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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Uniform convergence for a sequence of function

I have to show that if $f_n$ is a sequence of bounded functions that converges uniformly to $f$ on an interval I, then $f_n$ is uniformly bounded. I understand what uniformly bounded means, there ...
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42 views

The Least Characteristic of Shapes in $\mathbb{R}^n$

Fix the following notations: "Shape" denotes a closed curve in $\mathbb{R}^2$ or a closed surface in $\mathbb{R}^3$. $P$ denotes the circumference of a shape in $\mathbb{R}^2$. $A$ denotes the area ...
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1answer
28 views

Exterior differentiation, changing variables

Let $\eta=a(x,y,z)dx\wedge dy+b(x,y,z)dy\wedge dz+c(x,y,z)dx \wedge dz$. How to express $\eta$ in spherical coordinates, that is in form of $\Phi^{\ast}\eta=A(r,\theta,\varphi)dr\wedge ...
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40 views

Component of an Open Set: Polygonal Arcs

So I'm reading my Complex Analysis book, and I'm a little confused. Specifically, I'm puzzled by why it is that we need an open disk that does not intersect any of the line segments of the polygonal ...
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55 views

What's the most elegant way to show $x_{n+1}=\frac{2x_n^3+a}{3x_n^2}$ converges against $\sqrt[3]{a}$?

Let $1<a\in\mathbb{R}$, $x_0>\sqrt[3]{a}$ and $$\displaystyle x_{n+1}=\frac{2x_n^3+a}{3x_n^2}\;\;\;\;\;(n\in\mathbb{N}_0)$$ It's easy to show that it holds: $x_n>\sqrt[3]{a}$ ...
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83 views

Does $\sum_{n=0}^{\infty}\frac{4^n}{4^{n+1}}$ Diverge Or Converge?

I am told it diverge, however surely; $$\frac{4^n}{4^{n+1}} = \frac{4^n}{4\cdot 4^n} = \frac{1}{4}$$
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2answers
57 views

Derivatives and what is a good definition?

So I have a question in general about derivatives. I understand that the formal definition is something like $f$ is differentiable at $x=a$ if the limit exists where that limit is either the limit as ...
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2answers
62 views

Does there exist a subset $S\subset\mathbb R$ such that inf $\{a>0:S+a=\mathbb R-S\}=0$?

I founded the following question a good challenge in real analysis and topological properties of real line... Does there exist a subset $S\subset\mathbb R$ such that inf $\{a>0:S+a=\mathbb ...
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1answer
113 views

Why isn't the Dirichlet Function Riemann Integrable?

Can we find out the upper and lower sums based solely on the domain [a,b] ?
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72 views

Find the limit of the following: $\lim_{n \to \infty}(n^2+1)(\cos(\frac{1}{n})-1)$

$$\lim_{n \to \infty}(n^2+1)\left(\cos\left(\dfrac{1}{n}\right)-1\right)$$ Now I have been working on this for a while but don't know how to proceed, using L'Hopital isn't helping me, no matter ...
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54 views

Algebra of pseudo-differential operators

The class of pseudod-ifferential operators form an associative algebra of Fourier integral operators. Moreover, given symbols $a,b,c\in C^\infty$ (each associated to some pseudo differential ...
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1answer
294 views

MAX-HEAPIFY - why the worst case is when the bottom level is “half full”?

In the 3rd edition of 'Introduction to Algorithms', on page 155, when analysing MAX-HEAPIFY it says: The children's subtrees each have size at most 2n/3 - the worst case occurs when the last row ...
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1answer
55 views

Fréchet derivative, is this true?

I was just wondering whether the following statement is true: Let $H_1,H_2$ be Hilbert spaces and $\{e_n\}_{n\geq 0}$ be an orthonormal basis of $H_2$. Let $f:H_1\rightarrow H_2$ be an operator (not ...
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93 views

Some Topological Properties of Starlike Sets!

A subset $E$ of $\mathbb R^n$ is starlike if it contains a point $p_0$ (called a center for $E$) such that for each $q\in E$, the segment between $p_0$ and $q$ lies in $E$. For more information please ...
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57 views

existence of a special function

Whether there exists a function $f(x,y)$ defined on $[0,1]\times(0,1]$ satisfies the following conditions: for any $x\in(0,1]$, $f(x,y)$ is decreasing with respect to $y$ and ...
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114 views

Homework - Showing any continuous functions on a compact subset of $\mathbb{R}^3$ can be approximated by a polynomial.

$ X = \left\{(x, y, z) | \frac{x^2}{3} + \frac{y^2}{5} + \frac{z^2}{7} \le 1 \right\} $ Prove: If f(x,y,z) is continuous on X, then for any ϵ > 0, there exists a polynomial p(x,y,z) such that ...
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50 views

Analysis Limit Question Confusion

Suppose that $|r| < 1$. Show that $$1 + r + r^2 + \cdots + r^n = \frac{1 - r^{n+1}}{1 - r} $$ and find $\lim_{n \to \infty} (1 + r + r^2 + \cdots + r^n)$ Does anyone have any idea on how to ...
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241 views

Showing L infinity norm bounded by L2 norm on a manifold

I have the following problem that I'm working on: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality ...
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3answers
93 views

Proof a sequence converges to a limit

For a sequence $$a_n = \frac{\sin(n)+2}{4n^2-28}$$ How would you use the definition of a limit of a sequence to prove $a_n$ converges to $0$ I am really stuck with how this definition works, I ...
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1answer
66 views

how to prove $\lim\limits_{x \to 0+} f'(x)$ = $\lim\limits_{x \to 0-} f'(x)$ implies $f'(0)$ exists?

$f:(-1,1) \to R$ is continous and $f'(x)$ exists for all $x \in (-1,1)$ except $x=0$ if $\lim\limits_{x \to 0+} f'(x)$, $\lim\limits_{x \to 0-} f'(x)$ exists and same how to prove $f'(0)$ exists?
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240 views

Compact metric connected space

If I have a compact metric space $X$ such that for all $a,b \in X$, there are points $a:=x_1,...x_n=:b$ such that $d(x_i,x_{i+1})< \varepsilon$, then this space is connected. Somehow, I don't see ...
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1answer
114 views

double integral.

I just received this problems from a friend, and I think its a HW problem. its: $$ \int_1^e \int_{1+y^2}^5 \cos (x- \ln x) \ dx \ dy $$ I looked at it, and If I did graph the the region right then ...
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1answer
253 views

continuity and existence of all directional derivatives implies differentiable?

Let $f : A \to R$ be continuous Assume that all directional derivatives exists. Must $f$ be differentiable? I think f doesn't have to be differentiable, but i can't find a counterexample.
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67 views

Adding sin(x + a) + sin(x + b)

I'm trying to prove $$\forall a,b \in \mathbb{R} \exists c,d,e\in \mathbb{R}: f(x):=\sin(x+a) + \sin(x+b) = c \sin(x+d) + e$$ I attempted using $\sin(s) = \frac{e^{is} - e^{-is}}{2i}$ and ended up ...
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1answer
64 views

Infinite differentiability and power series expansion

Does every infinitely differentiable function have a power series expansion?Is this a theorem? Or is this an open question?
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68 views

how to show ${\partial ^2 f \over \partial x \, \partial y}= {\partial ^2 f \over \partial y\, \partial x}$ ??

$f(x,y)$ is real-valued function on $R^2$ $f$ is of class $C^1$ and $\dfrac{\partial ^2 f}{\partial x \,\partial y}$ exists and is continous. how to show $\displaystyle{\partial ^2 f \over \partial ...
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1answer
64 views

Does $\frac{n^a}{a^n}$ converge or diverge? $a$ is Natural Number

I Applied the Ratio test and it all cancelled apart from $\frac{1}{a}$ so this would suggest the series converges, however wolfram alpha says it diverges. Whats going on? Sorry this has been editied ...
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1answer
107 views

Help explain why the proof works - Gradient Estimate Using the Maximum Principle

I found the following proposition and proof in a textbook for one of my classes, and the end of the proof seems a little "unfinished" to me. In other words, as the author left it, I'm not entirely ...
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19 views

convergent series with unbounded terms?

Are there any sequences $\{a_n\}$ such that $\sum_n a_n <\infty$ but $\{a_n\}$ is unbounded? I want to say there aren't any but I can't think of any counterexamples. Just want to make sure.
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1answer
69 views

$\frac{1}{(1+s^{2}) (1+t^{2})}$ real analytic in $\mathbb R^{2}$ but not real-entire; why?

A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point $(s_{0}, t_{0})$ in there corresponds an expansion with ...
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33 views

Show that a function $\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty})$, $f \rightarrow f'$ is continuous

I have the following: Show that the function $$\phi:(V, \|\cdot\|_{C^1}) \rightarrow (W, \|\cdot\|_{\infty}),~~f \rightarrow f'$$ is continuous. With $$\|\cdot\|_\infty = \sup\{|f(x)| ~ \big| ~x\in ...
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102 views

Law of iterated logarithms for BM

The law of iterated logarithms for the standard Brownian motion asserts that $(\ast) \limsup\limits_{h \downarrow 0} \frac{B(h)}{\sqrt{2h\log\log(\frac{1}{h})}} = 1$ I'm trying to prove the ...
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1answer
36 views

Using Properties of a Dense Set to prove characteristics of a continuous function

1. If $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous and $f(x)=0$ for all $x$ in a dense set $E$, then $f(x)=0$ for all $x \in \mathbb{R}$ 2. If $f:\mathbb{R} \rightarrow \mathbb{R}$ and ...
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1answer
19 views

For what points $c$ in $\mathbb{R}$ is $f$ continuous?

Let $X \subset \mathbb{R} $ be a fintie set and define $f:\mathbb{R} \rightarrow \mathbb{R}$ by $f(x)=1$ is $x \in X$ and $f(x)=0$ otherwise. At which points $c \in \mathbb{R}$ is $f$ continuous? ...
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1answer
45 views

If a function $f$ is continuous $\implies$ $|f|$ is continuous.(Answered by Myself)

$(i)$ Given a function $f:E \rightarrow \mathbb{R}$ define $|f|:E \rightarrow \mathbb{R}$ by $|f|(x)=|f(x)|$ for $x \in E$ show if $f$ is continuous at $c \in E$ then so is $|f|$. $(ii)$ Now ...
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1answer
22 views

Convergence of a sequence of powers of functions

Let $f_n : [0,1] \to \mathbb{R}$, $n = 1,2,...$ be functions that converge uniformly to a function $f$, which is bounded. I wish to show that $f_n(t)^m$ converge uniformly on $[0,1]$ to $f(t)^m$ for ...
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106 views

When does $\displaystyle\lim_{n\to\infty}x_n^{1/n} = \alpha$ imply $\displaystyle\lim_{n\to\infty}\frac{x_n}{\alpha^n}$ exists?

Let $x_n$ be a sequence of numbers satisfying $$ 0 < x_n \leq x_{n-1} \leq \ldots \leq x_0 = 1, \quad n \in \mathbb{N} $$ as well as $$ \lim_{n \to \infty} x_n = 0, \quad \text{and } \quad ...
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1answer
74 views

a question about integral proof: $\lim_{n\to \infty} \int_{0}^\infty {n\cdot {\ln(1+{f(x)\over n}}})dx=\int_{0}^\infty f(x)dx$

A non-negative function ${\rm f}\left(x\right)$ is continuous in $(0,\infty)$ and $\displaystyle{\int_{0}^{\infty}{\rm f}\left(x\right)\,{\rm d}x}$ is convergent. Then, we need to prove $$\lim_{n\to ...
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55 views

Limit when x approaches a number $\ne \infty$

I was rushing through a analysis task book when I suddenly encountered $$\text{Find }~\lim_{x \to 3}\frac{x^2-9}{x-3}~\text{ by using the definition of limits.}$$ I thought, as the term equals to ...
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104 views

Proving connectedness

Suppose $S$ is a set, any pair of points of $S$ ($P,Q$ assume) can be contained in a connected subset of $S$. Show $S$ is also connected. I tried to use the polygonal chain theorem(Every open set ...
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1answer
85 views

Closed Graph Theorem

Let $(x_n)$ be a Schauder basis of $X$ and $(y_n)$ an equivalent one to $Y$. They are supposed to be equivalent, hence for every sequence $(a_n)$ the series $\sum_{n \in \mathbb{N}} a_nx_n$ converges ...
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43 views

On a Variational Inequality

Let $H$ be a Hilbert space with real inner product. Consider $f: C \rightarrow C$, where $C \subset H$ is closed and convex. I am not sure about the variational inequality problem: find $x \in H$ ...
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220 views

L1 convergence implies Lp convergence

We know that {f_n} is a sequence in L1 AND in Lp where p >1. We also know that f_n converges to f in L1. Does that imply f_n converge to f in Lp as well? Thank you.
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61 views

Show $f$ differentiable $\iff$ $f(x) -f(x_0) = \psi(x)(x-x_0)$

The questions are a) to show that $f(x)$ is differentiable at the point $x_0$ if and only if $f(x) -f(x_0) = \psi(x)(x-x_0),$ where $\psi(x)$ is a function that is contiuous at $x_0$ b) if ...
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2answers
31 views

Reciprocal Squareroot Birkhoff Integrable?

Is the reciprocal of the squareroot Birkhoff integrable over the unit interval: $$\int_{(0,1]}\frac{1}{\sqrt{x}}<\infty?$$ Then that would be an example of a function not Riemann but Birkhoff ...
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142 views

Can any function be upper bounded by a separable function?

Given a function $f(x,y)$, can we always find functions $h(x), g(y)$ such that $$f(x,y) \leq h(x) + g(y)$$ for all $x,y, \geq 0$? Note that I have placed no restrictions on the functions $f(x,y), ...
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1answer
64 views

What is $\|r-r'\|$ in cylindrical coordinates?

For two radius vectors $r,r'$ we have in Cartesian coordinates $$\|r-r'\|_2 = \sqrt{(x-x')^2+(y-y')^2+(z-z')^2}$$. Is there a similar expression for this in terms of the components in cylindrical ...
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2answers
49 views

If $f(x)=g(t)$ and $x=t^2$ what are $f'$ and $f''$?

If I have $f(x)=g(t)$ and $x=t^2$, can we conclude from this what $f'(x)$ and $f''(x)$ are in terms of g(t) and its derivatives?
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70 views

Using Taylor's Theorem to show $|x-tan(x)|\leq 1/300$ for $0\leq x \leq 1/10$

Using Taylor's Theorem deduce that for $0\leq x \leq 1/10$ $|x-tan(x)|\leq 1/300$ So my attempt; to get the taylors theorem about $x_0=0$ $f(x)=x-tan(x)$ $f'(x)=1-sec^{2}(x)$ ...
0
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1answer
45 views

For which $\alpha$ does this function have a positive solution?

For which $\alpha \in \mathbb{R}$ does $$e^{\alpha x}-1=x$$ have a positive solution Hint; Consider Derivatives at $0$. My attempt; Now firstly I will rewrite this to form the function ...