0
votes
1answer
25 views

A set, which appropriately scaled is expressible as sums of elements of a compact set is pre-compact

Assume $X$ is a Banach space and $K\subseteq X$ is compact. Let $C\subseteq X$ be such that $(\forall x\in C)(\exists x_1,x_2\in K)(2x=x_1+x_2)$ Does it follow that $C$ is pre-compact? In particular ...
1
vote
2answers
107 views

The $\cos(\sin 60^\circ)$

I stumbled across this question and I cannot figure out how to use the value of $\cos(\sin 60^\circ)$ which would be $\sin 0.5$ and $\cos 0.5$ seems to be a value that you can only calculate using a ...
3
votes
1answer
99 views

Does every countable subset of the set of all countable limit ordinals have the least upper bound in it?

I'm sorry if the question is that kind of trivial, I just feel uncertain about these ordinals all the time. Is the answer to the following question "yes": Denote by A the set of all countable limit ...
1
vote
1answer
100 views

Closed set in $l^1$ space

Let $$ X := \left \{ (a_n) : \sum_{n=0}^\infty |a_n| < \infty \right\}$$ with the metric $d(a_n,b_n) := \sum_n |a_n-b_n|$. Let $\delta_j^{(n)} := 1$ if $n = j$ and $0$ otherwise. Denote ...
0
votes
1answer
52 views

Extension of a metric defined on a closed subset

If $X$ is any metrizable space, $A$ is a closed subset of $X$. Let $d$ be a compatible metric on $A$ then $d$ can be extended to a compatible metric on $X$.
1
vote
1answer
88 views

Integral $\int_{-\infty}^{\infty} e^{−k^2/(k-k_0)^2}dk$

So I have here the integral $\int_{-\infty}^{\infty} e^{−k^2/(\Delta k)^2}dk$, but then $\Delta k$ is equal to $k-k_0$ so this equation becomes $\int_{-\infty}^{\infty} e^{−k^2/(k-k_0)^2}dk.$ The ...
2
votes
1answer
40 views

Concept Of Double Integration

Can someone explain how double integration is equivalent to calculating volume as single integration is calculating area.
2
votes
2answers
310 views

Non-trivial Topology

I can't understand the differences between a non-trivial topology and a trivial one. Whuat's the meaning of "non-trivial" topology? Is there a link with connection's properties? For example, could we ...
1
vote
1answer
97 views

Fundamental theorem of Morse theory for $\Omega(S^n )$

Using the Fundamental theorem of Morse Theory we can prove that $\Omega(S^n)$ is homotopically equivalent to a CW complex with one cell each in dimensions $o,n-1,2(n-1), \cdots$ and so on. But how can ...
0
votes
2answers
67 views

Which of these sets is a subspace of F?

Let $F = \mathbb{R}^\mathbb{N}$. I need to check which of these sets are subspaces of $F$: $F_1 := \{ x \in F:\ \text{$x$ is bounded}\}$, $F_2 := \{ x \in F:\ \text{$x$ is convergent}\}$, $F_3 := \{ ...
1
vote
3answers
681 views

A subgroup $H$ is normal in $G$ iff $H$ is the union of some conjugacy classes of $G$

How to show that: A subgroup $H$ is normal in $G$ iff $H$ is the union of some conjugacy classes of $G$.
2
votes
1answer
65 views

Complex Differential Equation: $f'(z)=bf(z) \iff f(z)=ae^{bz}$

Let $f\colon G\to\mathbb{C}$ be holomorphic on the domain $G\subseteq\mathbb{C}$ and $b\in\mathbb{C}$. Show that the two following statements are equivalent: 1) $f(z)=ae^{bz}$ on $G$ with ...
0
votes
2answers
133 views

Approximate a Discrete time dynamical system by a continuous one

Is there any method by which we can somehow "embed" a non linear discrete time system into a continuous time dynamical system? (Assume discrete time system here is a set of non linear difference ...
2
votes
1answer
140 views

What do countable transitive models of ZFC look like?

According to Cantor's Attic (link): Not all transitive models of ZFC have the $V_\kappa$ form, for if there is any transitive model of ZFC, then by the Löwenheim-Skolem theorem there is a ...
1
vote
2answers
179 views

motivation of additive inverse of a Dedekind cut set

My understanding behind motivation of additive inverse of a cut set is as follows : For example, for the rational number 2 the inverse is -2. Now 2 is represented by the set of rational numbers less ...
1
vote
1answer
2k views

Find the volume of the body bounded by $z = x^2 + y^2, z= 1-x^2-y^2$.

Again, I am new to volume of bodies and I am struggling with it. Find the volume of the body bounded by $z = x^2 + y^2, z= 1-x^2-y^2$. Now from a previous question, I know that I can do it by ...
1
vote
3answers
65 views

Factor Equations

Please check my answer in factoring this equations: Question 1. Factor $(x+1)^4+(x+3)^4-272$. Solution: $$\begin{eqnarray}&=&(x+1)^4+(x+3)^4-272\\&=&(x+1)^4+(x+3)^4-272+16-16\\ ...
13
votes
2answers
243 views

Prove that $m^{2013}-m^{20}+m^{13}-2013$ has at least $N$ prime divisors

for positive integer $N>1$,There always exists $m$ such that $$m^{2013}-m^{20}+m^{13}-2013$$ has at least $N$ prime divisors Thank you all, this is good problem, but I don't know how to solve it.
1
vote
1answer
55 views

Prove automorphism is trivial

I would like to prove the following: Let $L\subset L'$, where $L'$ is a quadratic extension of $L$, and $\rho\in\text{Aut}(L'/L)$, the automorphism group of $L'$ which fixes $L$. Also, let ...
0
votes
0answers
145 views

Pfaffian And Determinant

I am working in tilings using Pfaffian. There is a basic property namely: Let $ B$ be a $n\times n$ Matrix and $$ A = \begin{pmatrix} 0 & B\\ -B^T & 0 \end{pmatrix}$$ then ...
10
votes
1answer
382 views

Inequality $\frac{1-3ab}{1-2ac}+\frac{1-3bc}{1-2ba}+\frac{1-3ca}{1-2cb}\geq 0$

Let $a\ne 0$, $b\ne 0$ and $c\ne 0$ such that $a^2+b^2+c^2=1$. Prove that: $$\dfrac{1-3ab}{1-2ac}+\dfrac{1-3bc}{1-2ba}+\dfrac{1-3ca}{1-2cb}\geq 0.$$ My attempt to the solution: We get that $ab ...
0
votes
1answer
56 views

Hilbert transform and maximal function

What is the relation between the Hilbert transform (in $\mathbb{R}$) and maximal functions?
0
votes
0answers
124 views

Worst case lower bound for comparison sort

It is known that in the worst case comparison sort takes $\Omega(nlgn)$ of time for n elements. Let's say I'm given $n$ elements as input to sort. Each input sequence of $n$ elements has $n/k$ ...
3
votes
2answers
215 views

Textbook Questions to Do while Self-learning

I am working through Dummit and Foote's Abstract Algebra this summer in preparation for a class next year. However, this is my first time really trying to learn a subject through only a text. It seems ...
4
votes
3answers
167 views

How to show that $H \cap Z(G) \neq \{e\}$ when $H$ is a normal subgroup of $G$ with $\lvert H\rvert>1$

Let $G$ be a group of order $p^n$, $p$ a prime, $n>1$ and $H$ a normal subgroup of $G$ with $\lvert H\rvert>1$. Show that $H \cap Z(G) \neq \{e\}$.
4
votes
2answers
150 views

Minimal geodesics in $S^{n+1}$

Let $\Omega^d(M)$ the space of minimal geodesics on a smooth manifold $M$. How can I prove that if $M= S^{n+1}$, $\Omega^d(S^{n+1}) \simeq S^n$?
1
vote
1answer
133 views

What is $(V^*)^{\otimes l} \stackrel{f}{\rightarrow} S^l( (V^*)^{\oplus l} )\cong S^l( (V^{\oplus l})^* ) \stackrel{g}{\rightarrow} S^l(V^*)$?

Let's work over an algebraically closed field and let $V$ be a finite dimensional vector space. Then is there a natural chain of maps $$ (V^*)^{\otimes l} \stackrel{f}{\rightarrow} S^l( (V^*)^{\oplus ...
0
votes
1answer
220 views

Computing probability in roulette

Andy plays european roulette and bets the lowest amount of 10 dollar at red and he's going to double the amount until he wins. Which is the probability he wins before he has spent all his 1 000 dollar ...
1
vote
2answers
46 views

How to evaluate base indefinite integrals

I can find a list of known integrals anywhere, but how would I build this list myself? For example how can I prove that $\int x^2 dx = x^3 / 3$? I want to understand the general theory. It would be ...
4
votes
1answer
41 views

For what kind of a subset its sums equal $\mathbb{R}^4$

For short, suppose $a,b$ are real numbers. Let $A=\{(\cos(at), \cos(bt), \sin(at), \sin(bt))\mid t\in \mathbb{R}\}$. Let $B=\sum A=\{\sum_{i=1}^n x_i\mid x_i\in A, n \geq 1\}$. For what values ...
1
vote
2answers
113 views

Can we conclude from $V=\ker(T) \oplus\operatorname{im}(T)$ the invariance of both subspaces?

Can we conclude for an endomorphism $V \in \operatorname{End}(V)$ where V is a finite dimensional vector space from $V=\ker(T) \oplus \operatorname{im}(T)$ that nullspace and image are invariant ...
3
votes
3answers
217 views

what are first and second order logics? [duplicate]

The only knowledge I have on logic is due to a book I read a couple of years ago called Introduction to logic: and to the methodology of deductive sciences by Alfred Tarski. And in it he talks about ...
3
votes
4answers
79 views

$\neg P \implies \neg T$ and $P \implies \neg T$. Where do I go next?

I can't find any logic equivalence or inference rules on this. Personally, I feel that $\neg P \implies \neg T$ and $P \implies \neg T$ would mean that it follows that $\neg T$ is true regardless, and ...
6
votes
2answers
123 views

Prove that the series $\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$ converges

Let $f$ be a non-negative decreasing function on $[1,+\infty)$. Prove that the series $$\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$$ converges.
0
votes
2answers
82 views

A question about of normal subgroups

Suppose that $G$ is a finite group. Is it true to say that "If $M\lhd G$, then $M$ is containing a minimal normal subgroup of $G$"?
1
vote
1answer
63 views

problem of probability and distribution

Suppose there are 1 million parts which have $1\%$ defective parts i.e 1 million parts have $10000$ defective parts. Now suppose we are taking different sample sizes from 1 million like $10\%$, ...
0
votes
2answers
394 views

A question about bounds, least and minimal elements, and partial vs strict ordered sets

It's not very clear to me if the concepts of bounds, least elements and minimal elements (also, greatest elements and maximal elements, etc. ) apply only to partial orders or if the definition applies ...
4
votes
2answers
293 views

Do isomorphic structures always satisfy the same second-order sentences?

I know that if two mathematical structures are isomorphic, then they satisfy the same first-order sentences. The converse is false. This is probably a completely obvious question, but is it true that ...
0
votes
1answer
104 views

Norm inequality (supper bound)

Do you think this inequality is correct? I try to prove it, but I cannot. Please hep me. Assume that $\|X\| < \|Y\|$, where $\|X\|, \|Y\|\in (0,1)$ and $\|Z\| \gg \|X\|,\|Z\| \gg \|Y||$. prove that ...
6
votes
1answer
437 views

Motivation behind introduction of measure theory

Is the motivation behind the introduction of measure theory the Lebesgue integral? In order to evaluate such an integral we need the length of each of the horizontal strip of width $h$. I have a ...
3
votes
3answers
463 views

$f$ integrable, $g$ measurable, $f = g$ almost everywhere implies $g$ integrable

If $f\in L(X,\mathcal{x},\mu)$, that is: $f\colon X\to R$ is measurable; $\int f^+\,d\mu<+\infty$ and $\int f^-\,d\mu<+\infty$; $\int f\,d\mu=\int f^+\,d\mu-\int f^-\,d\mu$. If $g\colon X\to ...
0
votes
1answer
367 views

Calculating $1819^{13} \pmod{2537}$ using Fermat's little theorem

Can anyone make me understand how to calculate $1819^{13} \pmod{2537}$ using Fermat's little theorem? Here $p=2537$ and $p-1=2537-1=2536$. I am unable to understand how to express $1819^{13}$ in ...
0
votes
2answers
404 views

Find k such that f is density function

I have the following function: $f_X(x, \theta) = \left\{ \begin{array}{lr} k/x^3 & : x \leq \theta \\ 0 & : x > \theta \end{array} \right.$ and $\theta >0$. I ...
1
vote
1answer
42 views

Which sentences are preserved by equivalence of quasi-ordered sets?

Given a quasi-ordered set $(X,\lesssim)$, we can define that for $a,b \in X$ we have $$a \sim b \;\Leftrightarrow\; a \lesssim b \;\wedge\; b \lesssim a.$$ and thereby obtain a partially ordered set ...
1
vote
0answers
394 views

What does it mean by piecewise smooth boundary?

I will be highly obliged if anyone can give me any reference where i can get the definition of domain (in $\mathbb{R^n}$) with piecewise smooth boundary. My question is when a domain in ...
1
vote
0answers
53 views

Specific Modular Arithmetic Question with Exponentiation

Are there any theorems that can be used to reduce $1213^{797} \pmod {2591}$ without using a computer?
0
votes
2answers
266 views

Factorial related problems

How many zeros are there in $ 25!$ ? My answer was $6$. But i solved it by finding how many numbers are divisible by $5$ and $2$.here i was told to find out the zeros at the last end. But what is the ...
0
votes
1answer
135 views

Express $[\cos(x) + \sqrt3 \sin(x)]$ in the form $[r\cos(x-a)]$

Express $[\cos(x) + \sqrt3\sin(x)] $ in the form $[r\cos(x-a)]$, where $r>0$ and $ 0\leq360$, hence solve the equation $[\cos(x) + \sqrt3\sin(x)= \sqrt2]$ This is as far as i have completed. I ...
1
vote
1answer
114 views

Calculating radius for $\sum\frac{\sin{n^3x}}{n^2}$

Let $\displaystyle \sum\limits_{n=1}^{\infty}\frac{\sin{n^3x}}{n^2}$ be a series: a. Find where the series converge pointwise and where uniformally. b. Does its derivative is continuous? About A: ...
17
votes
5answers
601 views

A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$

This is a problem from "A Course of Pure Mathematics" by G H Hardy. Find the limit $$\lim_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$$ I had solved it long back (solution presented in my blog ...

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