All Questions

0
votes
0answers
8 views

How many 4 digits prime numbers can be formed from 0,1,…,9 without repeated digits?

I'm just curious about the prime numbers in combinatorics. Can we use the combinatorics rule to find the number of prime number from given number, for example from the above condition? My attempt: I ...
1
vote
0answers
4 views

Showing $\phi(f \cdot g) = \phi(f) + \phi(g)$

For $\phi \in C_n(X; G)$ a cocycle being thought of as a function from paths in X to G, I want to show: $\phi(f \cdot g) = \phi(f) \cdot \phi(g)$. What I'm not sure is how I'm supposed to relate a ...
-1
votes
1answer
11 views

Integration $\int \frac{x}{x^2-5x+6}dx$

Evaluate the Integral: $\int \frac{x}{x^2-5x+6}dx$ I solved twice and once I got $3log\left|x-3\right|-2log\left|x-2\right|+C$ and I tried again and changed one step and I got ...
0
votes
1answer
17 views

Nature of the series $\sum_{n=3}^{\infty} \dfrac{1}{(\log\log n)^{\log n}}$

Does $\sum_{n=3}^{\infty} \dfrac{1}{(\log\log n)^{\log n}}$ converge or diverge ? I tried some tests , but nothing conclusive is coming . Pleas help
1
vote
1answer
10 views

Two definitions - do they differ?

I have the following definitions which may be true or false but for me it seems like both true A) If ($a_{2n} - a_n$) converges to $0$ then $a_n$ converges. B) If $a_n$ converges then ($a_{2n} - ...
0
votes
0answers
3 views

Why we always chose minimum Ratio in Simplex method?

Why we always chose minimum ratio in simplex method? (linear programming problem)
4
votes
0answers
15 views

Pronunciation of `Rng` - the non-unital Ring

I chuckled the first time I heard that a Ring without a multiplicative identity (Ring without the i) is called a ...
0
votes
1answer
8 views

first and second projection

i have read that in the ordered pair $z=(x_1,x_2)$, an element of a direct product $Z=X_1 \times X_2$ of sets $X_1$ and $X_2$, the element $x_1$ is called the first projection and $x_2$ is called the ...
0
votes
1answer
8 views

what will be the PDF of the magnitude of this random variable x+j y?

if we have a complex random variable [x+j*y] where (j :sqrt(-1)) and x,y both have Gaussian distribution and statistically dependent , so what will be the distribution (PDF) of the magnitude of this ...
1
vote
2answers
16 views

To show following function is discontinous

Given $f(x) = [x + 1] (\sin(1/x))$, where[.] denotes greatest integer function ; when $x\in (-1,0) \cup (0,1)$ $$f(x) = 0 , \text{ otherwise}$$ Question is to show f has discontinuity of second ...
0
votes
0answers
8 views

Summing the sequence $a(n) = \sin(n x) \exp(-nt)$

Consider the sequence $a(n)$ defined by $a(n) = \sin(n x) \exp(-nt)$, where $n = 0, 1, 2, 3, 4, \ldots$. The parameter $x$ is a real number. Parameter $t$ is a positive real number. It is clear that ...
1
vote
1answer
5 views

Formula of regular 2m-gon inscribed in a unit cirlce

From pg 80 of Introduction to Calculus and Analysis I by R. Courant: If we let$f_m$ denote the area of the regular $m$-gon inscribed in a unit circle, the area of the inscribed $2m$-gon is given by ...
0
votes
1answer
12 views

Combinations of fruits and their “nutrients”

As a computer scientist and not a mathematician, I know not some of the formal language to describe my problem, so I'll present it in a word problem form. Maybe someone can help me hone my search and ...
0
votes
0answers
6 views

Question on weak star convergence in subspace

Let $X$ be some normed linear space and let $X^\ast$ denote its dual space endowed with the weak star topology. Let $U^\ast$ be some subspace of $X^\ast$. If I want to show that $\varphi_n$ ...
0
votes
0answers
5 views

Minimal polynomial in $T$-invariant subspace

I am stuck on the following problem. Problem: Let $V$ be a finite dimensional vector space over field $F$ and $T$ a linear transformation from $V$ to $V$. $W$ is an invariant subspace. Let $h_1$ be ...
0
votes
0answers
10 views

Proving 2 random variables differ with positive probability

Suppose that conditional on $x$, $y$ is normal with mean $x'\beta_0$ and variance $\sigma_0^2$. The log of the conditional density is then $$ ...
0
votes
1answer
8 views

Asymmetric simple random walk?

It comes from the book Probability: Theory and Example. I don't understand the part marked with red line. Why it cannot converge to an interior point of $(a,b)$? Can anyone help? Thanks so much!
0
votes
1answer
32 views

how can publish my log approximation formula

I've successfully found out a formula which can give log value of any base till 4-5 places after decimal I want to know whether it can get published because I've seen some journals which have ...
1
vote
1answer
20 views

A simple question about free group

Fix $r\in \mathbb{N}$ and let $\mathbb{F}_{r}=\langle g_{1}, ...,g_{r}\rangle$ be the rank-r free group. I have asked a question several days ago: Is $\mathbb{F}_{2}$ a subgroup of $\mathbb{F}_{3}$? ...
1
vote
0answers
21 views

Does there exist a continuous function between the following sets:

Does there exist a continuous function between the following sets: $A.f:(-1,1)\rightarrow (-1,1]$ which is onto and one-one $B.f:\{(x,y):y^2=4x\}\rightarrow \mathbb R$ which is one-one What ...
0
votes
1answer
8 views

Rational Canonical Form Confusion; Choosing Basis Which Gives the Rational Canonical Form.

I am reading the theory of finitely generated modules over a PID. One of the applications of the the theory is that one can derive the theory of rational canonical form of a linear operator on a ...
0
votes
3answers
13 views

Limit About Complex Variable

I am trying to see if I have right understanding of the limit. when the book mentions $$\lim_{z\to z_0} f(z) = L$$ this just means that $f(z_0) = L$ is this correct?
0
votes
1answer
31 views

det (AB)=det(A)det(B) is possible when A and B are ????

det (AB)=det(A)det(B) is possible when A and B are ???? what is the answer when A and B are ???? This is a Fill in Blanks that I found in my paper but I don't have this answer
0
votes
1answer
9 views

Prove existence/non-existence of a pdf given mean, std, range

Given: Mean = 100, Range = [4, 10000], std = 3000 Is it possible to prove whether a pdf exists or not that satisfies these values? If it does exist, what would be approximate shape of the ...
-1
votes
0answers
24 views

The convergence of the multiplication of two convergent series?

If we know that \begin{equation} {\sum\limits_{n=1}^{\infty} }a_n \end{equation} and \begin{equation} {\sum\limits_{n=1}^{\infty} }b_n \end{equation} are convergent What about their ...
0
votes
0answers
5 views

Composite residuosity statement.

Consider the following definition. A number $z$ is said to be $n$-th residue modulo $n^2$ , if there exists a number $y \in \mathbb{Z}_{n^2}^*$ such that $$z\equiv y^n \mod n^2$$ Let us take $n=6$ ...
0
votes
2answers
20 views

Solve this Differential Equation $[x\csc(\frac{y}{x})-y]dx+ydy=0$.

$[x\csc(\frac{y}{x})-y]dx+ydy=0$ My work: $[\csc(\frac{y}{x})-\frac{y}{x}]dx+\frac{y}{x}dy=0$ Let $u=\frac{y}{x}\rightarrow y=ux\rightarrow dy=udx+xdu$ $[\csc(u)-u]dx+u(udx+xdu)=0$ ...
2
votes
1answer
15 views

Uniform convergence to exponential exercise

Yesterday I encountered the following exercise in a tutorial sheet from the University of Lyon : define a sequence of functions $(f_n)$ (with $f_n:[0,\infty) \to {\mathbb R}$) by ...
0
votes
0answers
19 views

Every map from a compact Hausdorff space is continuous

Could someone help me figure out my mistake? I just proved that if $X$ is compact Hausdorff then $f: X \to Z$ is continuous. Here's my proof: Let $U$ be open in $Z$. Let $x \in f^{-1}(U)$. Since $X$ ...
1
vote
0answers
14 views

If the sum of two i.i.d. random variables is normal, must the variables themselves be normal?

It is well known that if two i.i.d. random variables are normally distributed, their sum is also normally distributed. Is the converse also true? That is, suppose $X$ and $Y$ are two i.i.d. random ...
2
votes
1answer
25 views

Equivalence classes of $\mathbb R$

Let $X$ be a locally compact, connected, locally connected, Hausdorff space. Considder $U_1\supseteq U_2\supseteq\cdots$ of open and non-empty connected subsets with compact frontiers such that ...
-2
votes
0answers
18 views

how to transform among each number

assume there are 16 numbers in base 3 the basic construction for numbers from 0 to 3^(3^2)-1 numbers is using 0..15 0 1 2 3 ... 14 15 ...
0
votes
0answers
7 views

How to find the interior of this set?

let $S=\{A\in M_n(\mathbb R):tr(A)=0\}$ The question is to check whether $S$ is Nowhere dense .I think the set is closed and hence the problem reduces to findind int(S).How to do that?
0
votes
0answers
11 views

Non-orthogonal basis

I have a set of complex vectors (maybe 10,000 vectors, each of which has maybe 200 elements). I know that each of the complex vectors is a linear combination of a small (maybe 10) collection of ...
0
votes
3answers
27 views

To show $f(x)$ is discontinuous at every point

$$f(x)=\begin{cases} 1 ,& \text {$x$ is rational} \\ 0 , & \text{$x$ is irrational}\\ \end{cases}$$ How do I show this function is discontinuous at every point. How to think about it ...
0
votes
0answers
17 views

Expected number of rooms with at least one man and woman?

here's the problem: "10 men and 10 women randomly go into 10 rooms. What is the expected number of rooms with at least one man and woman?" Here's my reasoning: So, I think that by linearity of ...
0
votes
0answers
14 views

BODMAS Order of operations

9÷3(6×4÷8) Here we all can apply BODMAS and solve the equation in parenthesis first giving us 3. But after this step shall we solve by 9÷3(3) = 9÷9 = 1 or 9÷3×3 = 3×3 = 9 . Please help
1
vote
0answers
11 views

Suppose $H:= \{\sigma \in G| \sigma(1) = 1\}$, if for any $j \in \{1,2,…,n\}$ $t_j\in G$ such that $t_j(1) = j$. Show that $|G| = n|H|.$

Let G be a subgroup of the symmetric group $S_n$ in n letters. Consider the following subset of G: $$H:= \{\sigma \in G| \sigma(1) = 1\}$$ Suppose that G acts on the set $\{1,2,...,n\}$ transitively ...
1
vote
3answers
33 views

Sum of convergent and non-convergent series, does it converge? And how to prove?

Series $a_n$ is convergent and $b_n$ is not-convergent. Will the sum $a_n + b_n$ converge? I think it will not converge, But how do I show it? I believe I have to use the definition. $|a_n - A| < ...
4
votes
1answer
22 views

Why $R^{op}$ used in the definition of “category of $R$-modules”?

Earlier today, I was thinking: "Oh, an $R$-module is just an additive functor $R \rightarrow \mathbf{Ab}.$" Anyway, I had a bit of a read over at nLab, and it says: For any small ...
0
votes
1answer
22 views

A question about analytic continuation of a function in real axis

A function $f(x)$ is an real function and analytic in an open interval of $x$-axis or the whole $x$-axis. Is there only unique way to analytically extend it to the whole complex plane? I know ...
1
vote
2answers
41 views

The convergence of a recurrcively defined sequence.

Let $a_1=\sqrt{2}$ and $a_n=\sqrt{2+a_{n-1}}$ determine the convergence of the sequence and find its limit. I know the sequence converges to $2$ and i can show this informally. But I don't know how ...
0
votes
1answer
20 views

Orthogonal Matrix with a specific row

I have an assignment with the following question: Does an Orthogonal Matrix exist such that its first row consists of the following values: ($1$/$\sqrt{3}$, ...
1
vote
3answers
26 views

Help understanding a proof about vector spaces

The exercise goes like this: -Let $W= {(x,y,z)|2x+3y-z=0}$ Then $W\subseteq\mathbb{R}^3$, find the dimension of $W$. -Find the dimension $[\mathbb{R}^3|W]$ This was a problem from my algebra exam, ...
0
votes
0answers
25 views

E(XY) = E(X).E(Y|X) . Is this true for mean = zero.

I know that Joint Probability density function for two random functions $X$ and $Y$ $$P(XY) = P(X)\cdot P(Y|X)\tag{1}$$ But I just read in a set of lecture notes that for E(X)=E(Y)=0 $$E(XY) = ...
0
votes
2answers
20 views

limits of integration in spherical coordinates.

Consider a cone centered about the positive z axis with its vertex at origin,a $90^{\circ}$ angle at its vertex,topped by a sphere of radius $6$.Compute the volume of region bounded by sphere and ...
0
votes
2answers
76 views

Multiplying a infinite number with a rational number?

Please do not down vote this question. It may be stupid, but I wonder. Why is it that we cannot multiply $3.99999\cdots$ by $4$ and write $16,....$?
0
votes
0answers
18 views

Find the eigenvalues and eigenvectors of T in V

Let $\mathbf{V}$ be the linear span of the functions 1, cos x, sin x. Let the operator T on V be given by the rule $T y(x)= y(x+\pi/4)$. Find the eigenvalues and eigenvectors of T in V. I'm not sure ...
0
votes
0answers
13 views

Find the volume of the 3D solid.

Let $\delta$ be the region bounded by the graphs of the functions $f(x) = x^2$ and $g(x) = 2 x^2$. Find the volume of the solid generated by revolving $\delta$ around the line $x = 1$.
0
votes
1answer
26 views

$\{(x,y)\in \mathbb R^2:xy=1\}$

To check which pairs are Homeomorphic? A.$\{(x,y)\in \mathbb R^2:xy=0\}$ B.$\{(x,y)\in \mathbb R^2:xy=1\}$ C.$\{(x,y)\in \mathbb R^2:xy=0,x+y\geq0\}$ D.$\{(x,y)\in \mathbb R^2:xy=1,x+y\geq 0\}$ I ...

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