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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3
votes
2answers
34 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|$...
0
votes
1answer
54 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
votes
1answer
34 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 $(...
2
votes
3answers
26 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?
0
votes
1answer
20 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} \...
1
vote
0answers
21 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 ...
0
votes
1answer
48 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, ...
1
vote
1answer
40 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= ...
2
votes
3answers
104 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\...
0
votes
0answers
37 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 ...
0
votes
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 ...
0
votes
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.
0
votes
1answer
18 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 ...
1
vote
1answer
41 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
votes
1answer
33 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 ...
0
votes
1answer
30 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 ...
4
votes
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 ...
1
vote
1answer
28 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 ...
0
votes
1answer
37 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 ...
0
votes
0answers
24 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
votes
2answers
34 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_{...
0
votes
0answers
25 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 ...
0
votes
0answers
20 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
votes
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
votes
1answer
43 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
vote
1answer
28 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 ...
11
votes
1answer
121 views

What is the use of hyperreal numbers?

For sometime I have been trying to come to terms with the concept of hyperreal numbers. It appears that they were invented as an alternative to the $\epsilon-\delta$ definitions to put the processes ...
0
votes
0answers
36 views

Proof of iff condition of harmonic map

How to compute the equation above the red line in the picture below ? Below picture is from the Harmonic maps and their heat flows. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
0
votes
1answer
48 views

Bilinear form vs two-form

I have a basic linear algebra question since I'm confused with the definitions: What is the difference between bilinear form and 2-form? I looked up in Wikipedia and it says that a linear form $B(v,...
1
vote
1answer
43 views

Prob. 2, Sec. 2.8 in Kreyszig's functional analysis text: What is the norm of these bounded linear functionals on $C[a,b]$?

Let $C[a,b]$ denote the normed space of all the continuous (real or complex-valued) functions defined (and continuous!) on the closed interval $[a,b]$ on the real line, where $a, b \in \mathbb{R}$ and ...
3
votes
1answer
119 views

Is the infinite decimal fraction $1.23456…n$ irrational?

How to prove that the number $ 1.23456\dots n$ is an irrational number? The number consist, of course, of natural numbers in increasing sequence.
1
vote
1answer
24 views

Corollary of the Birkhoff Kakutani Theorem: first countable topological vector spaces

http://planetmath.org/birkhoffkakutanitheorem A topological group $(G,*,e)$ is metrizable if and only if $G$ is Hausdorff and the identity $e$ of $G$ has a countable neighborhood basis. In ...
3
votes
4answers
36 views

How tounderstand the proof to show if a real number $w>0$ and a real number $b>1$, $b^w>1$?

Claim: If there are two real numbers $w>0$ and $b>1$, then $b^w>1$. One proof is that for any rational number $0<r<w$ and $r=m/n$ with $m,n\in\mathbb Z$ and $n\ne0$, $(b^m)^{1/n}>1$...
0
votes
0answers
14 views

Why $(d+1)^{2T}{4T+R \choose R}e^{-2R+14T} \le \epsilon$ given $R=cT \ln d + \ln \frac{1}{\epsilon}$?

I read a paper and I found that the only step I don't understand is this derivation from $$R=c T \ln d + \ln \frac{1}{\epsilon} \\\text{to}\qquad (d+1)^{2T}{4T+R \choose R}e^{-2R+14T} \le \epsilon$$ ...
5
votes
3answers
53 views

Proving the Well-Ordering Principle for Natural Numbers

I know the WOP is treated like axiom of a natural number, but I was curious if I can prove WOP defined for the set of natural numbers N by following: Suppose A is a subset of N, which then obeys all ...
0
votes
4answers
65 views

Are compact sets on $\mathbb R^n$ always connected?

I am unsure if compact sets on $\mathbb R^n$ are always connected. Can someone explain it to me?
1
vote
3answers
37 views

Compact set and a sequence of closed sets, the intersection of all of them is empty

I'm having a hard time to prove this. The problem is: Let $V \subset \mathbb{R}^d $, be a nonempty compact set and $(A_n)_{n\in\mathbb{N}}$ a sequence of closed nonempty sets in $\mathbb{R}^d$ with ...
1
vote
1answer
36 views

Convergence of the integral: $I_{\alpha }=\int _0^{\infty }\left(\frac{e^{-\alpha x}}{\sqrt{x}}\right)\:dx$

Study the convergence of the integral: $$I_{\alpha }=\int _0^{\infty }\left(\frac{e^{-\alpha x}}{\sqrt{x}}\right)\:dx$$ and calculate $I_2$. Ok so to study the convergence I'm using convergence ...
3
votes
1answer
65 views

sum of series using mean value theorem

Let $f(x)$ be a function which is differentiable on $[0,1]$ with $f(0)=0$ and $f(1)=1$. Show that for every $n\in \Bbb N$ there exists numbers $x_1,x_2,\ldots,x_n\in [0,1]$ such as $$ \sum_{k = 1}^n \...
0
votes
0answers
14 views

Convolution notation

I refer to notations like $$f*K_\epsilon(0)$$ in Convolution with Gaussian question. Do they mean $(f*K_\epsilon)(0)$, i.e. the convolution evaluated at zero or $f*(K_\epsilon(0))$, i.e. $f$ ...
19
votes
5answers
3k views

A very curious rational fraction that converges. What is the value?

Is there any closed form for the following limit? Define the sequence $$ \begin{cases} a_{n+1} = b_n+2a_n + 14\\ b_{n+1} = 9b_n+ 2a_n+70 \end{cases}$$ with initial values $a_0 = b_0 = 1$. ...
0
votes
0answers
49 views

Mean value theorem sum of $\frac{1}{f ' (x _k)}$ [duplicate]

Let $f$ be a function which is derivable in $[0,1]$ with $f(0)=0$ and $f(1)=1$. Show that for every $n$ in $\mathbb{N}$, there exists numbers $x_1,x_2,.....,x_n$ all in $[0,1]$ such as \begin{...
2
votes
2answers
72 views

Introductory Topology Book Recommendation for Economics

Would you please share your 2 cent on book recommendation for introductory topology book to graduate student in Economics. Have exposure to the first half of the yearlong analysis course in the ...
0
votes
1answer
39 views

$\succsim$ preorder on X being continuous imply lower contour set closed

$\succsim$ is preorder (i.e. preference relation) on X that is continuous. This implies the lower contour set is closed. Would you please share your 2 cent on my parenthesis explanation (e.g. line ...
0
votes
1answer
30 views

Convolution with Gaussian question

Let $K_\epsilon(x):=\dfrac{e^{-x^2/\epsilon^2}}{\epsilon\sqrt\pi}$ for $x\in \mathbb{R}$, $\epsilon>0$. Let $f\in L^\infty(\mathbb{R})$ where $f$ is of bounded variation on any interval $[a,b]$. ...
0
votes
2answers
40 views

Existence of sequence of polynomials such that $\lim_{n\to\infty} \int_0^1 |h(x) - p_n(x)|^2 dx = 0$

For a function $h:[0,1] \to \mathbb{R}$: $$h(x) = \begin{cases} 1~~\text{for}~~ x\in[0, \frac12] \\0 ~~\text{for}~~ x\in(\frac12, 1] \end{cases}$$ how could we prove the existence of sequence of ...
4
votes
1answer
47 views

Estimate the value of f at a given point

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a differentiable everywhere. Assume $f(-\sqrt2,-\sqrt2)=0$, and also that $|\dfrac{\partial f}{\partial x}(x,y)|\le |\sin(x^2+y^2)|$ and $|\dfrac{\...
0
votes
0answers
25 views

PDE with analytical solution in some cases

I studied PDE, if a classification of them means studying. I haven't studied solving methods for PDE so I'd like to have an elementary answer if it is possible. I got this one: $$ \frac{\partial \...
0
votes
1answer
29 views

X,Y position percent to degrees?

My english its not that good so im going to explain with picture. So the first circle is the info I can get, x % and y %, what I want to know its the degrees of the red dot. The second picture ...