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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34 views

Integration of a differential equation

I've got some problems with integrating a ODE, so maybe someone could add some words of advice. Given the following equation: $z''(x)-2\gamma z'(x) +p(x)z(x)=0$, $(1)$ and $\varphi(x)=z(x)e^{-\gamma ...
2
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1answer
42 views

Show that certain sequence used in the proof of Wallis product formula is decreasing

Define $I_0 := \pi/2, I_1 := 1$ and $$ I_{n+2} := \frac{n+1}{n+2} I_n. $$ This sequence is monotone decreasing, which could be seen by recognizing that $$ I_n = \int_0^{\pi/2} \cos^n(x) \mathrm d x ...
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0answers
30 views

Sums of nested radicals

Is there a known example of an infinite sum of finitely nested radicals that evaluates to a given value? Or an infinite sum of convergents of an infinite continued fraction? The finitely nested ...
2
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1answer
163 views

Reference request: Analysis, Algebra and Topology - Same author(s)/publisher(s), progressive order

Is there anywhere I can acquire a collection of all Mathematical undergraduate textbooks by the same publishing author, or authors(so that they are similarly written) and can be completed in a logical ...
2
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1answer
71 views

a question about analysis, how to construct $x={1\over q_1}+{1\over q_2}+ \cdots+{1\over q_N}$

Let $x$ be a positive rational number, strictly between $0$ and $1$. Prove that there is a finite strictly increasing list of positive integers $2 \leq q_1<q_2<\cdots<q_N $ such that ...
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1answer
52 views

Question about continuity

If $f:\mathbb{R}^+\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and we have that $u$ is continuous and satisfy $-(p(t)u'(t))'=f(t,u)$ Why $p(t) u'(t)$ and $u'(t)$ are continuous? where ...
3
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1answer
44 views

Sequence of $L^{2}$ functions satisfying an integral condition

I am working on the following problem: Suppose $f \in C^{\infty}([0, \infty) \times [0, 1])$ is such that $$C :=\int_{0}^{\infty}\int_{0}^{1}|\partial_{t}f|^{2}(1 + t^{2})\, dx\, dt < \infty.$$ The ...
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0answers
27 views

Finding roots of polynomials of arbitrary degree

I asked this question on MO, but it has been tagged off-topic. Is there any analogue of the method for expressing roots of polynomials of degree $5$ with elliptic and $\eta$-functions that ...
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0answers
26 views

Question about large sieve

I'm studing the large sieve inequality, following the method of Selberg. I'm reading the book "Opera de cribro", by Friedlander and Iwaniec. At page 154 I've found this ...
3
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2answers
44 views

Show that for a sequence of real numbers $(a_n)_n$ $\lim_n a_n=0$ implies $\frac{1}{n}\sum_{i=0}^{n-1}\lvert a_i\rvert=0$

Let $(a_n)_{n\in\mathbb{N}}$ be q sequence of real numbers with $\lim_{n\to\infty}a_n=0$. Show that this implies $$ \lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\lvert a_i\rvert=0. $$ This is my idea ...
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0answers
21 views

On limit of function and differentiability on endpoints of an open interval

Before asking a question I would first like to mention the definitions of limit of function and differentiality at x=p 1) Limit of function (f) at x=p Let E be domain of f and p be a limit point of ...
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1answer
31 views

Can essentially bounded function take infinite value on measure zero set?

I know that $\|f\|_{\infty}=esssup_{x\in X}(f(x))$, which means we can neglect measure zero sets in our definition of essential supremum. I am comportable when the function is bounded on all points ...
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1answer
38 views

No direct proofs of “if $ f: (X, d_X) \to (Y, d_Y)$ is continuous and $X$ is compact then $f$ is uniformly continuous.”

I am studying the theorem "if $f:(X,d_X)\to (Y,d_Y)$ is continuous and $X$ is compact, then $f$ is uniformly continuous." I am not looking for a proof, but I have an argument against any attempt at a ...
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3answers
126 views

How to prove the following limit: $\lim_{n \to +\infty} 4^n\left[\sum_{k=0}^n (-1)^k{n\choose k}\ln (n+k)\right]=0$?

How to prove the following limit: $$\lim_{n \to +\infty} 4^n\left[\sum_{k=0}^n (-1)^k{n\choose k}\ln (n+k)\right]=0?$$
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0answers
39 views

Derivative : tangent line and multiplication of derivative

Could anyone give me a hint how to prove the following statements ? I suppose that I have some general ideas of pictures of functions satisfying the condition should look like. But it seems impossible ...
2
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1answer
26 views

Construct continous function that passes through a countable set of points

Suppose I have two sequences, $t=(t_1,t_2,... )$ in $[0,1]$ and $y=(y_1,y_2,...)$ in $\mathbb{R}$. Is it possible to construct a continuous function $f:[0,1]\longrightarrow \mathbb{R}$ such that ...
1
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1answer
52 views

Is set measurable in Lebesgue sense? [closed]

Let $f_n :\mathbb R \rightarrow \mathbb R$ be a sequence of continuous functions. $$B=\{ x \in \mathbb R: \text{sequence } \{f_n (x) \} \text{ bounded}\}$$ Is B measurable in Lebesgue sense? Hmm ...
0
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1answer
34 views

Local extremes and $y^3-y+1=0$

Find local extremes: $$F(x,y)=y^4-8xy-4y+8x^2$$ $$F_x = -8y + 16x=0$$ $$F_y = 4y^3 - 8x-4=0$$ $$y^3-y+1=0$$ And I stopped in this moment... How to solve it?
4
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0answers
27 views

Differentiation in Besov-Zigmund spaces

This is my second question in a short time on Besov spaces. I apologize. I am having a rough time with them and I really need to understand this spaces quickly. The besov spaces ...
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1answer
36 views

Why such a complicated counterexample to differentiable function, which has discontinuous partial derivatives

Here a counterexample is given, that a differentiable function has not necessarily continuous partial derivatives, but I asked myself why such a complicated example is given? Would simply $$ f(x) = ...
2
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1answer
59 views

Question on $\sigma$-algebra of a Cartesian Product

Question: Let $\Omega_1$ and $\Omega_2$ be $2$ nonempty sets and let $\mathcal{A}$ be a $\sigma$-algebra of subsets of the Cartesian product $\Omega_1 \times \Omega_2$. Suppose that $\mathcal{S}_i$ is ...
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2answers
46 views

How to prove that an open ball is contained in some set?

Before asking my question, I'd ask you to read the following question : Proving a set is open I fully understand the geometric argument, and I can intuitively grasp that if were to choose an open ...
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3answers
43 views

triple integral and limits

$$\iiint\limits_H (x^2+y^2) \, dx \, dy \, dz \\ H=\{(x,y,z) \in R^3: 1 \le x^2+y^2+z^2 \le9, z \le 0 \}$$ I'm using Spherical coordinate system: $$x=r\cos \theta \cos\phi $$ $$y=r\cos \theta ...
2
votes
2answers
58 views

Find the $L^2[-\pi,\pi]$ projection of $f(x)$

I need to find the $L^2[-\pi,\pi]$ projection of $f(x)=x^2$ onto the space $V_n\subset L^2[-\pi,\pi]$ spanned by ...
2
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1answer
67 views

Limit using Stirling's formula

Consider the limit $$\lim_{n \to \infty} \left[n(1-2\log2) + \sum_{k=1}^{n} \log\left(1+\frac {k}{n}\right)\right] = \frac {1}{2} \log 2.$$ This can be shown by using Stirling's formula for $n!$. My ...
0
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1answer
21 views

Lebesgue's integral and measure

I have got one question, because I see a big black hole in my knowledge about measure theory and Lebesgue's integrals: $$lim \int_A \sqrt [n] {x_1x_2} dl_2(x_1 x_2), A = {x_1^2 + x_2^2 <1, 0 \le ...
0
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2answers
41 views

How find two differential operator $A$ and $B$ such $A\circ \dfrac{d}{dx}=B\circ x$

Question: show that there exists differential operators $A$ and $B$ where $$A=\sum_{k=0}^{n}a_{k}(x)\dfrac{d^k}{dx^k}\neq 0,B=\sum_{k=0}^{n}b_{k}(x)\dfrac{d^k}{dx^k}\neq 0$$ ...
2
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1answer
37 views

lebesgue's integral and theorems [closed]

I have got an exercise from Lebesgue's integral: $$\lim_{n \to \infty} \int_{A} x^n y^{2n} \, dl_2 (x,y), \ A=\{ (x,y) \in \mathbb{R}^2 \mid 4x^2+y^2 \le 1 \}$$ I do not really understand Lebegues's ...
0
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1answer
19 views

Finding extremal of a fixed end point problem. Optimisation

I want to find the extremal of the fix-end point problem $\int_1^2 \frac{\dot{x}^2}{t^3}$ with $x(1)=2,x(2)=17$ First I check the euler-lagrange equation is equal to $0$. We have: ...
3
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1answer
87 views

Iteration of continuous function on compact interval

Let $f:[0,1]\to [0,1]$ be continuous function. Moreover assume that for every $u\in [0,1]$ there exists $n(u)\in \mathbb{N}$ such that the nth iteration $f^{n(u)} (u) =0.$ Is this true that there ...
1
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1answer
24 views

When convolution of two functions has compact support?

It is well-known that, if $f$ and $g$ are compactly supported continuous functions, then their convolution exists, and is also compactly supported and continuous (Hörmander 1983, Chapter 1). Next, ...
0
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1answer
25 views

Calculate double integral on the specific field

Calculate integral $\int\int_D \frac{dxdy}{(x^2+y^2)^{3/2}}$ $ D=\{(x,y)\in R^{2} : x^2+y^2 \le1, -x \le y, x+1 \le y\} $ I draw a graph and this is a little part of a disk. As I calculate $x \in ...
1
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1answer
28 views

self-adjoint operator without eigenvalues?

I have a self-adjoint operator $d$ which acts on vector fields defined on $\mathbb{R}^n$. I am interested on its eigenvalues. That is, I study the equation $d(X)-\lambda X=0$. I have found that if ...
1
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0answers
12 views

Local minima: Sufficient conditions. Comparison of Calculus verses Calculus of Variations

My lecturer has written: Let $y=x^*+\epsilon \eta$ where $x^*,\eta,y\in \mathbb{R}^2$ $0\leq f(y) - f(x^*) = \epsilon V_1 + \epsilon^2 V_2 + O(\epsilon^3)$ $V_1 = \nabla f(x^*)\eta$ $V_2 = ...
1
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0answers
36 views

Values of $x$ for convergence

I was posed this problem, it took me a while to solve it – but, I did nevertheless. I shall pose it for all of you, too. In my opinion it is a great exercise. For what values of $x$ is the series ...
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0answers
19 views

Constrained optimization minima and maxima and non-degeneracy answer check

Find the critical points of $$\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\f{f(\1,\2,\3)}\def\l{\lambda}$$ $$\f=\1\2+\2\3+\3\1$$ subject to constraint $\1+\2+\3=1$ First I will construct the Lagrangian: $$L ...
3
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1answer
68 views

Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$.

Let $f(x)$ be a non-decreasing function on $[0, 1].$ You may assume that $f$ is differentiable almost everywhere. Prove that $\int_0^1f'(x)dx \leq f(1) - f(0)$. I am having a hard time with this ...
3
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0answers
38 views

Showing equality of sets in $C[a,b]$

The exercise states: Let $a,b\in\mathbb{R}$, $a<b$ and let $(C[a,b],\Vert\cdot\Vert)$ denote the vector space of continuous real functions on $[a,b]$ endowed with the uniform norm. Let ...
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0answers
18 views

bound available for the difference between left and right sides in Jensen's inequality [closed]

Is there any bound available for the difference between left and right sides in Jensen's inequality ?
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1answer
7 views

Is the sum of quasi concave functions quasi concave

Is the sum of quasi-concave functions a quasi concave function? I presume that's not in the case in general, but under which conditions is this true?
0
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1answer
45 views

Proving differentiability by using Caratheodory's Lemma

Let $I$ be an open interval and let $c\in I$. Let $f:I\rightarrow\mathbb{R}$ be continuous and define $g:I\rightarrow\mathbb{R}$ by $g(x)=\left|f(x)\right|$. Prove that if $g$ is ...
0
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0answers
46 views

Show that a function is solution to differential equation

I have a homogenous differential equation $a_0 y'' + a_1 y' + a_2 y = 0$ and a function $y(t) = t e^{\lambda_0 t}$. First I am assuming that $\lambda_0$ is a root in the characteristic polynomial. ...
1
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1answer
24 views

Negation of sequence convergence

I just to verify that the negation of sequence convergence of ($a_n$) to a limit, a would be something like: "There exists a $ \epsilon >0$ s.t. for all $n \in \mathbb{N}$ there exists an $n> ...
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0answers
39 views

The definition of compact set without using open cover

In history, I know the definition of compact set comes from Heine-Borel Theorem in $R$, so it is natural that we define the compact set w.r.t open cover. I am curious what if we use any other set as ...
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1answer
16 views

How to show this series has at most three nonzero consecutive terms and at least one not null?

Let $\psi\in C^\infty_0(\mathbb R^n)$ such that $1\leq \psi\leq 0$ and satisfying $$\psi(x)=\left\{\begin{array}{lcl}1&\textrm{if}&1\leq |x|\leq 2\\0&\textrm{if}&|x|<1/2\ ...
0
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0answers
48 views

Doubts on Hartogs' lemma

I have the following version of Hartogs' Lemma (which is used to build up the proof of Hartogs' extension theorem). Let $\{\phi_\nu\}_\nu$ be a sequence of subharmonic functions which are uniformly ...
0
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0answers
26 views

Interval for a maximal solution

So, I have this multiple answers question from a test for analysis, and it's driving me crazy for either I don't understand correctly the concept of maximal solution, or the options given are wrong. ...
0
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1answer
29 views

A circular path connecting two complex numbers

For any two complex numbers $z_1$ and $z_2$, $f(t)$= $z_1+t(z_2-z_1) $is a path in $ℂ$ where $t∊[0,1]$. The image of this path is a line segment. Is there a way of getting a similar path but to ...
6
votes
1answer
66 views

Can we characterize the space of functions which is real analytic but not real entire?

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 ...
2
votes
1answer
84 views

How to solve such a nonlinear ODE, the analytical solution of which is known!

I have the following ODE with initial/boundary value conditions: $$\left. \begin{aligned} \left(x^2-10 x-y^2\right)y\, y'(x)+(x-5) y^2 y'(x)^2-(x-5) y^2=0 ;\qquad (\text{ODE})\\ y(0)^2=25;\qquad ...