)$, is it correct that $$E[v_3\mid v_1 < \max\{v_2,v_3\}, v_3=\max\{v_2,v_3\}] ...
0
votes
0answers
12 views

Whether the code produced by CRC is a cyclic code?

Wikipedia says that CRC algorithm is based on cyclic codes, but it doesn't say that it is a cyclic code. If I understood correctly, a linear code of length $n$ called cyclic if and only if its ...
0
votes
0answers
10 views

recurrence relation for Strassen's matrix multiplication

Let $a$ and $b$ be constants. Solve the following recurrence relation for $T(n)$: $$T(n)= \begin{cases}7T\left(\frac{n}{2}\right)+ an^2 & n>2 \\ b & n \le 2\end{cases}$$
7
votes
1answer
253 views
+400

$F[t]$ 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 existential theory ...
0
votes
7answers
898 views

Perpendicular bisector

Show that BE is the perpendicular bisector to AC. I tried to prove this through Pythagoras, but the answer I got did not prove it was at a right angle, and therefore said it was not the ...
0
votes
1answer
17 views

Statements with multiple quantifiers

Suppose $P(x,y)$ is a predicate whose truth depends on $x$ ($c\in D$) and $y$ ($y\in E$). In the following statement,does the order of assigning values to $x$ and $y$ matter? For example, assign some ...
1
vote
0answers
35 views

Biased Random Walk with Variable Probability

Consider a random walk in which the probability to move forward in time $t$ is $p_t$ and the probability to move backward is $q_t=1-p_t$ with $p_t<q_t$ with $p_t<p_{t+1}$ and $q_t>q_{t+1}$. ...
2
votes
0answers
16 views

What are explicit maps in the following exact sequence?

Let $G$ be a group and $M,N$ be normal subgroups of $G$ such that $G=MN$. Then there is a natural exact homology sequence $Ker(M \wedge N \xrightarrow{\lambda} [M,N]) \xrightarrow{\rho} H_2(G) ...
4
votes
2answers
123 views

Solving ODE rigorously

I am given the ODE $$(f''(r)+\frac{f'(r)}{r})(1+f'(r)^2)-f'(r)^2f''(r)=0$$ and want to solve it rigorously for $r>0.$ So especially, I don't want to loose any solutions. $\textbf{Derivation of ...
4
votes
4answers
76 views

Calculate simple expression: $\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}$

Tell me please, how calculate this expression: $$ \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} $$ The result should be a number. I try this: $$ \frac{\left(\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - ...
0
votes
1answer
39 views

Getting the angle that is needed for covering a given distance on an ellipse's cirumference

In a small programming exercise I asked myself, I want to calculate various things about ellipses. The part I'm stuck with is the following: I want to calculate the angle that is needed cor covering a ...
0
votes
1answer
20 views

Find the analytic function

$f(z)=1 $ satisfies the condition Using Identity Theorem $f(z)=1$ can be only function that satisfies this. so option (b) is NOT true. Am I on correct path?
0
votes
0answers
25 views

Continuity at $(0,0)$ of $f(x,y)=2xy^2/(x^2+y^4)$ along the paths $φ(t)=(t,t)$ and $ψ(t)=(t^2,t)$

Let $f: \mathbb R^2→\mathbb R$, $φ: \mathbb R→\mathbb R^2$, $ψ: \mathbb R→ \mathbb R^2$ be given by $φ(t)=(t,t)$, $ψ(t)=(t^2,t)$, $t ∈ \mathbb R$ and $$f(x,y) = \begin{matrix} \frac{2xy^2}{x^2+y^4} ...
0
votes
1answer
42 views

Let $A, B$ and $X$ be sets. Prove that if $A ∪ B ⊆ X$ then $A ⊆ X$.

I have just started learning set theory and I've been trying to learn how to do proofs, however I really can't figure out I've been trying to answer a simple one: Let $A, B$ and $X$ be sets. Prove ...
1
vote
0answers
18 views

Symplectic structure on coadjoint orbits or on Lie algebra?

I often read (and hopefully understood) that there is a symplectic structure on the coadjoint orbits of a Lie group. Now, in my opinion this means that we even have a symplectic structure on the dual ...
0
votes
2answers
46 views

I need advice on calculating this limit of a function resulting in $-\frac14\pi$

I am looking for advice on solving this limit of a function. I am struggling to find the correct process: $$\lim\limits_{x\to-\infty}\operatorname{arccotg}\frac{x}{(x^2-4)^{\frac12}}$$
0
votes
1answer
120 views

Improper integral $\int_{0}^{\infty}\frac{x^n}{x^{m+n+1}} \ dx=\frac{n! {(m-1)}!}{(m+n)!}.$

How can I prove that $$\int_{0}^{\infty}\frac{x^n}{x^{m+n+1}} \ dx=\frac{n! {(m-1)}!}{(m+n)!}\quad ?$$ I tried to do induction on $n$ and on $m$, separately, but I could only do the base case ($n=1$ ...
3
votes
2answers
41 views

Almost Everywhere Convergence versus Convergence in Measure

I am having some conceptual difficulties with almost everywhere (a.e.) convergence versus convergence in measure. Let $f_{n} : X \to Y$. In my mind, a sequence of measurable functions $\{ f_{n} \}$ ...
1
vote
0answers
23 views

Vector field left-invariant then also its respective flow?

I was wondering whether left invariance of a vector field $X$ to a respective Lie group $G$ (so $dL(a)(x)(X(x))= X(ax))$ is transfered to the respective flow defined by $\frac{d}{dt} \phi^{t}(x) = ...
4
votes
3answers
100 views

Prove $\log_7 n$ is either an integer or irrational

I have been trying to prove a certain claim and have hit a wall. Here is the claim... Claim: If $n$ is a positive integer then $\log_{7}n$ is an integer or it is irrational Proof ...
2
votes
1answer
37 views

graded Hopf algebra and its dual

I am learning Hopf algebras, and there are two questions as follows: Is the tensor product of two Hopf algebras still a Hopf algebra? Let $A$ be an infinite dimensional algebra. Is the dual ...
1
vote
1answer
18 views

What is the difference between a simple “fraction” and a “common fraction”?

I have read about a common fraction in this statement (written in a text book): Ratio is the simplest form of a common fraction, in which the numerator denotes the antecedent and the denominator ...
2
votes
1answer
17 views

Generating function for tuples of objects based on their maximal size

This is a question which arose while working through Flajolet-Sedgewick's Analytic Combinatorics. In their terminology, the cartesian product of two combinatorial classes $\mathcal{A},\mathcal{B}$ ...
2
votes
2answers
34 views

Number of spanning trees in a complete split graph

A graph is a complete split graph if we can partition it into an independent vertex set and a clique, such that every vertex of the independent vertex set is adjacent to every vertex in the clique. ...
2
votes
1answer
17 views

Definition of differentiability at the point in multivariable calculus.

I'm self-studying the analysis from Zorich and the next definition of differentiability is given: $f:E\to \mathbb{R}^n$ is differentiable at the point $x$, which is a limit point of $E\subset ...
2
votes
2answers
97 views

A question on a Hopf algebra

Let $(H, \nu,\eta, \Delta, \epsilon, S)$ be a Hopf algebra. $S$ is the antipode. I am reading a proof of the fact $S(xy)=S(y)S(x)$. First, define maps $\nu, \rho$ in $\hom(H \otimes H, H)$ by ...
0
votes
1answer
37 views

what's the greatest volume of a cylinder using calculus?

I have a rectangle that has the perimeter of 38cm. I need to make this rectangle into a baseless cylinder and find the greatest volume of it, by deriving. so far i came with this: for the rectangle ...
0
votes
1answer
302 views

How To Solve for Percentage When The Only Given Values Are Mean and Standard Deviation

How To Solve for Percentage When The Only Given Values Are Mean and Standard Deviation For Example: The scores of students in Mathematics examination is normally distributed with a mean of 60 and a ...
-2
votes
5answers
9k views

How to reverse percentage?

If I have a value of 25%, and I want the value as it would be on the opposite side of the halfway point (75% in this case), what is a formula that can calculate this? I don't know the appropriate ...
0
votes
0answers
17 views

The highest direction of the trace operator

Let $W$ be a real and symmetric matrix ${m \times m}$ from the set $\widetilde{W_m}$, and $T:\widetilde{W_m} \rightarrow \mathbb{R}$ a function defined by $T(W) = trace(W^3)$. We are interested to ...
1
vote
0answers
15 views

Cauchy formula for repeated Lebesgue integration

Is there an equivalent of the Cauchy formula for repeated integration (https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration) for the following \begin{equation} f^{(-n)}(x) = \int_a^x ...
2
votes
2answers
45 views

Sketch the complex function: $z\overline{z}+(1+2i)z+(1-2i)+1=0$

Tried sketching the complex function: $z\overline{z}+(1+2i)z+(1-2i)+1=0$ I first simplified it by converting $z=x+yi$ I got: $(x+yi)(x-yi)+(1+2i)(x+yi)+(1-2i)+1=0$ Which gave me this implicit ...
0
votes
2answers
43 views

Differentiability of a function from the plane to the line at the origin

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a function such that $f(x,y) = (x^3 + y^3)^{1/3}$. I want to study the diffrentiability of $f$ at the origin. My claim is that it is not differentiable at the ...

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