0
votes
2answers
34 views

What is the logic behind the probability of getting 'four of a kind' in poker?

This hand ($5$ cards of $52$) has the pattern $AAAAB$ where $A$ and $B$ are from distinct kinds. The number of such hands is $\binom{13}{1} \binom{4}{4} \binom{12}{1} \binom{4}{1}$. The probability ...
0
votes
1answer
13 views

Form of Matrix for Reflection about a Line

I've seen a bunch of variations on the wonderful properties of this specific matrix. My textbook gives one algebraic form in particular that I'm having a bit of trouble verifying: Any help here? I ...
5
votes
0answers
51 views

if $f(x)$ is even and can be infinitely differentiable, how about $f(\sqrt{x})$

I have a question $f(x)$ is even and can be infinitely differentiable, how about $f(\sqrt{x})$ in [0,$\infty$)? can we say that the $f(\sqrt{x})$ also can be infinitely differentiable in ...
3
votes
2answers
21 views

Use Induction to prove $\forall m,n \in \Bbb Z_{\ge 0}, 1 +mn \leq (1 + m)^n$

Use Induction to prove: $$\forall m,n \in N, 1 +mn \leq (1 + m)^n$$ for integers $m,n\ge 0$. My biggest problem with this proof is ...
0
votes
0answers
10 views

How to navigate around a smooth surface?

Suppose I want to find the shortest path between two points in $\Bbb{R}^3$ with smooth obstacles in the way? I understand things like Dijkstra's algorithm for shortest paths on a graph. But what about ...
3
votes
1answer
23 views

Maximum / Minimum Question with 3 Variables?

I seem to be stuck in this problem, would need your help! Question: Assume I have : 147 of x, 174 of y, 238 of z A different amount of x, y ...
1
vote
0answers
15 views

Calculating Normals across a sphere with a wave-like vertex shader

This is a bit of a CS question, but more than not it's a 3D math problem. I've been trying to get the correct normals for a sphere I'm messing with using a vertex shader. The algorithm can be boiled ...
1
vote
3answers
35 views

If $A,B$ are proper subsets of connected space $X,Y$, then the set $(X\times Y)\setminus (A\times B)$ is connected

Let $A$ be a proper subset of $X$ and let $B$ be a proper subset of $Y$. Prove that $(X\times Y)\setminus (A\times B)$ is connected whenever $X$ and $Y$ are connected?
0
votes
1answer
11 views

If given $F(u_k)\to F(u)$ and $F'(u_k)\nabla u_k \to F'(u)\nabla u$ in $L^p$, why we have $F(u) \in W^{1,p}$ and $\nabla F(u)=F'(u)\nabla u?$

If $F$ continuous, $u_k \in C^{\infty}(\overline U)$, $u \in W^{1,p} (U)$. If given $F(u_k)\to F(u)$ and $F'(u_k)\nabla u_k \to F'(u)\nabla u$ in $L^p$, why we have $$F(u) \in W^{1,p}$$ and $$\nabla ...
0
votes
1answer
22 views

From nowhere dense perfect set to zero measure set.

I know that Cantor set is nowhere dense and perfect. But if I have a nowhere dense perfect set, can I call it a Cantor set? Also, I already proved that a certain subset of the real line is a ...
2
votes
1answer
42 views

Show $\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$ if $|z_1| <1$ and $|z_2| < 1$

Show $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$$ if $|z_1| <1$ and $|z_2| < 1$ Consider: $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right|^2$$ ...
1
vote
3answers
52 views

Set $A$ not closed under $\star$ then $A$ not a group under $\star$?

I am currently doing some exercises. I have been through some examples of solutions in other books that questioned me. I know well that $(A,\star)$ is a group if it satisfies the following points, ...
1
vote
2answers
23 views

Invertible element of $S$

Let $S=\mathbb{Z}[\sqrt{2}]$ = {$a+b\sqrt2|a,b\in \mathbb{Z}$} and $R = \mathbb{Q}[\sqrt2]$ = {$\alpha + \beta\sqrt2 | \alpha, \beta \in \mathbb{Q}$}. Consider $x=3+2\sqrt2$ and $y = 3+4\sqrt2$ ...
0
votes
1answer
21 views

How to factorize this.

We just started calculus and busy with limits. we were told that use a limit as long as it does not make the equation undefined. So the question is: $\displaystyle \lim_{x\to 0} \dfrac{2x}{x^2+x}$ ...
0
votes
1answer
13 views

Understanding the Product of $\Delta$-sets: $\Delta^1\times\Delta^1$

We learnt that a $\Delta$-set is said to be a sequence $K_\bullet$ of sets $\{K_n\}_{n\ge0}$ with defined "face" maps $d_i:K_{n+1}\rightarrow K_n$ for $0\le i\le n$ satisfying $d_id_j=d_{j-1}d_i$ ...
1
vote
0answers
31 views

Taking the Fourier Transform of a specific function.

I have this function: $$f(x) = \prod_{p\text{ is prime}} \left(1 - \frac{x^2}{p^2}\right)$$ Now, this function can be said to be an infinite degree polynomial with zeros on each of the primes and ...
1
vote
0answers
9 views

Frequency response of a linear, shift-variant system

I am working my way through recorded lectures and a textbook related to DSP, and have come across a question that I am not sure how to answer. This is probably just due to how new I am to these ...
0
votes
0answers
6 views

Embedding a $deg \geq 4$ curve in $\mathbb{P}^2$

Reading the paper by Kollar "The structure of algebraic threefolds", among some examples he does talking about intrinsic and extrinsic geometry over $\mathbb{C}$, he mentions that "an irreducible ...
2
votes
4answers
44 views

How can I square $-1 < x < 1$?

If I square $-1 < x < 1$, I get $1 < x^2 < 1$ which doesn't make any sense. What additional algebraic steps do I need to apply in order to get the proper inequality $0 < x^2 < 1$? ...
1
vote
2answers
57 views

Does $A\setminus B = A\setminus C$ imply $B=C$?

Let $A, B, C$ be sets with $B \subset C$ and $C \subset A$. Does $A\setminus B = A\setminus C$ imply $B=C$? I am not sure what the \ means, so I don't know how to solve this.
1
vote
3answers
62 views

An example of something that cannot be explained by means of posets [on hold]

My professor said this but I do not understand- can you help me? Give me an example of something that cannot be explained by means of posets; that is, a description composed of orderings of times. ...
1
vote
0answers
16 views

Change of variables for functions with constraints

I want to find critical points of say F = (x1-x2) + (y1^2-y2^3) + (z1^2-z2^3) with constraints x1^1 + y1^2 + z1^2 -1 = 0 and x2^1 + y2^2 + z2^2 -1 = 0 I can do this by finding critical points of ...
0
votes
1answer
25 views

How (the graphic of) a $\mathcal C^1$ but not $\mathcal C^2$ function looks like

We know examples of functions (obviously we are in the context of real valued functions) which are continous but not derivable; the simplest is $x\mapsto|x|$. In particular we have a precise graphic ...
1
vote
1answer
26 views

Integral $\int_0^{\infty}\cos(a_0+a_1x+a_2x^2)\frac{1}{x^2+\frac{1}{4}}dx$

Is this integral known to have a closed form? $$\int_0^{\infty}\cos(a_0+a_1x+a_2x^2)\frac{1}{x^2+\frac{1}{4}}dx$$ Is there anything special about it?
0
votes
0answers
14 views

Vector/Matrix Norm Limit at Infinity

Suppose A $\in$ $\mathbb{R}^{m×m}$ and $r^{(0)}$ $\in \mathbb{R}^{m}$ A contains only Non-Negative Elements and Each Column either sums to 0 or Sums to 1 $r^{(n+1)}$ = $Ar^{(n)}$ $\; \; \; n = ...
2
votes
1answer
16 views

Where is my mistake: determining the Lie algebra of complex orthogonal matrices

I tried to determine the Lie algebra of $O(3, \mathbb C)$ but I think there is a mistake but I can't find it. Here is my work: Let $\mathfrak o$ denote the Lie algebra of $O(3, \mathbb C)$. The ...
0
votes
0answers
6 views

Conditional convergence of series and product

Let $e_k = 0$ for $k$ is odd and $e_k = 1$ when $k$ is even. Set $b_k = \frac{e_k}{k} + \frac{(-1)^k}{\sqrt{k}}$. How do I show that the series $\sum b_k$ diverge while the corresponding product $\Pi ...
-1
votes
1answer
29 views

Prove that $\int_0^{\pi} \sin^nx\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cos(n+2)xdx=0$

Prove that $$\int_0^{\pi} \sin^nx\cdot\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cdot\cos(n+2)xdx=0$$ with $n \in \mathbb{N}$ I think it's true, but I can't prove.
3
votes
2answers
20 views

Find the length of the chord given that the circle's diameter and the subtended angle

A chord of a circle subtends an angle of 89 degrees at its centre. Find the length of the chord given that the circle's diameter is 11.4 cm. The problem I have here is that I can't visualise this ...
1
vote
1answer
11 views

Optimization of Frobenius Norm and Nuclear Norm

How to solve the following optimization problem, \begin{equation} \boldsymbol{\hat{x}} = argmin_{\boldsymbol{x}} \frac{1}{2} \| \boldsymbol{x - y} \|_F^2 + \lambda \| \boldsymbol{x} \|_{*} ...
1
vote
0answers
7 views

Rao-Blackwell improvement for a nonrandomized estimator

Context: please consider a parametric statistical model $(\mathcal{Y},\{P_\theta:\theta\in\Theta\})$ and suppose that we are estimating $g(\theta)$. Associated with this is the set of decisions ...
2
votes
2answers
19 views

Why do consecutive triangular numbers in pairs like $6$ and $10$ always add up to a perfect square?

I was a bit surprised by this when I thought of it. Look here: $$001, 003, 006, 010, 015, 021, 028, 036, 045, 055$$$$004, 009, 016, 025, 036, 049, 064, 081, 100$$As you just saw, $15$ and $21$ add to ...
0
votes
1answer
13 views

product of Matrices notation meaning

I am trying to workout what the meaning of the notation is below. $D$ is a matrix and it is the product from $1$ to $n$. However, the $k(i)$ notation of the matrix and the subsequent $k:(1,2,....n) ...
0
votes
1answer
15 views

Freely homotopic but not homotopic

I want to find a example of closed paths freely homotopic but not homotopic (I do not have many tools, like fundamental group, then has to be the simplest way possible). I thought at the following: ...
1
vote
1answer
9 views

Trajectory of a particle for $t\ge1$ given $r(t)$ for $0\le t \le 1$.

I have a question on the process for which to solving this question. It is a homework question, and I already have the answer, but I am not sure on the correct process to attaining that answer. The ...
1
vote
2answers
29 views

Prove that vector space and dual space have same dimension

As an exercise in my textbook, I need to prove that if $V$ is a finite dimensional vector space with dual space $V^*$ over $\mathbb{R}$, then dim$(V)$=dim$(V^*)$. Let $\omega\in V^*$ and let ...
1
vote
2answers
19 views

Prove that I can always write a number a, a>0 as any number c, c>0 to the power of some number (a=c^x)

I'm very new to math, I'm sorry if my question is stupid. I started to study math by my own so I can study Computer Engeneering. I'm studying logarithms and I try to come up with simple proofs of the ...
0
votes
1answer
30 views

how to proof that this function is zero

given $f$ continuous and diferentiable into $\mathbb{R}$ such that $\forall x\in\mathbb{R},|f'(x)|\le|f(x)|$ and $f(0)=0$ then proof that $f(x)=0$ atempt: taking $x>0$, since $f$ is ...
0
votes
0answers
7 views

Simple linear regression; Beta(0) = 0

In simple linear regression, $Y_{i}=\beta_{1}+\beta _{1}X_{i}+\epsilon_{i}$, what are the implications if $\beta_{0}=0$ such that $Y_{i}=\beta _{1}X_{i}+\epsilon_{i}$, assuming that $X=0$ is within ...
0
votes
2answers
10 views

number of one-one function;a set to itself

How do you find the number of all one-one function from a set to itself? If you are asked to find the number of all one-one functions possible from any set A to itself ,how do you do it?The following ...
0
votes
1answer
10 views

What is the range of this complex function: $f(z) = 2x^2+(1-x^2)(1+i)$ defined on $|z|\leq1$?

These range problems, I just don't get it. I tried to get this into a form where I could use the fact that $0\leq\theta\leq2\pi$, but I'm just not sure how to get it to that point. Any ${hints}$ ...
-4
votes
1answer
17 views

Need help with one of my homework questions [on hold]

Express the function in the form $f(g(x))$ $F(x) = (x^2 + 1)^{10}$
2
votes
1answer
17 views

Integration by parts $\int(x+y)e^{-x}dx$

What I'm trying to solve: $\int(x+y)e^{-x}dx$ Here's my professor's approach: $$u = x, du = e^{-x}$$ $$du = dx, dv = -e^{-x}$$ By doing parts: $(-xe^{-x}) - \int(-e^{-x})dx - ye^{-x} = (-xe^{-x}) ...
1
vote
1answer
29 views

Equality on functions in $ \mathbb{R}^n $

Let $ f,g : M \subset \mathbb{R}^p \to \mathbb{R}^q $ continuous. Given $ a \in M $, supose that all open ball centered in $a$ contains a point $x$ such as $f(x) = g(x) $. Show that $ f(a) = g(a) $. ...
0
votes
2answers
22 views

Continuity of the multiplication map $f\mapsto x^2 f(x)$ between normed spaces

Let $F:C[0,2]\to C[0,2]$ be the map defined by $(F(f))(x)=x^2f(x)$. Show that $F$ is continuous as a function from $(C[0,2],\|\cdot\|_{\sup})$ to $(C[0,2],\|\cdot\|_{2})$. I read this solution: ...
1
vote
0answers
21 views

What exactly is a torsion line bundle?

I am currently studying bigness of divisors, and its relationship between its numerical class and Kodaira dimension (using the aide of Lazarsfeld's Positivity in Algebraic Geometry I). I understand ...
0
votes
1answer
8 views

show that [T]β is a diagonal matrix

$V = P_1(R), T(a + b(x)) = (6a - 6b) + (12a - 11b)x$, and $β = \{3+4x, 2+3x\}$ Show that $[T]β$ is a diagonal matrix I am totally confused about how to write down the matrix form of this ...
1
vote
1answer
25 views

In a Category, Is the Set of Morphisms Between Objects Defined to Be All Possible Morphisms?

For instance, if I have a category $\mathfrak{M}$ whose objects are families of morphisms $\{f_i\colon A_i\to B\}_{i\in{I}}$, then if we consider two such objects, say $C=\{f_i\colon A_i\to B\}$ and ...
1
vote
1answer
23 views

What is the difference between Hom and Sheaf Hom?

I'm reading Hartshorne's book, and in 3.6 he begins to go into detail about Ext and sheaf Ext, which are derived functors of Hom and sheaf Hom respectively. Let $\mathcal{F,G}$ be sheaves of ...
0
votes
1answer
24 views

How can I reword this problem illustrating a scenario that needs Bayes Theorem to solve?

Taken from Stat Trek, an example explaining Bayes Theorm http://stattrek.com/probability/bayes-theorem.aspx Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent ...

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