2
votes
0answers
37 views

Real Analysis book with pictures and ideas of proofs

I am taking real analysis course in my graduate class of Maths. My classes will start in 3 months. I have studied real analysis but not very rigorously. Whenever I see theorem I have no idea on how ...
2
votes
1answer
24 views

Nature of an improper integral

I want to study the convergence of this integral at 0: $$ \int_0^{1}\frac{e^{\frac 1 t}}{\sqrt{t(1+t^2)}}\;dt. $$
1
vote
3answers
44 views

Finding the derivative of an inverse function.

Let $f(x) = (-x^2)/(x^2+1)$. If g(x) is the inverse function of f(x) and f(1)=-1/2, what us g'(-1/2)? Can someone explain how to do the above problem as I am not even sure where to start. Would I ...
-3
votes
1answer
57 views

Can someone help me why this equation equals zero?

I played around with some numbers and stuff and made this weird equation: $$\huge x^{- \frac{n}{n^{-x}}}$$ So the thing is, with every number I tried typing this into a calculator, I got 0. Can ...
3
votes
4answers
145 views

Finite dimensional algebra

Let $A$ be a finite dimensional algebra. Prove that an element of $A$ is invertible iff it is not a zero divisor. Let $a$ be an invertible element, then there exists an element $b$ such that $ab=1$ ...
-1
votes
1answer
24 views

Multiplying logarithms of different bases [on hold]

How do you multiply the following logs... $$\log_5(n) * \log_2(n)$$
0
votes
1answer
13 views

Need help solving a system of iterative differential equations

Here I have a system of differential equations: $u_{0}''=-1$ $u_{0}u_{0}''+u_{1}''=-1$ $u_{2}''+u_{1}''u_{0}+u_{0}''u_{1}=-1$ $u_{3}''+u_{2}''u_{0}+u_{1}''u_{1}+u_{2}u_{0}''=-1$ ...
-3
votes
2answers
42 views

Solve $2y^{(5)}-7y^{(4)}+12y'''+8y''=0$ [on hold]

Find the general solution of higher order linear differential equation? Find the general solution of Differential equation using auxiliary equation? $$2y^{(5)}-7y^{(4)}+12y'''+8y''=0$$
0
votes
1answer
16 views

Holder continuous functions embedded in Sobolev

For simplicity, I will consider $u\in W^{1,p}(\Omega)$, where $1<p<\infty$ and $\Omega\subset\mathbb{R}$ is open and bounded. I am able to show that \begin{equation} |u(x)-u(y)| \leq ...
0
votes
0answers
16 views

Lower bound for the chromatic number of $\mathbb{R}^n$

I'm going through a proof that of the following lower bound for the chromatic number of $\mathbb{R}^n$: $$\chi(\mathbb{R}^n) \geq (1.2 + o(1))^n$$ At some point in the proof we get that ...
1
vote
1answer
21 views

Calculating determinant matrix with size of n

we got the following matrix in order of $n$x$n$: $$\begin{pmatrix} 1 & 0 & . & . & . & 0 & 1\\ 1 & 1 & 0 & . & . & . & 0\\ 0 & 1 & 1 & 0 ...
0
votes
1answer
17 views

Permutations and Combinations 2

The word ARGENTINA include the four consonants R,G,N,T and the vowels A,E,I How many of the arrangements have a consonant at the beginning,then a vowel,then another consonant at the beginning,then a ...
0
votes
1answer
22 views

The genus of a certain kind of cubic

I have a cubic curve that looks like $$ a_0 x^3 + a_1 x^2 y + a_2 xy^2 + a_3 y^3 = b $$ with $a_0, a_1, a_2, a_3$, and $b$ all integers, and $a_0$ and $b$ nonzero. I'm not sure but I think in my ...
1
vote
3answers
23 views

Subgroups and subsets

I have some trouble with groups. Say we know that A is a subgroup of B. If we have some subset of A, say H, can we deduce that H is also a subgroup of B? Thank you. So if I have set of 2x2 real ...
0
votes
0answers
6 views

Calculating normalization constant in circle detection process

I'm doing some research in computer vision, and I need to calculate if two edge points correspond to the same circular object, but i have few questions. Formula is: where: pi and pj are two ...
2
votes
1answer
47 views

“Cascade induction”?

I refer to this answer. The answer is based on several simplification steps, all of them proven by induction. $$S_n = 2903^n - 803^n - 464^n + 261^n$$ $$T_n = 2642\cdot2903^n - 542\cdot803^n - ...
0
votes
1answer
23 views

Closed Form Summation Example

$$ \sum_{i=1}^n (ai +b) $$ Let $n \geq 1$ be an integer, and let $a,b > 0$ be positive real numbers. Find a closed form for the following expression. In other words you are to eliminate the ...
1
vote
0answers
7 views

Finding the frequency of a fan using a slow motion camera

This problem has me a little stumped. I'm not sure if my answer is correct and would just like to check: The camera shoots at 187 frames per second. The fan takes 33 frames to complete one revolution ...
0
votes
2answers
49 views

A Sequence That has No Upper Bound But Does Not Tend To Infinity

Let $a_n$ be a sequence which has no upper bound. Give an counterexample sequence for the statement $$\lim_{n\to\infty} a_n=\infty$$ Any hint?
1
vote
0answers
10 views

Spaces where distance between a point and a hyperplane is always reached

In this article I give an example of a Banach space where the distance between a point and a hyperplane is not reached. In the same article, I mention that for finite dimensional vector spaces or ...
0
votes
1answer
33 views

Partition on a Closed Set A= [2,3]

Is it possible to define a partition on a closed set,such that the union of the partitions will give [2,3] and their intersection to be empty?
0
votes
1answer
17 views

Two questions concerning ideal factorization and norm

$\bullet$ In $\mathbb Z[\sqrt{-5}]$ why is $(2)=(2,1+\sqrt{-5})(2,1-\sqrt{-5})$ Actually both ideals on the RHS contain $(2)$, but also their product ? Can we just multiply RHS in the normal sense; ...
1
vote
2answers
45 views

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$ Can I multiply and distribute the $ \ x^{7/2}\ $ and $ \ sec^2 \ $ together. What is the strategy to solve this problem.
3
votes
1answer
26 views

Help to prove that a group is cyclic

As part of my study of Abstract Algebra I'm trying to prove that $U_p$ si cyclic for $p$ a prime number. It's a classical result, but I'm trying to prove it following 4 steps stated as problems in my ...
1
vote
0answers
8 views

Extending a function from set without limit points

Problem: Let $D$ be a subset of (smooth) manifold $M$ without limit points in $M$. Let $f \colon D \to \mathbb R$ be any real-valued function. Can $f$ be extended to smooth real-valued function $g ...
0
votes
2answers
24 views

How to prove that eventually $(x^p/e^{x^q}) < 1/(x^2) $ for $p,q>0$

How to prove that eventually $x^p/\exp(x^q) < 1/(x^2) $ for $p,q>0$. I tried showing that $x^{p+2} > \exp(x^q)$ by using the Taylor expansion of e but this didn't really work.
0
votes
1answer
22 views

How to derive velocity out of acceleration in a circular motion?

A car starts moving in a circle with a radius of 200 meters. It has a constant tangential acceleration of $1{\text{m}\over {\text{sec}}^{2}}$. a. What is the angular acceleration? b. What is the ...
3
votes
2answers
48 views

Give an intuitive explanation for polynomial quotient ring, or polynomial ring mod kernel

I learned how to see quotient groups intuitively when I learned of a group mod its commutator subgroup. If we take a group and mod out all the elements that do not commute, we get a quotient group ...
0
votes
1answer
18 views

Representing a positive $x$ with a generating function

If we want to find the integer solutions of $x_1+x_2+x_3=n$ such that $x_1$ is positive using a generating function. We would first make the parentheses for each $x_i$, for $x_{2,3}$ it would be the ...
-1
votes
1answer
14 views

Domain of compostions of linear mappings

Let $T$ be a linear transformation from $\Bbb R^3$ into $\Bbb R^2$ and $S$ be a linear transformation from $\Bbb R^2$ into $\Bbb R^3$. Is the mapping $ST$ a linear transformation from $\Bbb R^3$ into ...
-3
votes
1answer
30 views

Polynomial and a field [on hold]

How to prove that if a polynomial $$f(x) = ax^3+bx^2 +cx +d,$$ where $a,b,c,d \in K$, where $K$ is a subfield of $\mathbb{C}$, has a root in $K(\alpha)$ then $f$ has a root in $K$. $\alpha \in ...
1
vote
0answers
18 views

Maps that preserve tensor rank

Suppose we have some tensor product of vector spaces. By tensor rank, I mean the minimal number of simple tensors required to write down an element of this tensor product of spaces. Is there much ...
1
vote
0answers
25 views

Compute the values of two infinite products whose factors are the same

I have the following question: How to prove that $(1-\frac{1}{2})\cdot (1+\frac{1}{3})\cdot (1-\frac{1}{4})\cdot (1+\frac{1}{5})\cdot (1-\frac{1}{6})\cdot (1+\frac{1}{7})\cdot ...
0
votes
0answers
9 views

Prove the Lagrangian reminder term $lim_{x\rightarrow x_0}\frac{\xi_n-x_0}{x-x_0}=\frac{1}{n+2}.$

For Taylor expansion $$f(x)=f(x_0)+f^{'}(x_0)(x-x_0)+\cdots+\frac{f^{(n+1)}(\xi_n)}{(n+1)!}(x-x_0)^{n+1},$$ if $f^{(n+2)}(x_0)\neq 0,$ how to prove $$lim_{x\rightarrow ...
1
vote
1answer
26 views

To show that a function defined by integral is absolutely continuous

Let $$ F(x)=\int_{[0,x]\times[0,x]}f,\quad x\in[0,1] $$ Here f is a Lesbegue-integrable on the unit square $[0,1]\times[0,1]$. I need to show that $F$ is absolutely continuous and express the ...
-5
votes
0answers
22 views

About Ito's theorem exemple [on hold]

Find an exemple of Lie algebra $g$ that is metabelian and there are not two abelian algebras $A$ and $B$ such that $g=A+B$.
0
votes
3answers
26 views

Looking for good intro book on differential equations

I am looking for a good book to study ordinary differential equations. My background is that I have successfully completed calculus 1 through 3. So this included derivatives and integrals, ...
0
votes
1answer
21 views

Solving an equation with complex numbers

I want to use complex numbers to solve the following problem: $x^2 = 95 - 168i$. I am sure there are a few ways of doing this but the way I want to do it is to let $x = a + bi$ and then solve for $a$ ...
0
votes
0answers
8 views

Smooth map from one manifold to boundary of another

Problem: Given a smooth function $f \colon M \to N$ (where $M, N$ are smooth manifolds) such that $\operatorname{im} f \subseteq \partial N$, show that when considered as a function $M \to \partial$ ...
0
votes
0answers
18 views

Is there an alternative encoding scheme to binary where similarity of pattern correlates with size of number?

If I compare binary for 7 111 and binary for 8 1000 there is no correlation between these two patterns that suggests that ...
1
vote
1answer
24 views

Evaluating the sum of a partial geometric sequence using Sigma notation

I have a worksheet from my instructor with this problem on it, but the solution he has given is different from what I got, and I don't know why. I'm not sure how to input the Greek letter sigma, but ...
-1
votes
4answers
23 views

finding roots of cubic equation and the values of constants

$x^3+px^2+qx+30=0$ where $p$ and $q$ $\in R$, has a root $1+2i$. $1)$ Find the other non-real root. $2)$ Find the third root of the equation. Hence, or otherwise, find the values of $p$ and $q$.
0
votes
1answer
15 views

Polynomial in $\mathbb{Z}_2[x]$ that is reducible but has no roots a prime $p$ for which $x+10$ divides $x^4+x^3+x+1$ in $\mathbb{Z}_p[x]$

First, I am suppose to find a prime $p\geq 4$ where $x+10$ divides $x^4+x^3+x+1$ in $\mathbb{Z}_p[x]$. Second, I am supposed to find a fifth degree polynomial in $\mathbb{Z}_2[x]$ that is reducible ...
1
vote
1answer
17 views

Initial value problem through origin

$\frac{dz}{dt}=8t*e^z$, Through the origin I have never done an initial value problem before, but I took it to mean that it gave me the initial value of the differential equation (0, 0) and that I ...
0
votes
1answer
10 views

Darboux sums inequality with relation to Sup|f'(x)|

Assuming f is continuous on [a,b] and differential on (a,b) and assuming f ' is bounded on (a,b) ; denote k = sup(|f '|) prove that, for all P a partition of [a,b]: 0 ≤ U(f,P) - L(f,P) ≤ k(b-a)Δ(P) ...
3
votes
2answers
139 views

Simple 2 equations and 2 unknowns

I am reading the second partial derivative test example, but I am suck on the following step: $$f(x,y) = -x^3 + 4xy - 2y^2 + 1$$ And we have the partial derivatives as follow... $$f_x(x,y) = -3x^2 ...
0
votes
1answer
21 views

Definition of an absolutely continuous random variable

Just what is the proper definition of an absolutely continuous random variable? It's supposed to be something like: $$\mathbf{P} (A) = \int_A f d \mu$$ for some Borel set $A$. But what is $\mu$? Is ...
1
vote
1answer
8 views

Convex combination and convex set

From where does $tx + (1-t)x'$ originate from? I am selfstudying an economists book, and this is popping up all of a sudden. I get that it's a line between $x$ and $x'$, but why? And is $tx' + (1-t)x$ ...
1
vote
1answer
56 views

As the limit of $n$ goes to infinity, prove that $x^n = 0$ if $\operatorname{abs}(x)<1$. [duplicate]

As the limit of $n$ goes to infinity, prove that $x^n = 0$ if $\operatorname{abs}(x)<1$. So I want to prove it this by observing that $\operatorname{abs}(x) < 1$ which means ...
0
votes
0answers
24 views

Sequence of measurable functions on a null set

I am trying to do this problem: Let $(X,\Sigma,\mu)$ be a measurable space and let $E \in \Sigma$. Let $(f_k)_{k \in \mathbb N}:E \to \mathbb R$ be a sequence of measurable functions such that for ...

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