1
vote
1answer
44 views

Mean value theorem for integrals: how does the sign matter?

The mean value theorem is that $\int_a^b f(x)g(x) dx = f(\xi) \int_a^b g(x) dx$ for some $\xi \in [a,b]$ if g(x) does not change sign on $[a,b]$. However, I can't see why the sign matters. There is a ...
0
votes
1answer
27 views

Integrals in Reverse

I'm asked what solid this integral represents(the integral is used to obtain the volume of a solid, we are given this). I see that since we have the 2pi, this is probably a volume obtained by using ...
0
votes
2answers
102 views

Finding the norm of a linear functional

This is a basic question of functional analysis, but I want to know how to... Find the norm of the linear functional $f$ defined on $C[-1,1]$ by $$f(x)=\int_{-1}^0 x(t) \, dt - \int_0^1 x(t) \, ...
1
vote
1answer
80 views

Question about Lebesgue Covering Dimension

Suppose we have a metric space equipped with two different metrics: $(X,d), (X, d')$. What must be true of the metrics: $d, d'$ in order for $X$ to have the same Lebesgue covering dimension? A ...
5
votes
0answers
54 views

When does a variety have a point over a finite field for sufficiently large primes p?

Let $X$ be an algebraic variety over the rational numbers. Suppose that $X$ has positive dimension. I would like to say that $X(\mathbb{F}_p)$ is non-empty for sufficiently large primes $p$. One idea ...
0
votes
1answer
242 views

What does the ${\uparrow}{\uparrow}$ symbol mean?

The question asked is calculate the value of: $\dfrac{2\mathbin{{\uparrow}{\uparrow}}3}{2^{100}}$ and I have no idea what the ${\uparrow}{\uparrow}$ symbol means.
1
vote
0answers
42 views

Collinear points if and only if difference of angles is 90

In $\triangle ABC$, where $\angle BAC > 90^{\circ}$ let $H, O$ be the ortho-center and circumscribed center respectively. Let $K$ be a point on the line $AH$ such that $AH=AK$. Prove that $C, K$ ...
0
votes
1answer
54 views

Finding the Newton map

Start with $p(x)=(x-x_0)^k g(x)$. I need to find the Newton map, which is $Np(x)=x−p(x)/p'(x)$. Is $p'(x)=k(x-x_0)^{k-1}g(x)+(x-x_0)^kg'(x)$? I'm having a tough time with $k$ and $x_0$.
0
votes
1answer
110 views

Question about the ideal $I=(xy,yz,zx)$ in the ring $\mathbb C[x,y,z]$. [duplicate]

Given the ideal $I=(xy,yz,zx)$ in the ring $\mathbb C[x,y,z]$. I want to compute $V(I)$, which is the intersection of all ideals containing $I$. And I also want to prove that $I$ can't be generated by ...
1
vote
2answers
41 views

Is this correct the rational numbers?

Determine the rational numbers $a,b$ , if , $($$2a-b$$)$ - $2b\sqrt3$ $=$ $3$ + $2\sqrt3$ I'm thinking that $-2b\sqrt3$ = $2\sqrt3$ $=>$ $ b = -1 $ $2a-b$ resembles with $3$ , and i solved ...
1
vote
1answer
95 views

discrete math basic question

Al has 75 days to master discrete mathematics. He decides to study at least one hour every day, but no more than a total of 125 hours. Assume Al always studies in one hour units. Show there must be a ...
1
vote
1answer
58 views

Random distribution of colored balls into boxes.

This is an abstraction of a real problem I have: I have a large number of balls that are either Red or Blue ($n = 9*10^7$) and a bunch of containers ($c = 3*10^7$). I've calculated that the ...
1
vote
0answers
52 views

Bounded variation in the context of Feller's paper on Muntz' Theorem

The paper I have posted a picture of is a paper of Feller. He shows that the functions $f_k$ are Laplace transforms of $C^\infty$ functions $u_k$. In order to execute his suggested proof, I ...
0
votes
1answer
36 views

Let $T$ be exponential with parameter $\lambda$. Let $X$ be discrete defined by $X= k$ if $k \leq T < k+1$, $k=0,1,2,\dots$. Find the pdf of $X$.

To be honest, I am lost on this question. Here is what I have so far: $$ \ F_T(t)=- e^{-\lambda t}=P[T\le t] \ $$ $$ \ P[X=k]=P[k\le T \lt k+1] \ $$ I am not sure how to go about finding the pdf for ...
0
votes
1answer
54 views

Two theorems about absolute continuity, what am I missing? Since one is much nicer than the other.

Look at these two theorems: and It seems to me that the second theorem is much better than the first since we don't need to check the condition $f(x) = f(a) + \int_a^x f'(t)$. And also every ...
7
votes
1answer
259 views

A problem of Ramanujan's interest: closed form of $1 + 2\sum_{n=1}^{\infty} \frac{\cosh(n\theta)}{\cosh(n\pi)} $

I am Brian Diaz, and I am new to the math.stackexchange community. I have been struggling with attempting to find a closed form of the following series: $$ \varphi(\theta) = 1 + 2\sum_{n=1}^{\infty} ...
1
vote
0answers
26 views

Which functions could be discussed in a calculus course?

In a typical calculus course, students are exposed to many functions. Some have their origin in a field of study, while others are simply combinations of polynomials, trig functions, exponentials, and ...
0
votes
3answers
64 views

How to solve $\lim_{n \to \infty} \frac{x^n}{1+x^n}$

How do you find the limit of $$\lim_{n \to \infty} \frac{x^n}{1+x^n}$$when n goes to infinity? I've no idea how to solve this type of questions. I think when $x=1$ it equals to a half and when $x$ ...
-1
votes
1answer
75 views

Biking uphill and downhill

During an interview, I was asked "If you can bike 20 mph uphill and 30mph downhill, and you have 1 hour to bike, how far or how long should you ride uphill before turning back." While a very ...
1
vote
1answer
92 views

Is the number of subsequential limits of a sequence always countable

I know that a sequence can have many different subsequential limits but is the number of subsequential limits always countable? How do we know?
4
votes
2answers
117 views

Tough trigonometric identity

Prove that $$\cot 13^o\cot 23^o \tan 31^o\tan35^o\tan41^o = \tan 75^o$$ I managed to rearrange it to the form $$\tan 31^o\tan 35^o\cot 49^o = \cot 15^o\tan 23^o\cot 77^o$$ and in this form we have ...
0
votes
1answer
76 views

Making a Piecewise Function into a single expression

A phone company gives a 25.00 dollar flat fee up to 200 minutes then .07 dollars for every minute afterwards. Build a function to find the price of any amount of minutes. Not in a piecewise function, ...
0
votes
1answer
159 views

determine values given equation of line parallel to x axis and y-intercept.

Determine the values of a and b for which the line $(a+2b-3)x+(2a-b+1)y+6a+9=0 $ is parallel to x axis and y-intercept is -3. Also write the equation of the line. here is what i have tried. $ eq : ...
2
votes
1answer
51 views

What is the difference between the algebraic function fields and the fields itself

I'm studying this book and I don't understand exactly what's the difference between the algebraic function field $F/K$ and $F$ itself. Thanks
0
votes
1answer
251 views

Why is Wolfram Alpha wrong?

I calculated $$\tan 75^o - [\cot 13^o\cdot \cot 23^o \cdot \tan 31^o \cdot \tan 35^o\cdot \tan41^o]$$ and I got a nonzero answer: ...
0
votes
1answer
27 views

Possible Number Combos That I can not figure out [closed]

I am wondering, I have 4 QB's, 8 RB's, 12 WR's, 4 TE's, 4 K's, 4 Def, I can only play 1 QB, 2 RB's, 3 WR's, 1 TE, 1 K, 1 DEF for a total of nine players. How many different combinations do I ...
0
votes
1answer
64 views

Prove this inequality by math induction

$$\sum \limits_{k=1}^{n-1} k^p < \frac{ n^{p+1}}{p+1} < \sum\limits_{k=1}^n k^p $$ I know how to prove it by using Riemann Sum, but it I was thinking if there is anyway to do it by mathematical ...
7
votes
1answer
194 views

Properties of the solutions to $x'=t-x^2$

Let $f_c$ be the solution to $$ \left\{ \begin{array}{c} x'=t-x^2 \\ x(0) =c \end{array} \right. $$ I'm trying to prove: If $c \geq 0$ then $f_c(t)$ is defined for all $t>0$ There is a ...
3
votes
0answers
246 views

True statements about natural numbers that are undecidable in Peano Arithmetic assuming consistency of PA only

I am looking for statements $P$ of Peano Arithmetic ($\textsf{PA}$) that have the following properties: They are as concrete and simple as possible. It is provable by finitistic means that neither ...
0
votes
1answer
33 views

The convergence in Bounded Variation functions

Given a sequence $u_n\in BV(\Omega)$ and $u\in BV(\Omega)$, where $\Omega\in R^n$ is open. We assume $u_n\to u$in $L^1_{loc}(\Omega)$ and we also assume that $$ \lim_{n\to\infty}|D ...
1
vote
1answer
57 views

What is the 'type' of a natural transformation

Let $C,D$ be categories with objects $O_C,O_D$ and morphisms $M_C:O_C\times O_C\to Type_0$, $M_D:O_D\times O_D\to Type_0$. Let $F,G:C\to D$ be functors. A natural transformation $\eta$ associates to ...
0
votes
2answers
53 views

Area Of Triangle . Given two equation and point.

What does the equation $ x^2 -y^2 =0$ represent ? If the line $y-2=0 $ intersects $x^2-y^2=0$ at points A and B and if O be the origin , then find the area of OAB. i don't understand the question ...
0
votes
3answers
61 views

Real numbers determine rational numbers

Determine the rational numbers $x,y$ ,knowing that $x(1+\sqrt2)^2$ + $y(1-\sqrt2) = 1 $ My result is $3x$ + $y$ - $y\sqrt2$ = $1$ I'm not sure how to continue this.
0
votes
1answer
126 views

how to draw this DFA?

{w belongs to all string patterns as a^i b^j a^k | i+j=even and j+k =odd} draw a DFA and find its regular language. please note here, i have put comma in between the format of aba string just for ...
0
votes
1answer
111 views

Does calculating d/dt of someething mean the same as calculating the derivative?

Probably a dumb question but I missed college for a week due to sickness. The exercise I have to do is: d/dt eight root of t^7. Does this simply mean I have to calculate the derivative of the eight ...
2
votes
1answer
89 views

Find $E[S]$ where $S$ is the standard deviation of a random sample from a $N(\mu,\sigma^2)$ population.

Recall that for a $N(\mu,\sigma^2)$ population $W=\frac{n-1}{\sigma^2}S^2\sim \chi^2(n-1)$. [a] Find $E[S]$ where $S$ is the standard deviation of a random sample from a $N(\mu,\sigma^2)$ population. ...
1
vote
1answer
55 views

The solution of Poisson's equation

From Evan's book, we know that, in $R^n$, the function $u$ defined as $$ u(x):=\int_{R^n}\Phi(y)f(x-y)dy,$$ where $f\in C^2(R^n)$ and $\Phi$ is the fundamental solutions of Laplace equation. Then we ...
0
votes
1answer
47 views

Proof on existence of the natural numbers, crucial step.

I am trying to understand/reconstruct the proof given by my Professor addressing the existence of natural numbers. However there is one step in particular I don't understand and the more I think about ...
3
votes
2answers
36 views

Suppose that $f(x) \ge 0$ and $\lim_{x \to c} f(x) = L$. Prove $\lim_{x \to c} \sqrt{f(x)} = \sqrt{L}$

Suppose that $f(x) \ge 0$ in some deleted neighborhood of $c$, and that $\lim_{x \to c} f(x) = L$. Prove that $\lim_{x \to c} \sqrt{f(x)} = \sqrt{L}$ under the two different assumptions on $L$: $L=0$ ...
2
votes
1answer
103 views

Show that there are infinitely many primes $p$ such that $p = 1 (\mod q)$ in a very specific way

I friend of mine has shown the following: Let $n \in \mathbb{N}$, and $q$ an odd prime number. Any $p$ dividing $1 + n + \cdots + n^{q-1}$ satisfies $p \equiv 1 (\mod q)$, whenever $ n \not \equiv ...
2
votes
2answers
370 views

What are {} and [] functions?

I see these functions in Bondy-Murty book about graph theory... In book written: "The complete m-partite graph on n vertices in which each part has either [n/m] or {n/m} vertices is donated by T m,n. ...
2
votes
3answers
171 views

Showing the limit of a function tending to zero

I need to show that: $$\lim_{x\to0}\frac {2\sigma^2(1-e^{-\theta x^2})} {x^2}=2\sigma^2\theta$$ By plugging in arbitrary values for the constants and trying different values for x I can see that ...
1
vote
1answer
48 views

Heat equation: Why are these ratios of functions constant

One can solve the heat equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} $$ by a separation of variables such that $u(x, t) = f(x)g(t)$. Substituting this into the ...
0
votes
2answers
39 views

If $z'\le az+b$ then $z(t)\le z_0+bt$

If $z$ satisfies; $z'\le az+b$, $\ z(0)=z_0>0$ with constants $a,b$ why is true that $z(t)\le z_0+bt$, if $a=0$ It is clear that it can't be justified only by integrating. We had only Gronwall ...
1
vote
1answer
153 views

Proving every 3-Regular graph with no cut-edge has a 1-factor

I have the proof in my textbook, but I'm stuck on a line (or two). Here's some context: Let $S \subseteq V(G)$. Count the edges between $S$ and the odd components of $G-S$, $o(G-S)$. Since $G$ is ...
1
vote
0answers
25 views

On an existence of a certain compact set in a Polish group

Let $\mu$ be a Borel probability measure defined on a Polish group $G$. Does there exists an equivalent measure $\lambda$ and a compact set $F$ with $\lambda(F)>0$ such that for each $X \subseteq ...
11
votes
0answers
216 views

Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points

The polylogarithm is defined by the series $$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$ There are relations connecting values of the polylogarithm at certain rational points in the ...
0
votes
1answer
174 views

F be the smallest subfield of the real numbers which contains irrational a. Prove that F is countable.

Let a be an irrational number and let F be the smallest subfield of the real numbers which contains a. Prove that F is countable.
6
votes
1answer
209 views

Can bipartite graphs have the following properties?

Let G be a bipartite graph with at least $3$ vertices. Can it have the following properties simultaneously ? $1$) Every vertex is start vertex of some hamiltonian path. $2$) It contains no ...
0
votes
3answers
69 views

There is at most one way to represent a number as $a+b\sqrt 2$ with rational $a,b$

If $a,b,c,d\in\mathbb Q$ and $a+b\sqrt 2= c + d\sqrt 2$, then prove $a=c$ and $b=d$ ? I don't have any idea to solve this , it's freaking me out.

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