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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7 views

Integrability of a composed function on $R^{n}$

Let $f: A \rightarrow B$ continuous on a block $A$ such that $|f(x)-f(y)| \geq c|x-y|$ $c > 0$ and $x,y \in A$. Prove that for all $g:B \rightarrow \mathbb{R}$ integrable $g\circ f: A \rightarrow \...
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12 views

Integrability question with a function on a box in $\mathbb{R}^2$

Let $f:\mathbb{R} \to\mathbb{R} \ be \ bounded,$ $\phi: \mathbb{R}^2 \to\mathbb{R}^2 $ be defined as $\phi(x,y)=(x,y+f(x))$ Prove that if for every bounded box $B\subset \mathbb{R}^2, \phi(B)$ ...
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1answer
5 views

how to prove this inclusion abot the discontinuity set of a composed function?

Let $D_{\phi}$ be the set o discontinuity points of $\phi$. How can i prove that if $g\circ f$ makes sense then $D_{g\circ f} \subset D_{f} \bigcup f^{-1}(D_{g})$? for $ x \in D_{g\circ f} \...
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0answers
9 views

Variant of local inversion theorem in special case

Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be defined by $F(x,y)=(x+2y+x^2\ ,\ y-x^3+y^2)$. Then show that for $p_0=(4,1)$ and $p_1=(1,1)$ there exists $\delta>0$ such that for every $\vec y\in B(p_0,\...
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1answer
16 views

if the metric $d_1$ is complete, and $\lim_{n \to \infty} d_1(x_n,x)=0$ iff $\lim_{n \to \infty} d_2(x_n,x)=0$, is $d_2$ complete?

two metrics $d_1, d_2$ on $X$, For all $x_n$ and $x$ from $X$ it holds : $\lim_{n \to \infty} d_1(x_n,x)=0$ iff $\lim_{n \to \infty} d_2(x_n,x)=0$ Is it true that $(X, d_1)$ complete implies that $...
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1answer
23 views

Help, why these are two different results of integral of $\sqrt{z}$ on unit circle depending the choice of Branch cut

everyone, I want test the effect of different choice of branch cut for contour, So I find a simple function, i.e. $\sqrt{z}$ with $z=re^{i\theta}$ on 1st Branch as $$I=\oint_{UnitCircle}{\sqrt{z}dz}$$ ...
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0answers
19 views

Looking for interesting applications of Ergodic Theory in mathematics

I would like a list of nice applications of ergodic theory, in mathematics or probability theory. I think there are several applications of ergodic theory to continued fractions, number theory in ...
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27 views

looking for a probability function which satisfies the following conditions

I am looking for a continuous probability function of$f(a,p,x)$ which satisfies the following conditions $a$ is a positive constant $0 \le p \le 1$ is a positive constant $x > 0$ is the variable $...
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0answers
9 views

Differentiation product of functions in multidimensional Analysis

$k: \mathbb{R}^d \to \mathbb{R}^{m\times m}, k(x)=g(x)f(x)^T$. Proof that $k$ is differentiable and apply for $v,x\in \mathbb{R}^d$: $$k'(x)v=\left(\sum\limits_{l=1}^d\big(v_l f_i(x) \partial_l g_i(...
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0answers
25 views

$X := \prod_{i\in I} X_i \: \: $ Show that X is (path-)connected, if $X_i$ is (path-)connected $\forall i \in I$

Let $I$ be an indexset and $(X_i, \mathcal T_i)$ a topological Space for $i\in I$. Let $X = \prod_{i\in I} X_i$ have the product Topology. Now i have to show the following two things: $X$ is ...
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2answers
189 views

Explanation needed for a statement about power series convergence

I got a task in front of me but I don't really understand it. If someone could explain, I think I would be able to solve it myself. $P(x) = \sum_{k=0}^{\infty}a_{k}x^{k}$ is a power series. There ...
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1answer
25 views

What's wrong with this proof?- reverse triangle inequality

Subtract this inequality $-|y|\leq y \leq |y| $ , from this inequality $-|x|\leq x \leq |x|$ to get $-(|x|-|y|) \leq x-y \leq |x|-|y|$. Using the property $-a \leq x \leq a \implies |x| \leq a$, we ...
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0answers
18 views

set equalities proofs

I'm teaching my self topology with the aid of a book, the problem i'm on ask to prove the following: Let $X$ be a topological space and $B$ be a subset of $X$. Prove the following set equalities.(...
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11 views

What is necessity for integral to be well-defined in defining solutions?

When trying to give a notion of solutions for differential equations with non-local terms, e.g., integral of unknown functions, to guarantee that the integral is well-defined, i.e., finitely ...
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1answer
16 views

Find all radiuses of convergence for this series - is my approach correct?

I'm supposed to find all radiuses of convergence for this power series: $\sum_{k=0}^{\infty} \frac{k^{2}}{3^{k}}x^{k}$ I've worked with ratio test: $\frac{{}\frac{(k+1)^{2}}{3^{k+1}}}{\frac{k^...
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0answers
30 views

Prove $\int_\Omega f(x) \,dx=f(x_B) \int_\Omega1 dx+ \mathcal O(\int_\Omega1 dx \cdot \sup_{x,y\in\Omega}\|x-y\|_2^2)$?

Let $\Omega \subset \Bbb R^n$ be a convex domain and $f: \Omega \to \Bbb R $ and $f \in \mathcal C^2(\Omega)$. Let $x_B $ be the barycentre of $\Omega$ with $$x_B:= \frac{\int_\Omega x \,dx}{\int_\...
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2answers
32 views

What's the series and what's the radius of convergence of this (power) series?

Find the convergence radiuses of this power series: $1 + n + n^{4} + n^{9} + n^{16} + n^{25} + n^{36} + ...$ First of all, I'm surprised it says $radiuses$ instead of $radius$. I know you find ...
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1answer
41 views

Problems with the Banach fixed point theorem

At the moment, I am studying for an exam and I came across the following exercise: Consider the map $f:[0,1] \to \mathbb{R}$, $f(x)=1-\arctan(x)$. Prove the following statements: a) $f$ has a ...
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2answers
30 views

two metrics on X such that lim d1(xn,x)=0 <=> lim d2(xn,x)=0, does it imply the identity of the two induced topologies?

Two metrics $d_1, d_2$ on $X$ For all $x_n, x$ from $X$ it holds: $$\lim d_1(x_n,x)=0 \iff \lim d_2(x_n,x)=0$$ Does it imply that the topology induced by $d_1$ is the same as the topology induced by $...
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0answers
43 views

Property of a continuous function

Let $f$ - a continuous function and $$\lim_{x \to \infty} \left( \int_0^x f(t)dt + f(x) \right) = 0$$ Prove that $$\lim_{x \to \infty} f(x) = 0$$
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2answers
33 views

Hints on showing Cauchy sequence converges

Let $T>0$ and $L\geq0$. Let $C[0,T]$ be the space of all continuous real valued functions on $[0,T]$ with the metric $\rho$ defined by $$\rho(x,y)=\sup_{0\leq t\leq T}e^{-Lt}\left|x(t)-y(t)\right|$...
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1answer
48 views

Compute definite integral by hand [on hold]

How can I compute $$\int_0^1 \frac{x^3t}{(x^2+t^2)^2} \, \mathrm{dt}$$ by hand?
4
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1answer
28 views

Find the radius of convergence of this power series: $\sum_{k=0}^{\infty } \binom{2k}{k}x^{k}$

Given: $\sum_{k=0}^{\infty } \binom{2k}{k}x^{k}$ I started by forming it: $\binom{2k}{k} = \frac{(2k)!}{k!*(2k-k)!} = \frac{(2k)!}{k!*k!}$ Now the problem is, I cannot write $2! * k!$ instead of $(...
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3answers
22 views

Exponential limit convergence for each $x$

I have $f_n(x)=\left( 1+\frac{-e^{-x}}{n} \right)^n$, what about the convergence to $f(x)=e^{-e^{-x}}$? is it true $\forall x?$ I say yes, but how can I show this? Is continuity of $f_n$ enough?
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0answers
9 views

Calculate the gradient of a function that is written with abstract vectors

:) I am supposed to calculate the gradient of the following function: $$f(\mathbf{w})=\sum^{n}_{i=0}\log(1+\exp(-y_i\mathbf{w}^T\mathbf{x}_i))+\frac{1}{b}\sum^{n}_{i=0}w_i^4$$ Where $\mathbf{x} \...
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0answers
19 views

Transformation of the gradient

For a function $f\in C^2$, $f:\mathbb{R}^n\to\mathbb{R}$ and a point $x\in\mathbb{R}^n$ with $\nabla^2f(x)$ positive definit one can calculate the new point $x^+=x+s$ as follows: Change the ...
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0answers
79 views

Are all important function spaces vector spaces?

EDIT: I definitely agree with Mike Miller that the question as written originally/below is too general. Is everything an analyst could ever care about locally homeomorphic to a T1 topological ...
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0answers
36 views

Show that for any $F \subset \mathbb{R}^n$ closed exists $f \colon \mathbb{R}^n \to \mathbb{R}$ of class $C^{\infty}$ such that $F= \Sigma (f) $.

Show that for any $F \subset \mathbb{R}^n$ closed exists $f \colon \mathbb{R}^n \to \mathbb{R}$ of class $C^{\infty}$ such that $F= \Sigma (f) $. $\Sigma (f) = \{p \in \mathbb{R}^n;$ $D_f(p)=0\}$. ...
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0answers
21 views

(Terminology_Taylor Series) “expand at $x_0$, evaluate at x, affine approximation”

I am reading one-variable calculus book where it explains Taylor series and little confused with the following terms: (1) Expand $f(x)$ at $x_0$ (2) Evaluate $f(x)$ at x (3) Best Affine, ...
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1answer
39 views

It's true that $ |\log^2(z)| \leqslant |\log(R)|^2 + |i \arg(z)|^2 $ where $z \in \mathbb{C}$

In some residue integral, when one have to prove that an integral vanish at infinity, I've found in some textbooks the inequality: $$ |\log^2(z)| \leqslant |\log(R)|^2 + |i\ \arg(z)|^2 $$ Where $z= ...
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3answers
92 views

Help, Where is wrong when I do same complex integration using two different contours

everyone! please give few hit. I want take the integral $$I=\int_{0}^{\infty}{\frac {dx}{ \sqrt{x}(1+{x}^{2})}} $$ by using the Residue Theorem. I choice two contours in complex plane with $z=r e^{i\...
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0answers
36 views

Baby Rudin existence of smooth function for every closed set as it's zero set

Problem 21 of chapter 5 asks whether a function can be found for every closed set $F$ in $R$, with it's zero set precisely $F$ having derivatives of all orders. The following solution problem is ...
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0answers
15 views

Equivalent definition of Lebesgue measurability in terms of additivity?

When introducing measurability, we noted that we wanted the following property to hold for $A, B \in \mathcal{P}(\mathbb{R})$ $m(A \cup B) = m(A)+m(B)$ (additivity) We then defined a set A to be ...
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1answer
15 views

Does specific function exist? [duplicate]

Check if exists function $f(x,y):R^2->R$ such that f(x,y) has directional derivatives in point (0,0) in each direction and (0,0) is point of discontinuity.
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1answer
17 views

Derivative of dot product with transposed function

According to this post Derivative of dot product I have a similar task: $$\langle f(x),g(x) \rangle = f(x)g(x)^T=j(x)$$ I have to show: $j'(x)=g'(x)f(x)^T+g(x)^Tf'(x)$ I know how to ...
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1answer
40 views

Convergence of $\sum_{n=0}^\infty n^{1/n}-1$ and $\sum_{n=0}^\infty (1/n!)^{1/n}$

$$\sum_{n=0}^\infty n^{1/n}-1$$ $$\sum_{n=0}^\infty (1/n!)^{1/n}$$ Hi. I am working on calculus now. While studying convergence test part, I ran into those problems... Wolfram alpha says they both ...
2
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1answer
32 views

Proving that a ball is open in Euclidean metric space.

I've come across an exercise in an analysis book that is presented as follows: We define the function || $\cdot$ ||$_1$ : $\mathbb{R}^2 \rightarrow \mathbb{R}$ by $||x||_1 := max(|x_1|,|x_2|)$. The ...
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1answer
29 views

Counterexamples to complex function theory results for Banach space valued functions

I'm wondering what results of complex function theory still hold true when considering analytic functions mapping from the complex plane to some complex Banach space. For instance, it can be shown ...
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5answers
78 views

Inequality : $\displaystyle \sum_{k=1}^n x_k\cdot \displaystyle \sum_{k=1}^n \dfrac{1}{x_k} \geq n^2$

I have to show the inequality of $$\left(\sum_{i=1}^n x_i\right)*\left(\sum_{i=1}^n \frac{1}{x_i}\right) \geq n^2.$$For $x_1, ... x_n \in \mathbb{R_{>0}}$ and $ n \geq 1$. I wanted to show this ...
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1answer
24 views

function is right-differentiable at zero iff certain integral finite

Suppose $f$ is nonnegative and integrable and $\int_\mathbb{R} f(x) \; dx < \infty$. For $t \ge 0$, define $$ g(t) := \int_\mathbb{R} e^{-tx^4 \sin \left( \frac{1}{1+x^2} \right)} f(x) \; dx $$ I ...
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1answer
36 views

How Changing the order of integration(Elementary proof of the prime number theorem)?

I'm studying the exchange of integration order, I need help, any hint? For every real number $\rho \geq 0$, write $V(\rho)=e^{-\rho}R(e^{\rho})=e^{-\rho}\psi(e^{\rho})-1$ where $\psi(x)$ is the ...
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0answers
23 views

What's the best separation you can get on set sums?

Given a set $S = \{ a_1 < a_2 < \ldots < a_n \}$ of real numbers, we want to maximize the separation between any sum of n elements with replacement and the total of the set. That is, let $C =...
2
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2answers
32 views

Solve the following (logarithmic) function for x

$x^{log_{2}x}+16x^{-log_{2}x} = 17$ Looks horrible, I started by removing the exponents: $e^{ln(x)*log_{2}x}+16e^{-ln(x)*log_{2}x}=17$ | ln() $ln(x)*log_{2}x-16ln(x)*log_{2}x=ln(17)$ $ln(x)*log_{...
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0answers
24 views

(Order Relation) Monotone Continuous Complete Preorder on $\mathbb{R^L_+}$ has $y\geq x\rightarrow y\succsim x$

I am trying to show a monotone continuous complete preorder on $\mathbb{R^L_+}$ has $y\geq x\rightarrow y\succsim x$. Can you please share your 2cent on the below proof? Thank you! Point of ...
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0answers
19 views

Proof of Rudin's Theorem 7.29

In the baby rudin, I have some difficulties in understanding the concept of uniform closure of an algebra in Definition 7.28 and Theorem 7.29. The definition of the uniform closure is: Let $\...
2
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2answers
42 views

Solve the following (logarithmic) function for $x$

$(\log_{3}x)^{2} - 3\log_{3}x + 2 = 0$ We may not use many rules, so I would start by ignoring the ^(2), ignore -3* but take ...
0
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1answer
42 views

Min and max for a multi variable function

We have $A=\left[-1,1\right]^2$ Find $minf\left(A\right)\:maxf\left(A\right)\:f\left(A\right)$ for function: $$f:A\rightarrow \mathbb{R}$$ $$f\left(x,y\right)\:=\:x^3+xy+y^3$$ So I'm guessing A is a ...
1
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1answer
27 views

Laplace Operator Times Function

I'm just going through some proofs of a PDE book and have a question about one of them. It is stated that: $$ \int_U w \Delta w \text{ d}x = -2 \int_U |Dw|^2 \text{ d}x $$ Where $w$ is a solution of ...