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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19 views

find special limit in infinite and compute

find $$\lim_{n\to\infty}\left( \frac{a^\frac{1}{n}}{n+1}+ \frac{a^\frac{2}{n}}{n+\frac{1}{2}}+\cdots +\frac{a^\frac{n}{n}}{n+\frac{1}{n}}\right)$$ when $n\to\infty$ please help me to find it.when ...
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1answer
16 views

Prob. 6, Sec. 21 in Munkres' TOPOLOGY, 2nd ed: How to show directly that this sequence of functions does not converge uniformly?

For each $n = 1, 2, 3, \ldots$, let $f_n \colon [0,1] \to \mathbb{R}$ be defined by $$f_n(x) \colon= x^n \ \ \ \mbox{ for all } \ x \in [0,1].$$ Then $$ \lim_{n \to \infty} f_n(x) = \begin{cases} ...
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13 views

Corollary of Inverse Function Theorem?

Recall the inverse function theorem: Theorem. Lef $U\subseteq \mathbb R^n$ be an open subset, $p\in U$ and $f\in C^1(U, \mathbb R^n)$. If $Df(p)\in \textrm{Aut}(\mathbb R^n)$ there exists an open ...
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23 views

Show that a sequence (($x_n, y_n$)) in $X \times Y$ is $e$-Cauchy if the component sequences ($x_n$) and ($y_n$) are $d_X$-Cauchy and $d_Y$ -Cauchy.

How to solve this? Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces and let $e$ be a product metric on $X\times Y$. Show that a sequence (($x_n, y_n$)) in $X \times Y$ is $e$-Cauchy if the component ...
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1answer
14 views

Why is $\Delta u$ bounded, if $u\in C^2(\overline{\Omega})$ and $\Omega\subseteq\mathbb{R}^n$ is a bounded domain?

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $u\in C^2(\overline{\Omega})$. Why must $\Delta u$ be bounded?
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1answer
51 views

$\sum_{i=1}^n\frac{1}{y_i^2}=1$ and $y_{min} \rightarrow \infty$, prove $\lim_{n \rightarrow \infty }\sum_{i=1}^n e^{-y_i}=0$

we have two conditions, with $y_{n,i}>0$: $$\sum_{i=1}^n\frac{1}{y_{n,i}^2}=1$$ $$min(y_{n,i}) \rightarrow \infty \quad as \;n \rightarrow \infty$$ do we have $$\lim_{n \rightarrow \infty ...
1
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1answer
12 views

Preimage surfaces.

Let $U\subset\mathbb{R}^{m+n}$ open set, $f:U\longrightarrow\mathbb{R}^n$, $f\in C^{k}$, and $c\in\mathbb{R}^n$. Set: $$M=\{p\in U; f(p)=c\textrm{ and ...
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3answers
120 views

What is the limit $\lim_{n\to\infty}\frac{1^n+2^n+3^n+\dots+n^n}{n^{n+1}}$?

I want to determine the limit given below: $$\lim_{n\to\infty}\frac{1^n+2^n+3^n+\ldots+n^n}{n^{n+1}}$$ I have tried to solve thise several times ,but with no results.I have tried using lema stolz ...
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11 views

Is the function $U\colon Y\to Y'$ finite-to-one?

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and on it the map $T\colon X\to X$ which describes the following dynamics: For $x\in X$, which is a bi-infite sequence, let $x(i)$ denote the i-th position. ...
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1answer
12 views

How to solve this for $k_i$?

Let $\epsilon >0$ and $$ \max_{0\leqslant i\leqslant n-1}2^{-k_i}\leqslant\epsilon. $$ How can I solve this for $k_i$? It's the max that confuses me. -- $2^{-k_i}\leqslant\epsilon\iff k_i > ...
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1answer
27 views

Convergence of $\partial_{x_j} u(x,t)$ when $u$ converges in $L^2$ norm.

I hope you can help me with this question. We take $u(x,t)\in L^\infty_{loc}(\mathbb{R},H^1(M))\cap Lip_{loc}(\mathbb{R},L^2(M))$, the derivatives $\partial_{x_j} u $ exist and are continuous, i.e ...
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1answer
40 views

What can we say about the inner product of two Cauchy sequences?

Let $(x_n)$, $(y_n)$ be two Cauchy sequences in an inner a real or complex product space $X$, and let the sequence $(\alpha_n)$ be given by $$ \alpha_n \colon= \ \langle x_n, y_n \rangle \ \ \ \mbox{ ...
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1answer
27 views

For differentiable function where $f'(0)=a$ and $f'(1)=b$ we have that for all $c\in(a,b)$ there exists a $y$ such that $f'(y)=c$.

So what I'm trying to prove: Assume a function $f\colon\mathbb{R}\to\mathbb{R}$ is differentiable and $f'(0)=a$ and $f'(1)=b$. Prove that for any $c\in(a,b)$ there exists a $t\in\mathbb{R}$ such that ...
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1answer
44 views

Is the Inner Product a uniformly continuous function?

I know it's continuous but is it uniformly continuous?
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2answers
74 views

Why $\int _c^df^{-1}\left(y\right)\:dy+\int _a^b\:f\left(x\right)dx=b\cdot d-a\cdot c$? [duplicate]

Why $\int _c^df^{-1}\left(y\right)\:dy+\int _a^b\:f\left(x\right)dx=b\cdot d-a\cdot c$ ? where f is an bijective function and $f(a)=b,f(c)=d,$ I don't understand graph... I can't see on graph this ...
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1answer
27 views

Angle-doubling map is mixing

Let $$ T:\mathbb{S}^1\to\mathbb{S}^1\\ x\mapsto 2x $$ be the angle-doubling map on the circle. We know that this transformation is ergodic. We want to prove that is mixing. I have to show that $$ ...
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2answers
74 views

Graph connected does not imply $f$ is continuous [on hold]

Show an example of a function $\newcommand{\R}{\mathbb{R}} f: \R \times \R\to \R$ such that $f$ is not continuous, but its graph $$ \Gamma_f := \left\{\bigl((x, y), f(x, y)\bigr) \mid \text{$(x, y)$ ...
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2answers
17 views

How do I evaluate the Wronskian for this equation

Martin Braun - Differential equations and their applications Chapter 2.1 p.137 Let $y_1,y_2$ be solutions of Bessel's equation $t^2y'' + ty' + (t^2-n^2)y=0$ on the interval $(0,\infty)$ with ...
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2answers
45 views

Expanding a function

Is it possible to expand a function $$ f(x,y) = \dfrac{\sin (xy)}{\sqrt{x^2 + y^2}} $$ so it will be continuous on $\mathbb{R}^2$? Now, the denominator should not be equal to $0$, so for the domain, ...
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2answers
23 views

Continuous multivariable function without limits in a point

I am curious, if there can be a function $f(x,y)$, which is continuous in a point $[0,0]$, but for which iterated limit $\lim _{x \to 0} \lim _{y \to 0} (f(x,y))$ does not exist. Is it even possible ...
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0answers
22 views

Application of the Operator norm $\|.\|_O$ on the differential $df \in \hom(\mathbb{R}^n, \mathbb{R}^m)$

This question origins from my Analysis II Script which gives the following statement (without proof): Lemma Let $U \subset \mathbb{R}^n$ be convex and $f \in C^1(U, \mathbb{R}^k)$ then we have $$ ...
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1answer
80 views

The importance of being real

Let $\Sigma$ be a collection of holomorphic, one-to-one function from some simply connected region $\Omega$, which map $\Omega$ into the open unit disc $U$. Fix $z_0 \in \Omega$ and put $$\eta = ...
0
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1answer
6 views

Do the elements of a sequence converging to a point in the intrinsic core of a convex cone belong to the intrinsic core of the set eventually?

Let $X$ be a general Banach space and let $C\subset X$ be a convex cone. Consider a sequence $x_n$ in the affine hull of $C$ such that $x_n\to x$ for some $x\in icr(C)$, where $icr(C)$ denotes the ...
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37 views

Finiteness of the set of zeros

Please help me with the following problem: Let $f$ be a continuous function, linearly bounded, i.e. $$x+A<f(x)<x+B$$ We also now, that there exists an $x_0>0$ such that $$f(x)=x+C,\quad ...
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1answer
17 views

What is the difference between a trajectory and an orbit in dynamical systems?

In dynamical systems, the integral curves for a differential equation that governs a system are referred to as trajectories or orbits, then what is the difference bewteen they?
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1answer
21 views
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3answers
50 views

$\lim_{x\to\infty}{f(x)}=\lim_{x\to\infty}{g(x)}\Rightarrow\lim_{x\rightarrow\infty}{\frac{f(x)}{2^x}}=\lim_{x\rightarrow\infty}{\frac{g(x)}{2^x}}$?

Is this statement true? Why? $$\lim_{x \rightarrow \infty}{f(x)} =\lim_{x \rightarrow \infty}{g(x)} \quad \Rightarrow\quad \lim_{x \rightarrow \infty}{\frac{f(x)}{2^x}}=\lim_{x \rightarrow ...
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0answers
17 views

How to prove that $C([0,1])$ is not dense in $L^{\infty}([0,1])$ [duplicate]

Prove that $C([0,1])$ is not dense in $L^{\infty}([0,1])$.
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2answers
30 views

Show restriction map is a contraction/lipschitz mapping

For $C[a,b]$ (set of all continuous real valued functions), define $d(f,g) = \int^{b}_{a}|f(x)-g(x)|dx$ If $[c,d]$ is a subinterval of $[a,b]$ and the mapping $r:C[a,b] \rightarrow C[c,d]$ ...
2
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1answer
33 views

Divergent succession, but with convergent sum average.

An example of a sequence $a_n$ such that: $$a_n\rightarrow\pm\infty$$ but $$b_n=\frac{\sum_{k=1}^{n}a_k}{n}$$ converge.
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35 views

An interesting question about sequence.

Let $a^{(k)}=(a_j^{(k)},j=1,2,3,...) \in l_{\infty}$ be a sequence such that $\|a^{(k)}\|\le M$ for all $k-1,2,3,...$. Show that there exists a sub-sequence $a^{k_m}$ and $a\in l_\infty$ such that ...
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0answers
39 views

Normal space is compact

I know that a compact Hausdorff space implies Normal, but does the converse holds? I.e. If a space is normal, it is compact and Haudorff. (Although $T_4$ imlicitly implies $T_2$)
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1answer
36 views

Show that a set is not open

Suppose $U_1$ and $U_2$ are both nonempty subsets of $\mathbb R$ such that $U_1 \cap U_2 =\emptyset $ and $U_1\cup U_2 = \mathbb R.$ Consider points $p \in U_1\ \text{and}\ q \in U_2.$ Without loss ...
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1answer
26 views

$p$-adics with the least upper-bound property

It is well-known (I believe?) that the $p$-adics do not admit an ordering in the 'usual sense', the "usual sense" being a total order that is compatible with the field operations. I do not want to ...
2
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0answers
29 views
+50

Nonlinear heat equation $u_{t} = \Delta(u^{4})$

Consider the nonlinear heat equation $u_{t} = \Delta(u^{4})$ in $\{x \in \mathbb{R}^{3}: |x| < 1\}$ with $u = 0$ on $\{x \in \mathbb{R}^{3}: |x| = 1\}$. The problem I am working on is to show that ...
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1answer
25 views

Unsolved regular value problem in $\mathbb{R}^n$

I want to show that if $F : \mathbb{R}^n \rightarrow \mathbb{R}^{n-k}$ is a $C^1$ function and $rank(DF) = n-k$ then $M:=F^{-1}(\{0\})$ defines a manifold. My idea: Without loss of generality I ...
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1answer
32 views

Show that $\sum_{n = 1}^\infty n^qx^n$ is absolutely convergent, and that $\lim_{n \rightarrow \infty}$ $n^qx^n = 0$

I'm having trouble with proving the following for my math study: Let $x$ be a real number with $|x| < 1$, and $q$ be a real number. Show that the series $\sum_{n = 1}^\infty n^qx^n$ is absolutely ...
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0answers
31 views

Prove the uniform convergence

Let $a_n$ be a monotonic sequence convergent to a. Let f : R $\to$ R be a continous and monotonic function. Then we define a series of functions as follows : $$f_n(x) := f(x+a_n)$$ Prove that the ...
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0answers
18 views

Do smooth solutions of $u_{t}(x, t) = \Delta u_{t} + u$ satisfy $\sup_{0 \leq t \leq T}\|u(\cdot, t)\|_{L^{2}_{x}} = \|u(x, 0)\|_{L^{2}}$?

Let $u(x, t) : \mathbb{R}^{d} \times [0, T) \rightarrow \mathbb{R}$ be a smooth solution to $$u_{t}(x, t) = \Delta u_{t} + u$$ with $u(x, 0) = u_{0}(x) \in L^{2}(\mathbb{R}^{d})$. Furthermore, suppose ...
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1answer
27 views

Is the following a valid characterisation of complete metric spaces?

A metric space $(X, d)$ is called complete if and only if every Cauchy sequence converges. Now does the following hold: A metric space is complete if and only if every sequence $(x_i)_{i\in\mathbb ...
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12 views

Does this have the same limit superior?

Let $\gamma_{n,g}$ be the number of ways of putting down g $\ell$'s on the discrete interval $[0,n-1]$ with the $\ell's$ separated by at least two $0$'s. Let $\gamma_n=\sum_{g=0}^n \gamma_{n,g}$. ...
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1answer
26 views

Formula for the Beta function for natural m, n

Using only the definition $$B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt$$ for the Beta function $B(x, y)$, it's symmetry $B(x,y) = B(y,x)$ aswell as the fact that $(x + y)B(x + 1, y) = xB(x, y) ...
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33 views

Why can we choose such a $m$? Where does the $m$ come from?

Let $a_{n,g}$ be the number of ways of putting down g $\ell$'s on the discrete interval $[0,n-1]$ with the $\ell's$ separated by at least two $0$'s. Let $a_n=\sum_{g=0}^n a_{n,g}$. Suppose that $$ ...
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2answers
33 views

Is $C_0(\mathbb R)$ separable?

Let $C_0$ be the Banach space of all continuous real value functions whose limits in $\pm \infty$ is zero, with the supremum norm. Is this space separable?
2
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1answer
14 views

Upper bound for Estimation Lemma

I am struggling with the following question using the Estimation Lemma: Let $ \gamma$ describe the semi-circle $Re^{it}$, where $ 0 \le t \le \pi$, and $ R \gt 3$. Show that $$\int_\gamma ...
0
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2answers
43 views

Is this integral correctly calculated?

The problem is that I can't use wolframalpha to check this because he is worried about integration limits: I have $a>0$ and $t \in (-1,1).$ $$(1-t)^{\frac{a}{2}} \int_0^t \frac{1}{(1-x)^{a+1} }dx= ...
0
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2answers
30 views

Where is the following sequence convergent/absolute convergent?

I have the following sequence: $\sum_{n=1}^\infty x^n\tan \frac{x}{2^n}$ Any idea how to decide this question? It is obvious that $x^n$ goes to infinity if $|x|>1$, but how does the $\tan ...
3
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2answers
38 views

Sup and inf of $n \sin(1/n)$

If $n$ is a natural number then, what is the supremum and infimum of $n\sin(1/n)$? is the question I want to solve. I drew $sin(x)/x$ graph and I think that the supremum is $1$ and infimum is ...
0
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2answers
47 views

Companion Books for Rudin's PMA

S.E friends, I am a college sophomore with double majors in mathematics and microbiology. I wrote this email to seek your recommendation on selecting the introductory analysis textbook, particularly ...
1
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1answer
9 views

Example of a smooth $f$ such that $\sup_{t \in [0,1]}(f(t)/M - t)$ is not attained at $t = 0$

Let $f: [0, 1] \rightarrow \mathbb{R}$ be a non-negative smooth function which is not identically zero. Let $M := \sup_{t \in [0, 1]} f(t)$. Is there an example of an $f$ such that $$\sup_{t \in [0, ...