# All Questions

97 views

### Center of Mass using Line Integrals v.s. Triple integrals

What is the difference between these two formulas?(Below) One uses triple integrals and the other uses a single line integral. How do I know when to use which? Maybe I'm not grasping the concept as ...
434 views

### Find the limit. (If an answer does not exist, enter DNE.) $\lim _{x→∞} (\sqrt{9x^2 + x} − 3x)$

I was following this explanation until the 5th step. The most important misunderstanding is if $1x/1x$ is $1$, and one can multiply a quotient by $1$ and not change its value. However from what I can ...
43 views

### Transformations solving recurrence with generating functions

Why is $\sum_{n ≥3} a_{n-1} z^{n-1}$ equal to $(A(z) − 1 − z)$? $\sum_{n ≥3}a_{n-2} z^{n-2}$ equal to $(A(z) − 1)$?
75 views

### What is the error in this proof?

Note: This problem is from Discrete Mathematics and Its Applications [7th ed, prob 49, page 583]. This proof seems legit to me. If you know R is transitive, you ...
64 views

### How to solve this sum problem?

For the first radical section. $$\sqrt{1\times 2\times 3\times 4 + 1} - 1 = 1 + 3 + 1 - 1= 5 - 1 = 4$$ The second radical section. $$(\sqrt{2\times 3\times 4\times 5 + 1}) = 4 = 4 + 6 + 1 - 4 = 7$$ ...
44 views

### Determine inversible operator

How do I determine if the linear operator $f: P_2 \rightarrow P_2, f(p) = p+p'+p''$ is invertible? I suppose that I should find a function that is undoing $f$ but I don't know if there is some ...
51 views

### A small detail about regular functions on affine irreducible varieties

Let $K$ be a field, $X\subseteq\mathbb{A}^n$ an affine irreducible variety, and $U \subseteq X$ an open subset. Suppose that there are $f,g,f',g' \in K[x_1,...,x_n]$ such that $fg'=gf'$ over $U$ as ...
27 views

### Transformation matrix for a 3d->2d projection

We know $\mathbf{\hat{y}} = X\mathbf{w}$ and $A$ is the subspace in which $\mathbf{\hat{y}}$ lies (so the columns of the $X$ matrix define the subspace $A$). $\mathbf{\hat{y}}$ (2-dimensional vector) ...
197 views

### How to prove that the additive group of a finite field of order $p^n$ is isomorphic to $Z_p^n$?

Let $\mathbb{F}$ be a finite field of cardinality $p^n$ where $p$ is prime. How to prove that the additive group of $\mathbb{F}$ is isomorphic to $Z_p^n$?
87 views

228 views

### Compute coordinates of a point in 3D-Euclidean Space

My question concerns the computation of a point’s coordinates in three-dimensional Euclidean Space. I have a point P in three-dimensional Euclidean Space whose coordinates are unknown. My goal ...
84 views

74 views

### Why does a set of m elements have 2$^m$ subsets?

Note: This example is from Discrete Mathematics and Its Applications [7th ed, prob 2, pg 576], shout out to @crash. I understand why $A \times A$ has $n^2$ elements(because every member of set $A$ ...
149 views

### Formula for factorial?

I need an equation that defines factorial without using factorial, that also works for $0$. I have seen factorial defined like this: $$n! = 1\cdot2\cdot3\cdot4\cdots n$$ But if we plug $0$ into that, ...
548 views

### How can one find the area of the blue shaded region?

Here 3 circles are touching each other. Now how can one find the area of the blue shaded region in the given picture?
58 views

### What is the correct name for a “summable” number?

My math/CS teacher mentioned a function to me a few days ago (I don't remember the context), but didn't know the real name for it, so he just called it a summable function. We didn't really go into ...
251 views

### X Approaches Infinity! What does that Mean ? lim t → ∞ [(t^1/2) + t2] / 4t − t2 What should be my Strategy at Solving these types of problems.

I have had a tutor. He cannot convince me how one should solve limit approaching infinity problems. What is the strategy? Do you follow the limit laws if "the number x is approaching" is not in the ...
146 views

### Value of an integral involving the fractional part function

I have difficulties in evaluating the double integral defined in the following. Let $$\left\{ t \right\} = t - \lfloor t \rfloor,$$ $t> 0$ be the fractional part function, where the ...
79 views

### Show that $\sum_{n=-\infty}^\infty (x+n\pi)^{-2} = \sin^{-2}(x)$

Can you show that $$\sum_{n=-\infty}^\infty (x+n\pi)^{-2} = \sin^{-2}(x)$$ It is noted that if you make the substitution $x=x+2\pi$ the relation remains unchanged, but how can you show that ...
96 views

### True or False $A - C = B - C$ if and only if $A \cup C = B \cup C$

True or False $A - C = B - C$ if and only if $A \cup C = B \cup C$ I am in an introduction to proofs class. I think this is a true statement. I have began the proof and realize I have to do this ...
64 views

### Proof of an Limit

Using the formal definition of convergence, Prove that $\lim\limits_{n \to \infty} \frac{3n^2+5n}{4n^2 +2} = \frac{3}{4}$. Workings: If $n$ is large enough, $3n^2 + 5n$ behaves like $3n^2$ If $n$ ...
37 views

### Prove that every odd cycle is $3$-choosable without using the fact that $\chi_l(G) \leq 1+ \Delta(G)$

Prove that every odd cycle is $3$-choosable without using the fact that $\chi_l(G) \leq 1+ \Delta(G)$ Here is my attempt Let $G$ be an odd cycle from by vertices $(v_1,v_2,\ldots,v_n,v_1)$. Since $G$...
74 views

### What is the value of this Infinite Product of prime numbers expression? [duplicate]

What is the value of: $$\prod_1^\infty \frac{p_i^2}{p_i^2 -1 }$$ Where $$p_i$$ are the prime numbers: 2, 3, ...
28 views

### Interpretation of integral.

The height, in centimeters, of a bicycle pedal is given by $h(t)=30+16\sin t$ where $t$ is the time. Evaluate and interpret the following integral \begin{align} \dfrac{1}{2\pi}\int_0^{2\pi} h(t)\,dt. \...
127 views

### Suppose $f(x)$ is a rational function such that $3 f \left( \frac{1}{x} \right) + \frac{2f(x)}{x} = x^2$ for all $x \neq 0$. Find $f(-2)$. [closed]

Suppose $f(x)$ is a rational function such that $$3 f \left( \frac{1}{x} \right) + \frac{2f(x)}{x} = x^2$$ for all $x \neq 0$. Find $f(-2)$.
34 views

### Finding a solution to a congruence using Eulers Theorem

Suppose $\phi(m) = 1000$, and I wanted to find a number $a$ thats less then $2000$ that is not divisible by $7$ and satisfies $a \equiv 7^{3003}$ (mod $m$). What would be a solution to this and how ...
132 views

### about path connected covering spaces.

Let $p:E\rightarrow X$ be a covering space. It is well known that if $X$ is connected, then all the fibers have the same cardinality. This can be seen as a simple consequence of the fact that the ...
483 views

### what is the relation between Limit and Derivative?

We have that: \begin{align} \lim_{x \to 2} x^2 &= 4\\ D_{x = 2}\left(x^2\right) &= 4 \end{align} In both cases the result is $4$. So limit and derivative are always the same? If not, what ...
43 views

20 views

### If $X \sim N(\mu,\sigma^2)$, then $\int^t_sxf(x)dx=\sigma [f(s)-f(t)]+ \mu [F(t)-F(s)]$?

Here is my work, kindly let me know if this is correct: \begin{align*}\int^t_sxf(x)dx=&\int^{\frac{t-\mu}{\sigma}}_{\frac{s-\mu}{\sigma}}(\sigma z+\mu)\frac{\phi(z)}{\sigma}\sigma dz \\=& \...
59 views

### Semigroups: Product Rule [closed]

Given a Banach space $E$. Consider C0-semigroups: $$S,T:\mathbb{R}_+\to\mathcal{B}(E)$$ Then the product rule holds: $$(TS)'(t)x=T'(t)S(t)x+T(t)S'(t)x$$ How to prove this from scratch?
113 views

### Polynomial maximization

If $x^4+ax^3+3x^2+bx+1 \ge 0$ for all real $x$ where $a,b \in R$. Find the maximum value of $(a^2+b^2)$. I tried setting up ...
Consider the PDE \begin{align} u_t + (vu)_x &= 0,\phantom{u_0(x)} \quad x \in \mathbb{R},\, t>0 \\ u(x,0)&=u_0(x),\phantom{0} \quad x \in \mathbb{R} \end{align} Let $s \to z(s,x,t)$ ...
Can someone help me to prove there are infinitely many solutions to the Diophantine equation: $$x^2 − 3y^2 = 1$$ using the method of ascent. We can do this by showing how, given one solution $(u, v)$, ...