Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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6 views

Application of the IVT

It can be shown using the IVT that on any circle on the surface of the earth, there is a pair of opposite points P and Q such that the temperature at P is equal to the temperature at Q. Is it also ...
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3answers
26 views

Let $\{a_n\}$ be a sequence with limit $\alpha$, and define $b_n=a_{n+1}$ where $n\in \mathbb{N}$. Show that $\{b_n\}\rightarrow \alpha$.

Let $\{a_n\}$ be a sequence with limit $\alpha$, and define $b_n=a_{n+1}$ where $n\in \mathbb{N}$. Show that $\{b_n\}\rightarrow \alpha$. What I have: Since $\{a_n\}\rightarrow \alpha$ we know that ...
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3answers
22 views

Find the Inverse Modulus using Euclid's algorithm

I asked this before, but unfortunately, I didnt know the methods, nor was the questions phrased properly. Find the inverse of $4258 \pmod{147}$ Using Euclidean Extended Algorithm. Begin By Stating ...
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1answer
24 views

Bounding summations

Show that $\sum k2^k = \Theta( k2^k)$. I tried to use mathematical induction to prove the bound, but it didn't work. There are other ways that can be used to prove this bound, like bounding the ...
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3answers
65 views

Proof of an inequality by induction

Let $n \in \mathbb N^+$. Show that if $x_1, x_2, ... , x_n$ are $n$ real numbers such that $-1 \le x_i \le 0$ for each $1 \le i \le n$, then $$(1 + x_1)(1 + x_2)...(1 + x_n) \ge 1 + x_1 + x_2 + ... + ...
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0answers
16 views

Lebesgue integral of cartesian product of functions

Given two Lebesgue Integrable functions $f,g$, is there a notion of the integral $$\int_A f \times g \, \, dx_1 \times dx_2 ?$$ Is this even a definable notion? I couldn't find anything on the ...
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22 views

Multiplication on Reals as equivalence classes of cauchy sequences is well defined

So I understand the solution for the proof that multiplication of equivalence classes of cauchy sequences is well defined using boundedness of cauchy sequences and a chain of inequalities. I just ...
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2answers
22 views

Let $\{b_n\}$ be a sequence with limit $\beta$. Show that if $B$ is an upper bound for $\{b_n\}$, then $\beta \leq B$.

Let $\{b_n\}$ be a sequence with limit $\beta$. Show that if $B$ is an upper bound for $\{b_n\}$, then $\beta \leq B$. What I have: Assume that $\beta>B$, so $\beta-B>0$. Since $\{b_n\}$ ...
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0answers
33 views

Inflection point and 2nd derivative

Is it possible a function $f:\mathbb{R} \rightarrow \mathbb{R}$ to have an inflection point somewhere but that it is not two times differentiable at that point? If so, then can we have a form of that ...
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2answers
38 views

How to prove the closure $\bigcup_{n\ge 1} F_{n}$ is totally bounded and closed

Let $(X,p)$ be a metric space. Write $F$ for the set of subsets of $X$ which are closed, bounded, and non-empty. For each integer $n\geq1$, write $F_{n}$ for the set of subsets of X which are finite ...
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13 views

Role of functional equations in current panorama of pure mathematics

It seems that currently functional equations are greatly explored as a research field. I would like to know what is the importance and role of such a field in the panorama of the current development ...
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1answer
29 views

If $a_n$ and $b_n$ are equivalent sequences and $a_n$ is bounded then so is $b_n$.

This is what i know; If $(a_n)$ is an infinite sequence of which is bounded then we can say; $|a_i| < M $ for all $i \geq 0.$ since $a_n$ and $b_n$ are equivalent sequences, we can say that for ...
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2answers
74 views

How do I solve this Olympiad question with floor functions?

Emmy is playing with a calculator. She enters an integer, and takes its square root. Then she repeats the process with the integer part of the answer. After the third repetition, the integer part ...
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51 views

Terrence Tao, Analysis 1. Exercise 5.3.2. Real Numbers and Cauchy Sequences.

Let $ x = \lim_{n\rightarrow\infty}a_n, y = \lim_{n\rightarrow\infty}b_n$, and $ x' = \lim_{n\rightarrow\infty}a'_n$ be real numbers. Then $xy$ is also a real number. Furthermore, is $x=x'$, then $xy ...
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2answers
55 views

Prove that $(f_n)_n$ is uniformly convergent.

Let $g$: $[0,1]\to\mathbb{R}$ be continuous and $g({1})=0$. Define $f_n(x)= x^{n}{g(x)}$. Prove that $(f_n)_n$ is uniformly convergent.
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1answer
28 views

prove $({f_n})_n$ is uniformly convergent on ${[0,1]}$

The real function ${g}$ is continuous on $[0,1]$ .we define ${f_n}$ on ${[0,1]}$: $$f_n(x)=\frac{{{g(x)\sin^{n} (x)}}}{{{1+nx}}}$$ prove $({f_n})_n$ is uniformly convergent on ${[0,1]}$ .
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1answer
23 views

Looking for example of a surjective homomorphism on $(\mathbb R,+)$ which is not an automorphism

Give example of a surjective function $f:\mathbb R \to \mathbb R$ such that $f(x+y)=f(x)+f(y) , \forall x,y \in \mathbb R$ but $f$ is not injective . I think I have to do something with basis of ...
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0answers
15 views

Exponential estimate/inequality

I have a vector $x=(x_1,\dots, x_n)\in \mathbb{R}^n$ and some variance $\sigma^2 >0$. I know that the following inequality is wrong (but I present it because it would make world nicer in my view) ...
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1answer
42 views

absolutely convergent & conditionally convergent [on hold]

Prove that $$\sum_{n=1}^{\infty}\left(\sum_{m=1}^{n}\frac{{{1}}}{{{m}}}\right)\frac{{{\sin(nx)}}}{{{n}}} $$ for $x = {k\pi}$ , $k\in \mathbb{Z} $ is absolutely convergent. & for $x ...
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77 views

Explain about absolute convergence and convergence [on hold]

$$\sum_{n=1}^{\infty}\frac{{{(-1)^{n}\sin (n)}}}{{{n}}} $$ Explain about absolutely convergent and convergent. Thx a lot.
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2answers
48 views

Prove$\sum_{n=0}^{\infty} \frac{n}{(n+1)^2}-\frac{1}{n+2} $ is convergent [on hold]

prove $$\sum_{n=0}^{\infty}\frac{{{n}}}{{{(n+1)^{2}}}}-\frac{{{1}}}{{{(n+2)}}} $$ is convergent.
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0answers
14 views

Simplification of integral region (no integration skills needed)

We have the following "formula" or simplification for integrals: Let $f_i:[0,1] \rightarrow \mathbb{R}^{d\times d}$ for $i=1,\dots,n$ and $g_j:[0,1] \rightarrow \mathbb{R}^{d\times d}$ for ...
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2answers
41 views

$ \sum_{n=1}^{\infty} \frac{{{n+1}}}{{{n}}}{a_n} $ is absolutely convergent. [on hold]

prove that if $ \sum_{n=1}^{\infty} {a_n} $ is absolutely convergent $ \sum_{n=1}^{\infty} \frac{{{n+1}}}{{{n}}}{a_n} $ is absolutely convergent.
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1answer
22 views

prove that Radius of convergence is 1 [on hold]

Let's assume that $${\left\{ {{a_n}} \right\}_{n \in {\Bbb N}}}$$ is a positive sequence number. and let $$ \mathop {\lim }\limits_{k \to \infty }{A_k}=\sum_{n=0}^{k} {a_n} = \infty $$ if $$ \mathop ...
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1answer
25 views

$\lim_{x\to x_0 ;x\in X} f ( x)$ exists if f is a uniform continuous function and $x_0$ is an adherent point

Proposition: Let $X$ be a subset of $R$, let $f:X\to R$ be a uniformly continuous function, and let $x_0$ be an adherent point of $X$. Then $\lim_{x\to x_0 ;x\in X} f ( x)$ exists. Proof Take any ...
1
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1answer
12 views

Composed of non differentiable functions

It will be possible to find a function $f:\mathbb{R}\rightarrow \mathbb{R}$ non-differentiable at zero such that $f\circ g$ is differentiable at zero where $g:\mathbb{R}\rightarrow \mathbb{R}$ is ...
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13 views

Mean value theorem for sequences

This is a problem I am trying to solve. Given a sequence $x_n$ defined $x_{n+1}=F(x_n)$. Assume $\lim_{n \to \infty}x_n=x$ and $F'(x)=0$. Need to show that $$x_{n+2}-x_{n+1}=o(x_{n+1}-x_{n}).$$ ...
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0answers
18 views

Local existence for semilinear wave equations

After having done some research, I could not find a reference for the following. Suppose I have a problem of the following type, on $(t,r) \in \mathbb{R} \times \mathbb{R}^2$: $$ \begin{array}{ll} ...
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1answer
39 views

How can I evaluate the definite integrals with limits?

How can I evaluate the following limit $$\lim_{n\to\infty}\int_{0}^{1}\left(\frac{x^2+x+1}{4}\right)^n\sin(nx)dx$$
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1answer
20 views

How do I compute this Milnor number

I need to compute $\mu (x^5+y^5)=5$ on the point $p=(0,0)\in\mathbb{C}^2$. By definition, for $f\in\mathbb{C}[x,y]$, I have $$ \mu(f)=\dim\dfrac{\mathcal{O}_{(0,0)}}{<\dfrac{\partial f}{\partial ...
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1answer
29 views

Identity of Bernoulli Numbers and Bernoulli Polynomials

Consider the Bernoulli Polynomials $B_n\in\mathbb{R}$ given as the coefficients of the series: $$\frac{t}{e^t-1}=\sum\limits_{n=0}^{\infty}B_n\frac{t^n}{n!}$$ and the Bernoulli polynomials gven by ...
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1answer
19 views

Suppose a 2-adic metric is defined. Showing that if $d(x,y)$ has a midpoint, then $x=y$

Let $\mathbb{Z}$ be the integers. Recall 2-adic metric $$ d(x,y) = \begin{cases} 0 & x=y \\ \frac{1}{2^{n}} & x \ne y\ \text{and}\ 2^{n} \text{is the largest power of 2 that ...
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0answers
26 views

Evaluate an integral

I want to prove the following integral $I$ is finite: Let $d \in \mathbb{N} $ and $r>d$ \begin{align*} I=\int_{S} \frac{1}{|x-y|^{2r}}dxdy \end{align*} where ...
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1answer
41 views

Let $f$ and $g$ differentiables such that $|f'(t)| \le g'(t)$, for all $t \in [0,1].$ Prove that $|f(1)-f(0)| \le g(1)-g(0)$

Let $f:[0,1] \rightarrow \mathbb{R}^m $ and $g:[0,1] \rightarrow \mathbb{R}$ differentiables such that $|f'(t)| \le g'(t)$, for all $t \in [0,1].$ Prove that $$|f(1)-f(0)| \le g(1)-g(0)$$ Comments ...
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0answers
13 views

Test functions are dense in $L^p$?

I was wondering about the following: If we say that the test functions are dense in $L^p$, does this imply that there is also always a sequence of them converging pointwise and in $L^p$ norm to such a ...
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0answers
27 views

Question about contractible set .

Please if i have a contractible and closed set $A$ in $X$ thene $A$ is closed and there existe a continuous function $H:[0,1]\times A\rightarrow X$ such that $H(0,u)=u, H(1,u)=p\in X.$ If i ...
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37 views

Example of compacts $K_1$ and $K_2$ such that $dim_H(K_1 + K_2) > dim_H(K_1) + dim_H(K_2)$

guys! I'm looking for an example of compact sets $K_1$ and $K_2$ that show Hausdorff dimension doesn't satisfy in general the inequality $$ dim_H(K_1 + K_2) \leqslant dim_H(K_1) + dim_H(K_2). $$ ...
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2answers
47 views

the choice of 2 when proving the limit when $x\to\infty$

Suppose that $f$ is a continuous function on $\mathbb{R}$ and $\lim_{x\to -\infty}f(x)$ and $\lim_{x\to -\infty}f'(x)$ exist. Show that $\lim_{x\to -\infty}f'(x)=0$ A common way to show this is ...
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1answer
28 views

Injective holomorphic function is conformal(i.e. nonzero derivative)

STATEMENT: If $f:U\rightarrow V$, where $U,V$ are open subsets of $\mathbb{C}$, is holomorphic and injective, then $f'(z)\neq 0$ for all $z\in U$. Proof: We argue by contradiction, and suppose that ...
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4answers
42 views

Prove a Continuous Distribution Function is Uniformly Continuous

Let $F$ be the distribution function for a random variable $X$ and it is given that $F$ is continuous over the entire real line. Prove that $F$ is uniformly continuous over the real line. My ...
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10 views

Finite Variation Function.

Let $V$ be a right continuous BV function and put $V_t = \int_0^t a_s dC_s$ where $C$ is increasing and right continuous. Is it true that if $V$ is continuous then $\int_0^t |f_s a_s| dC_s < ...
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7 views

Divergence theorem and almost everywhere smooth boundary

Let $\Omega \subset \mathbb R^n$ be an open set whose boundary is almost everywhere regular and oriented ($\mathcal C^2$ class). For each vector field $F \colon \Omega \to \mathbb R^n$ ($\mathcal C^1$ ...
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1answer
29 views

Pass the lower limit to $-\infty$ for an integral of positive function

Hello I have an very elementary calculus problem. Let $\phi(\eta)$ be a real value function satisfying \begin{equation} \phi(-\infty)=1,\quad \phi(+\infty)=0, \end{equation} Let $g$ be a positive ...
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15 views

Taylor polynomial to find an approximation

Use the Taylor polynomial of degree 5 to give an approximation for ln(2) This may seem really simple but I have no idea how to do it, please help.
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1answer
58 views

Gauß and mean curvature

I was wondering whether the Gauß, mean curvature and shape operator of a surface actually depend on the chosen parametrization? Under a reparametrization of $f: \Omega \subset \mathbb{R}^2 ...
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46 views
+50

Isometric and conformal map

We defined conformal and isometric maps for surfaces $f,g: \Omega \subset \mathbb{R}^2 \rightarrow S \subset \mathbb{R}^3$. Under a reparametrization of $f$ I understand a diffeomorphism $\Phi : M ...
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3answers
26 views

Questions about coerciveness and convexity

I just have a few yes/no questions, and would really appreciate if you could correct me where I am wrong, and for what fundamental flaw I have. 1. Would the set of coercive functions a linear space? ...
1
vote
2answers
105 views

Prove log(xy) = log x + log y

I would like some help with proving the following theorem, which I found in some lectures notes on analysis: If $x, y > 0$ then $\log(xy) = \log x + \log y$. Hint: Let $f(x) = \log(xy)$. ...
2
votes
1answer
29 views

Prove that $(\|x\|^p_X + \|y\|^p_Y)^{1/p}$ is a norm

Let $X$ and $Y$ be normed spaces equipped with the norms $\|\cdot\|_X$ and $\|\cdot\|_Y$, then prove that the following defines a norm on $X\times Y$ for $1\le p < \infty$: $\|(x,y)\| := ...
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0answers
18 views

When has the Fourier transform for some values equal values?

Definition We take a function $F : \mathbb T^n \rightarrow \mathbb R$ that is even ( $F(x)=F(-x)$) and continuous (hence bounded), where $\mathbb T^n$ is the $n$-dimensional Torus. Now we define the ...