0
votes
0answers
19 views

Number of polyhedron diagonals

Suppose that I have a polyhedron with given number of faces, edges and vertices are given. Is there a formula that gives me the number of polyhedron diagonals, ...
4
votes
2answers
22 views

$G$ be a finite group of order $n$ , $H$ be a proper subgroup of order $m$ such that $(n/m)!<2n$ ; $G$ is not simple

Let $G$ be a finite group of order $n$ , $H$ be a proper subgroup of order $m$ such that $(n/m)!<2n$ ; then how to show that $G$ is not simple ? I have proceeded by Cayley's theorem , $\ker f$ is ...
1
vote
2answers
47 views

Can Two Different Polynomials Agree on an open interval? [duplicate]

Question: For a high degree polynomial $P_1$ , can we have another polynomial $P_2$ that is a part of $P_1$ (or they agree on open interval)? TBN: This question is partially answered in ...
3
votes
0answers
45 views

Another integral related to Fresnel integrals

How would we prove this result by real methods ? $$\int_0^{\infty } \frac{\sin \left(\pi x^2\right)}{x+2} \, dx=\frac{1}{4} \left(\pi-2 \pi C\left(2 \sqrt{2}\right)-2 \pi S\left(2 ...
0
votes
1answer
22 views

$U(\mathbb{C}^n)$, $SU(\mathbb{C}^n)$ connected subsets of $M_n(\mathbb{C})$?

As the title suggests, is $U(\mathbb{C}^n)$ a connected subset of $M_n(\mathbb{C})$? How about $SU(\mathbb{C}^n)$?
1
vote
2answers
20 views

Create some new numbers using $n$ arbitrary positive real numbers

Known facts: Let $a_i$, $b_i$, $i=1, \ldots, n$ be positive real numbers such that $a_1+ \cdots + a_n = a_1b_1 + \cdots +a_nb_n = 1$. Then $$b_1^{a_1}b_2^{a_2} \cdots b_n^{a_n} \leq 1.$$ ...
2
votes
1answer
47 views

How to remember these probability results?

If $A,B$ and $C$ are $3$ events, then $P$(Exactly one of $A,B,C$ occurs)$=P(A)+P(B)+P(C)-2[P(A \cap B)+P(B \cap C)+P(A \cap C)]+3P(A \cap B \cap C)$ $P$(Exactly two of $A,B,C$ occur)$=P(A \cap ...
-2
votes
0answers
19 views

Geometry midpoint [on hold]

John wants to center a canvas which is 8 ft wide on his living room wall which is 17 ft wide. Where on the wall should John mark the location of nails if the canvas requires nails every 1/5 of its ...
0
votes
3answers
35 views

$a^{\log_g b} = b^{\log_g a}$?

$g^{\log_g a} = a$, because it equals $a^{\log_g g}$. Does this mean that $a^{\log_g b} = b^{\log_g a}$? Note: thanks whoever edited it to proper markup
5
votes
3answers
78 views

Can we find uncountably many disjoint dense measurable uncountable subsets of $[0,1]$?

Can we find uncountably many disjoint dense measurable uncountable subsets of $[0,1]$? Obviously we may as well assume all the subsets have measure $0$. If I didn't specify the subsets were ...
2
votes
1answer
16 views

Do approximate identities remain approximate identities if one adjoins 1 to a C* Algebra?

If we have a C* Algebra $\mathscr{U}$ without an identity we can adjoin an identity $\mathbb{1}$ in the following way: We take $\mathscr{\tilde U}$ to be the set $\{(\alpha,A); \alpha \in ...
0
votes
1answer
20 views

Is every tensor the gradient of a vector?

I guess no, since every vector is not the gradient of a scalar - I assume ! Please confirm or guide in this regard .
0
votes
0answers
23 views

Showing $pk+1|p^p-1$ implies that $k$ is even

Suppose $p$ is an odd prime such that $pk+1$ divides $p^p-1$. Prove that it is not possible for $k$ to be odd. Here's my solution: Assume to the contrary that $pk+1$ does divide $p^p-1$ We can ...
0
votes
1answer
23 views

$1 \le p < q < \infty$ implies $L^q \subset L^p$

Suppose $1 \le p < q < \infty$ and $(X,\mu)$ is a Lebesgue measure space. Also suppose $X$ is of finite measure. Prove that $L^q \subset L^p$. First, we use Holder's inequality and find ...
0
votes
1answer
21 views

Basic conditional probability question [on hold]

$\sum_{c}p(a|c)p(c|b)=p(a|b)$. Does this equation hold true? If it is true, how to prove it mathematically?
0
votes
2answers
91 views

How do mathematicians find the underlying idea?

While reading through the lecture notes here (http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week2.pdf , page 22, last paragraph), I came across the following " Thus there must be some ...
1
vote
0answers
45 views

Why is $\frac{\partial }{\partial y}\int M dx = \int \frac{\partial M}{\partial y}dx$

$M$ is a function of $x$ and $y$. I'm getting this question from looking at the solution of the exact equation $M \mathrm{dx} + N\mathrm{dy} = 0$.
1
vote
4answers
38 views

Integrating linear/trigonometric

I have the following question- $\int$ $\frac{x}{1+cosx}dx$ Do I do integration by parts or is there some other method? Thanks for the help.
0
votes
4answers
41 views

An example of why $f(f^{-1}(B))\neq B$

Let $f:X\rightarrow Y$ be a function and $B\subseteq Y$ a subset of $Y$. I know (and have proven) that $f(f^{-1}(B))\subseteq B$. I've also found an example where $f(f^{-1}(B))\neq B$ for $B= ...
0
votes
2answers
23 views

Let $V=\mathbb{R}^\mathbb{R}$, let $W$ be the subset of $V$ consisting of all monotonically inc or dec functions. Is $W$ subspace of $V$?

Let $V=\mathbb{R}^\mathbb{R}$ and let $W$ be the subset of $V$ consisting of all monotonically-increasing or monotonically-decreasing functions. Is $W$ a subspace of $V$? Any solutions or hints are ...
-1
votes
0answers
6 views

If $X$ is a polish space, how do we find an equivalent metric under wich $X$ is a totally bounded?

According to Stroock and Varadhan, If $X$ is a polish space, then one can choose an equivalent metric under which the space is totally bounded (see Stroock and Varadhan - Multidimensional diffusion ...
1
vote
5answers
81 views

Spaces $X$ in which every subset is either open or closed, and only $\varnothing$ and $X$ are clopen

Let $(X, \tau)$ be a topological space. Then $X, \varnothing \in \tau$ and are both clopen. But I wonder if it is possible to construct a topological space $X$ in which all subsets are either open or ...
1
vote
0answers
12 views

Comparison of bivariate generating functions

Suppose we have two bivariate ordinary generating functions describing two integer sequences which have indicies $a,b$ and $c, d$ respectively. Is there a straightforward way to determine, from ...
-3
votes
0answers
19 views

Parametrize the given curve. [on hold]

$x^2 + y^2 = 121\;$ satisfying the condition $\;\displaystyle c(0) = \left(\frac{11}2, \frac{11\sqrt{3}}2\right)$ Thank you! 
0
votes
0answers
16 views

Showing that $(X_n)$ obeys the Markov Property.$

Consider a process $(X_n)_{n\geq0}$ where we define $X_0 = 0$ and for $n \geq 1$: $$X_n = X_{n-1} + Z_n$$ Where $Z_n$ for $n \geq 1$ are independent random variables on $\{ -1, 1 \}$ with ...
1
vote
3answers
22 views

The number of times will an individual child goes to the cinema before a group is repeated.

$1.)$ A mother with $7$ children takes $3$ at a time to a cinema.She goes with every group of $3$ that she can form.How many times can she go to cinema with distinct groups of $3$ children? ...
1
vote
1answer
28 views

A question about 2.1 Proposition on Folland's Real Analysis

Definition of measurable space: If $X$ is a set and $\mathcal{M} \subset \mathcal{P}(X)$(Power set of $X$) is a $\sigma$-algebra, $(X, \mathcal{M})$ is called a measurable space and the ...
1
vote
0answers
11 views

Inverse Laplace Transform with a functional

I don't know if it is possible, but I would appreciate if someone help me to obtain the inverse Laplace transformation of the function $F(s)=as/(s^2+w(s)^2)$. Where $s$ is a complex number and $w$ is ...
0
votes
0answers
22 views

Find $C^1$ class function such that

Given: $$g: \mathbb{R}^3 \rightarrow \mathbb{R}, g(x,y,z)=z^3-3xyz-x-8$$ Decide whether in the neighbourhood of the point $(x,y)=(0,0)$ there exist $C^1$ class function $z=z(x,y)$, such that ...
2
votes
1answer
31 views

Does multiplication by a positive definite matrix preserve eigenvalues?

Let $A$ be a positive definite matrix and let $B$ a matrix. Then, $AB$ is similar to $A^{\frac{1}{2}}BA^{-\frac{1}{2}}$, which is in turn similar to $B$, so I get that $AB$ and $B$ are similar. ...
1
vote
1answer
14 views

SVD of a matrix from SVD of its columns

Assume a matrix A, and I know the left singular vectors of SVD(A(:,i)), i=1,2,...,# of columns, is there a simple/fast transformation to obtain the left singular vectors of SVD(A) (the whole matrix)?
1
vote
4answers
49 views

What is the probability of drawing 3 balls such that none of them is red?

Given a bag containing $8\ \color{red}{red}$ balls and $4\ \color{green}{green}$ balls, what is the probability of drawing $3$ balls at random such that $\mathbf {none}$ of them are ...
1
vote
0answers
23 views

Differentiating the definite integral of x*f(x)

I'm trying to differentiate the integral below. I was wondering how I could approach it. z ~ N(a , $b^2$ $\cdot$ $x^{-2}$) $\frac{d}{dx} \int _{-\infty} ^ {z^*(x)} z \cdot \Phi ^{\prime} (z) dz$ ...
1
vote
0answers
24 views

Lagrange multiplier vs KKT

Suppose task 1: maximize $f(x, y)$ subject to $g(x, y) = 0$ and $h(x,y) = 0$ Suppose task 2: maximize $f(x, y)$ subject to $g(x, y) \geqslant 0$ and $h(x,y) = 0$ According to wiki for the first ...
3
votes
2answers
41 views

Use Taylor's Theorem with $n=2$ to prove that the inequality $1+x<e^x$ is valid for all $x\in \mathbb{R}$ except $x=0$.

Use Taylor's Theorem with $n=2$ to prove that the inequality $1+x<e^x$ is valid for all $x\in \mathbb{R}$ except $x=0$. Taylor's Theorem: $$ f(x)=\sum_{k=0}^n{1\over ...
1
vote
0answers
18 views

3-sigma approximation

I am making a system involving a sensor who has to be really precise. I found on their datasheet a diagram that shows the typical performance of the sensor. There's the mean value, the +3 sigma, ...
1
vote
0answers
21 views

color conversion from RGB to YIQ

I want to convert RGB color to YIQ. AS my knowledge the formula is below: To practice this math i went a to this link Color Conversion. I enter here RGB values 32,65,32. I found the result is YIQ = ...
3
votes
3answers
64 views

Is it right to say that if two vectors, $A$ and $B$, have same $L^p$ norms, for all $p$, then $A = B$?

Is it right to say that if two vectors, $A$ and $B$ (all elements of $A$ and $B$ are positive), have same $L^p$ norms, for all p, then $A = B$ ?. Thanks.
1
vote
1answer
28 views

Computing limits example: Swaping limit to $0$ into infinity.

I have found the following example: $$ \lim_{x\to 0^{+}} \frac{e^{\frac{-1}{h}}}{h} = \lim_{z\to\infty} ze^{-z} = 0 $$ Could you explain to a kid nice and slowly why does the limit of $x$ to ...
4
votes
1answer
46 views

Why is the Lagrange Multipliers Theorem not working?

Consider the function $h: K \to \mathbb{R}$ where $K := \{x \in \mathbb{R}^3:x,y,z \geq 0, x+2y+3z\leq 6\}$. $h$ is defined as: $$ h(x) = xe^{(x+2y+3z)} $$ Find the supremum and the ...
4
votes
1answer
11 views

Poisson Distribution of Underfilled Bottles

3 bottles per case are underfilled on average. What is the chance that at least 4 underfilled bottles will be contained in a random case? Using the formula: $1-\left( \dfrac{3^0 e^{-3} }{ 0!} + ...
2
votes
1answer
38 views

Why $\dfrac{\partial \sigma}{\partial u}=\dfrac{\partial \sigma}{\partial \bar{u}}$?

According to Elementary Differential Geometry by A N Pressley: (${\bf \sigma}_u=:\dfrac{\partial \sigma}{\partial u}$). The above text several times assuming that $\dfrac{\partial ...
0
votes
1answer
32 views

Expected % of heads flipping coins of different odds

So this is an analogy for a real world example but for simplicity. So if I were to flip a normal coin ten times I would expect heads 50% of the time or 5 head results. I could then compare this to the ...
2
votes
2answers
34 views

Integral values satisfying a inequality

Consider the following inequality : $$\frac{x^2+a^2}{a(4+x)} \ge 1$$ I am trying to find the range of integral values of $a$ for which this inequality holds for all $x$ belongs to $(-1,1)$ I ...
2
votes
0answers
23 views

Recovering a basis from an isomorphism with the dual space.

Let $V$ be a finite dimensional vector space, then given a basis for $V$ constructing an isomorphism $V \rightarrow V^*$ is easy, but how about the reverse direction? Given an explicit isomorphism ...
0
votes
1answer
28 views

SVD decomposition of matrix

Is it correct to say that a matrix $A$ and the matrix $A^HA$ have the same eigenvectors? Proof: $$ A= U \Sigma V^* \\ A^HA= U \Sigma^2 U^H $$ Am I correct?
1
vote
1answer
27 views

Topics to master (be literate at) before differential equations?

Good evening, I'm really enthusiastic about learning differential equations because it was said that D.E. is the most important tool of mathematics "can be used for modelling real-world physical ...
8
votes
4answers
577 views

Minesweeper probability

I ran into the situation pictured in the minesweeper game below. Note that the picture is only a small section of the entire board. Note: The bottom right 1 is the bottom right corner tile of the ...
2
votes
0answers
18 views

Can the determinant of an integer matrix with $k$ given rows be the gcd of the determinants of the $k\times k$ minors of those rows?

I'm interested if the following is true: Let $n\geq k\geq1$ be integers and let $$A=\begin{pmatrix}a_{11}&\cdots&a_{1n}\\ \vdots&\ddots&\vdots\\ ...
1
vote
4answers
46 views

Vector Functions of One Variable

Question A particle moves along the curve of the intersection of the cylinders $y=-x^2$ and $z=x^2$ in the direction in which $x$ increases. (All distances are in cm.) At the instant when the ...

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