0
votes
1answer
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 ...
0
votes
2answers
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 ...
-1
votes
1answer
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 $(A(z) − 1)$?
2
votes
1answer
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 ...
2
votes
1answer
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. $$(\sqrt{2\times 3\times 4\times 5 + 1}) = 4 = 4 + 6 + 1 - 4 = 7$$ ...
1
vote
2answers
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 ...
0
votes
1answer
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 ...
0
votes
1answer
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) ...
0
votes
2answers
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 $Z_p^n$?
1
vote
0answers
87 views

Identify subsets of $\mathbb{N}$ with their characteristic functions

If $G\subseteq2^{\mathbb{N}}$ is comeager then exist is a partition $\mathbb{N}=A_0\cup A_1$, $A_0\cap A_1=\emptyset$, and sets $B_i \subseteq A_i$ for $i \in \{0,1\}$, such that for $A \subseteq \...
-3
votes
1answer
38 views

statement is true for L^2 not for L^1

A problem about L^p space in Bass's real analysis book. Define $g_n(x) = n\chi_{[0,n^{-3}]}(x)$: (1) Show that if $f \in L^2([0, 1])$, then $\int_{0}^{1}f(x)g_n(x)dx\rightarrow 0$ (2)Show that ...
2
votes
2answers
811 views

Formula to best fit a rectangle inside another by scaling

I am very week in Math. I am a web programmer, and usually my work does not involve too much math - its more of putting records into database, pulling out reports, making those fancy web pages etc etc....
1
vote
1answer
78 views

Strange Double sum 3

Could you explain how to get the following double sum: $$\sum _{j=0}^{\infty } \sum _{k=0}^{\infty } \frac{2 (-1)^{k+j}}{(j+1)^2 k!\, j! \left((k+1)^2+(j+1)^2\right)}=(\gamma -\text{Ei}(-1))^2$$ where ...
0
votes
1answer
33 views

Continunity of a two variable function in Apostol's analysis

In discussing how the concept of differentiability implying continuity cannot be applied to functions of several variables, Apostol proceeds to give an example to demonstrate why. The function he uses ...
1
vote
1answer
83 views

Throwing three dice and analyzing results

Three dice are being thrown. $A$-exactly one of three numbers thrown will be $1$. $B$-different number is thrown on each die. What is: $P(A\cap B)$ $P(A\mid B)$ $P(B\mid A)$ Are $A$ and $B$ ...
2
votes
3answers
243 views

What is continuity, geometrically?

Suppose I build a function such that its graph is a unique line that can be drawn without lifting the pen(everywhere or in a specific range.) Is that function continuous?
2
votes
3answers
155 views

Field axioms: Why do we have $ 1 \neq 0$?

In the definitions of a field, we have $ 1 \neq 0$. I know that in regular multiplication $0 \times 1=0$ but for reciprocal we don't have inverse of $0$. But all the spaces and different definitions ...
4
votes
2answers
52 views

If $A \subsetneq C$ then $A \subsetneq B$ or $B \subsetneq C$. Contrapositive?

If $A \subsetneq C$ then $A \subsetneq B$ or $B \subsetneq C$. Is the contrapositive of this statement If $A \subseteq B$ AND $B \subseteq C$ then $A \subseteq C$. I asked because I think the ...
0
votes
1answer
135 views

finding the polar set

the question say find the polar-duals of the following sets in $\mathbb{R}^2$ 1) $\{(x,y):x\geq 2\}$ 2) $\{(x,y):x\leq 2\}$ 3) $\{(x,y):x=2\}$ the answers are $\{(x,0):x\leq 0\}$ , $\{(x,0):0\leq ...
1
vote
2answers
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 ...
0
votes
1answer
84 views

A very simple geometric/visual example of what a simplex looks like

I was trying to understand what a simplex was intuitively by constructing an example. Consider only points in $\mathbb{R}^2$. From wikipedia the definition seems to be: Choose k+1 points $u_0, ..., ...
4
votes
3answers
101 views

How to evaluate $\int\frac{x}{\sqrt{x^2+x+1}}dx\;?$

How do you integrate, $$\int \frac{x}{\sqrt{x^2+x+1}} dx $$ I am trying to use trig substitution, but I am having trouble finding a perfect square which works.
2
votes
1answer
115 views

Fourier series: Understanding a proof

Let $f:[0,2\pi]\to\mathbb{R}$, continuous, such that for all $n\in\mathbb{Z}$:$$\int_0^{2\pi} f(x)e^{i(n+\frac{1}{2})x} dx = 0$$ Prove that $f(x)=0$. The solution: We can rewrite the integral as: $...
1
vote
2answers
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 $A$ ...
1
vote
4answers
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, ...
5
votes
2answers
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?
2
votes
1answer
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 ...
0
votes
1answer
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 ...
3
votes
1answer
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 ...
1
vote
1answer
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 ...
1
vote
4answers
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 ...
2
votes
3answers
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 $n$ ...
2
votes
1answer
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 $G$...
2
votes
1answer
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, ...
0
votes
1answer
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. \...
0
votes
2answers
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 $f(-2)$.
0
votes
4answers
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 ...
2
votes
1answer
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 ...
1
vote
3answers
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 ...
1
vote
1answer
43 views

Artin-Wedderburn theorem and square dimension

Let $A$ be a finite-dimensional simple algebra over $\mathbb{C}$ of dimension $n$. By Wedderburn's theorem, we have that $A$ is isomorphic to a matrix ring $M_r(\mathbb{C})$, which is of dimension $r^...
2
votes
1answer
70 views

Kronecker Products and Powers notation

We are all familiar with the notation for powers, which represent repeated multiplication: $$x^n = \underbrace{x \times x \times \cdots \times x}_{n \text{ times}}$$ Is there something similar to ...
1
vote
1answer
84 views

The merchant and the fake coin [duplicate]

Next is a riddle that I found interesting and I decide to share it with you. Try solve it by yourself before reading the answer. A merchant has 13 fair gold coins with one fake among them. The fake ...
0
votes
1answer
47 views

Solve integral analytically

Could this integral be solved analytically? $$ \int_0^a \frac{\cos(\omega t+\phi)}{b - \sin(\omega t + \phi)} \, dt $$ So far I solve it numerical. If anyone could hint substitutions or ...
3
votes
1answer
70 views

Showing two forms on a manifold are equal

Let $\alpha$ and $\beta$ be two forms on a manifold $M$. To show that they are equal, does it suffice to show that for arbitrary $p\in M$ there exists some chart such that $\alpha_p=\beta_p$. I was ...
6
votes
1answer
788 views

What are these math symbols?

I'm studying linear algebra and all of a sudden the symbol $\dot{+}$ appears. For example: $a*(v \dot{+} w) = a*v \dot{+} a*w$ Any idea what it might be? Also two more symbols. they are on top of $...
1
vote
0answers
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 \\=& \...
-1
votes
1answer
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?
2
votes
1answer
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 ...
0
votes
1answer
39 views

Question on Method of Characteristics & Characteristic Curves

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 $s \to z(s,x,t)$ ...
0
votes
1answer
97 views

Pell-Type Diophantine Equation Solving using the method of ascent [duplicate]

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 $(u, v)$, ...

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