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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60 views

Question about notation in differential equations.

In general, an ordinary differential equation is in the form $$ \begin{cases} x'(t) = f(t, x(t)) \\ x(t_0) = x_0 \end{cases}. $$ When proving the existence and uniqueness theorems, an operator $T$ was ...
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3answers
79 views

Evaluation of the integral.

$$I\left(n,\epsilon\right)=\int_{-{\rm i}\infty}^{+{\rm i}\infty} \frac{{\rm e}^{\epsilon z}}{\left(z+\epsilon\right)^n}\,{\rm d}z$$ The integration is taken along the imaginary axis, an integer ...
1
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1answer
129 views

contraction point?

This is an interesting question I saw in a book online: Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence ...
0
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2answers
63 views

$A_k\subset\mathbb{R}$ such that $\lim\sup A_k=\mathbb{R}$ but $\lambda(A_k)=1$ (Lebesgue measure) for all $k$.

Construct a sequence of measurable sets $A_k\subset\mathbb{R}$ such that $\lim\sup A_k=\mathbb{R}$ but $\lambda(A_k)=1$ (Lebesgue measure) for all $k$. My thoughts: Since $\lim\sup ...
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1answer
196 views

Prove that $x$ is any positive real number greater than $0$, $x>0$, then exists $N$ in the natural numbers such that $\frac{1}{N^3}<x$

Prove that $x$ is any positive real number greater than $0$, $x>0$, then exists $N$ in the natural numbers such that $\frac{1}{N^3}<x$ My steps: Well I begin with $N\in\mathbb{N}$ and ...
3
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3answers
120 views

Please more help me to find the convergence interval and the sum -by using residue theory- of the series.

The sum is that $$\sum_{n=0}^\infty \binom{3n}{2n} x^n$$ First of all, I need to check whether the sum converges or not and if it is convergent, which points? I am using ratio test. $$ ...
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2answers
209 views

An exercise in Rudin's RCA

Would you please give me some help on the following problem? Suppose $1 \leq p \leq \infty$, and $q$ is the exponent conjugate to $p$. Suppose $\mu$ is a positive $\sigma$-finite measure and $g$ is a ...
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2answers
406 views

Necessary and sufficient conditions for a polynomial $p$ to satisfy $\|x\|\to\infty\implies p(x)\to\infty$?

I'm looking for a necessary and sufficient conditions (I'm not even sure these exist) for a polynomial $p:\mathbb{R}^n\to\mathbb{R}$ to be "radially unbounded", that is $$\|x\|\to\infty\implies ...
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0answers
56 views

Prove C[0, 1] is complete with metric a different metric

would any one tell me whether C[0,1] is complete under the following metric $$ \sup_{t\in [0, T]}e^{-Lt}|x(t)-y(t)| $$ and how to prove the claim I know some reasoning on how to prove C[0, 1] is ...
2
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1answer
219 views

Complex Analysis - Taylor Series

The question reads as follows: Let the function $f$ be given by $f(z)=\frac{z}{2}+\frac{z}{e^z-1}$ if $z\neq0$ and $f(z)=1$ if $z=0$. Show that $f$ is analytic at $z=0$ and that $f(z)=f(-z)$. ...
4
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1answer
107 views

$f_1, f_2 : \mathbb{R} \rightarrow \mathbb{R}$ nonconstant, continuous, with period $1, \sqrt{2}$, respectively, then $f_1 + f_2$ is not periodic

I've been working on this problem for several hours, but I keep getting stuck. Suppose $f_1, f_2 : \mathbb{R} \rightarrow \mathbb{R}$ periodic with period $1, \sqrt{2}$, respectively, and that each of ...
0
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1answer
219 views

Why is this function not locally Lipschitz?

I was reading an exercise, and supposedly this function: $$\chi \colon \Bbb R\times\Bbb R\to\Bbb R, \quad \chi (t,x)=3x^{2/3}$$ is not locally Lipschitz (in the second variable). In the notes this ...
3
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1answer
171 views

the solution of Fredholm´s integral equation

Be $\lambda \in \mathbb{R}$ such that $\left | \lambda \right |> \left \| \kappa \right \|_{\infty }(b-a)$. Prove that the solution $f^*$ of the integral equation of Fredholm $$\lambda f ...
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2answers
120 views

sequence of Lebesgue measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim \lambda(A_k)=1$, but $\lim \inf A_k=\emptyset$.

Give an example in $\mathbb{R}$ of a sequence of Lebesgue measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim \lambda(A_k)=1$, but $\lim \inf A_k=\emptyset$. My thoughts: By definition, ...
0
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1answer
58 views

Proving $m\lambda(E_m)\le\sum_{k=1}^{\infty}\lambda(A_k)$

Assume $A_1,A_2,...$ are measurable sets. Let $m\in\mathbb{N}$, and let $E_m$ be the set defined as follows: $x\in E_m\iff$ $x$ is a member of at least $m$ of the sets $A_k$. Prove that $E_m$ is ...
2
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1answer
90 views

Does the property $a\neq a$ exist somewhere in mathematics?

Whenever I read about properties of the real numbers, I'm always presented the property $a=a$ sometime ago I didn't know why they stated such obvious properties but then, after reading some abstract ...
2
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1answer
791 views

Why is the matrix norm $||A||_1$ maximum absolute column sum of the matrix.

By definition, we have: $||V||_p=\sqrt[p]{\displaystyle \sum_{i=1}^{n}|v_i|^p}$ $||A||_p=sup\frac{||Ax||_p}{||x||_p}$, $x\not=0$ and if $A$ is finite, we change sup to max. However I don't really ...
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355 views

Optimal assumptions for a theorem of differentiation under the integral sign

Let us consider the following integral: $$I(x)=\int_\Omega f(x, \omega)\, d\omega, $$ where $\Omega$ is a measure space and $f\colon \mathbb{R}\times \Omega \to \mathbb{R}$ is such that $f(x, ...
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4answers
126 views

Two form of derivative $ f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

Why I can write formula derivative $$ f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$$ in this form: $$ f'(x)=\frac{f(x+h)-f(h)}{h}+O(h)$$ I know, that it's easy but unfortunately I forgot.
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2answers
87 views

Prove $\int_0^1|f(t)-g(t)|dt \le (\int_0^1|f(t)-g(t)|^2dt)^{1/2} \le \sup_{t\in[0,1]}|f(t)-g(t)|$

Let $C[0,1]$ be the set of all continuous real-valued functions on $[0,1]$. Let these be 3 metrics on $C$. $p(f,g)=\sup_{t\in[0,1]}|f(t)-g(t)|$ $d(f,g)=(\int_0^1|f(t)-g(t)|^2dt)^{1/2}$ ...
1
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2answers
107 views

How to prove that $x/y$ is continuous in R

$f:R^2$ \{y=0} $\Rightarrow R$ , $f:(x,y)\Rightarrow x/y$. Prove (formally) that $f$ is continuous. I think what I should show is that any point that belongs to an open ball of radius $\epsilon$ of ...
0
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1answer
33 views

Asymptotic behaviour of real sequences

Let's say we have two real sequences $(a_n)_{n\in\mathbb{N}}$ and $(c_n)_{n\in\mathbb{N}}$ with $c_n\in o(\frac1n)$ (i.e. $c_n(\frac1n)^{-1}\xrightarrow{n\rightarrow\infty}0$). And for all ...
1
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1answer
51 views

Derivation of weak form of Euler Lagrange Equation

In Giaquinta's and Giusti's 1982 paper entitled "On the regularity of the minima of variational integrals", they look at the following quadratic functional: \begin{equation} ...
5
votes
1answer
410 views

Set of discontinuous points

Suppose $f$ is function from $\mathbb{R}$ to $\mathbb{R}$. Let be the set $\mathbf{A}$ that contains all the discontinuous points of $f$.Is $\mathbf{A}$ Borel Measureable?
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1answer
63 views

Question on Discrete metric space

Let $X = \{1,1/2,1/4,...,1/2^n,...\} \cup \{0\}$ and $Y = \{X\} - \{0\}$. Is $Y$ dense in $X$? The metric is the usual. If yes, why a separable discrete metric space is then countable? In this ...
6
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1answer
216 views

How prove this $\lim_{x\to+\infty}(f'(x)+f(x))=l$

let $f(x)$ is continous and $f'(x)$ is continous on $[0,\infty)$,show that $$\lim_{x\to+\infty}(f'(x)+f(x))=l$$ if and only if: $\displaystyle\lim_{x\to+\infty}f(x)=l$ and $f'(x)$ is uniformly ...
0
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2answers
56 views

Order properties of R

Let $S \subset \mathbb{R}$ be such that $a,b \in S \Rightarrow ab, a+b \in S$ for all $x \in R$, exactly one of the following holds $x \in S$ or $x=0$ or $-x \in S$ Show that $S = \{x \in ...
0
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1answer
74 views

why $\frac{\partial f(x,y)}{\partial x}+f(x,y)\frac{\partial f(x,y)}{\partial y}=0 \Rightarrow f(x,y) \equiv \text{constant}$

Assume $f(x,y) \in C^{(1)}(\Bbb{R}^2)$, if$$\frac{\partial f(x,y)}{\partial x}+f(x,y)\frac{\partial f(x,y)}{\partial y}=0$$. Show that $f(x,y) \equiv \text{constant}$ My approach: For every solution ...
1
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1answer
132 views

Prove that if $f$ is eventually monotone and eventually bounded $\Rightarrow \lim_{x\rightarrow \infty} f(x)$ is finite

If the function $f$ is defined on an unbounded above domain $D \subseteq \Re $ and is eventually monotone and eventually bounded, then $ \lim_{x\rightarrow \infty} f(x)$ is finite I tried to workout ...
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0answers
154 views

baby rudin 2.33, relative compactness

my question is relative to baby rudin theorem 2.33 which states; $$ \ suppose \ K \subset Y \subset X. \ then\ K \ is\ compact\ relative \ to\ X \ iff\ K\ is\ compact\ relative \ to \ Y.$$ ...
0
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1answer
49 views

Relation between $p$-superharmonic functions and concave functions

If I understood correctly what I read. In the one-dimensional situation the $p$ -superharmonic functions are exactly the concave functions and in several dimensions, the concave functions are ...
1
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1answer
3k views

The definition of locally Lipschitz

I am given this definition: A function $f:A\subset\mathbb R^n\to\mathbb R^m$ is locally Lipschitz if for each $x_0\in A$, there exist constants $M>0$ and $\delta_0 >0$ such that ...
1
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1answer
38 views

inf A of $n^22^{-n}$

Let $A=\{n^22^{-n}, n \in \mathbb{N}\}$. Find $\inf A$, $\sup A$. I tried starting by proving that $\frac{n^2}{2^n} \leq 1/n$ by induction. After, I showed that $\frac{n^2}{2^n} \geq 0$. By the ...
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2answers
22 views

Formally prove: $\lim_{n\to\infty}x_n=L_1\Longrightarrow\lim_{n\to\infty}x_{n+k}=L_1,\forall k\in\mathbb{N}$

OK, so I'm given the following: $$\lim_{n\to\infty}x_n=L_1\iff\forall\epsilon>0,\exists N(\epsilon)\in\mathbb{N}\ni\forall n>N(\epsilon),\ \left|x_n-L_1\right|<\epsilon$$ I just have no ...
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1answer
86 views

convergence sequence

Suppose that $g$ is continuous on an interval $[a,b]$ and that $g(x) ∈ [a,b]$ for all $x ∈ [a,b]$. (a) Use the intermediate value theorem to prove that is at least one number $c ∈ [a,b]$ with ...
2
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1answer
88 views

Let $\mu$ be the measure constructed in the riesz representation theorem. Is $\mu(\partial A)=0$?

I am currently self-studying Rudin's real complex analysis. The Riesz representation theorem in the book states: Let $X$ be a locally compact Hausdorff topological space. Let $T:C_c(C)\rightarrow ...
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2answers
269 views

Proof check: a sequence in $\mathbb{R}$ which does not converge to $x$ contains a subsequence $\{x_{n_i}\}$ for which no subsequence converges to $x$

Is the following proof correct? Can it be improved? Let $x\in \mathbb{R}$ and suppose that $\{x_n\}$ is a sequence that does not converge to $x$. Does there exist a subsequence $\{x_{n_i}\}$ ...
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3answers
265 views

How prove this nice limit $\lim_{n\to+\infty}\frac{1}{n}\sum_{k=1}^{n}f(\{ka\})=\int_{0}^{1}f(x)dx$

let $f(x)$ is Continuous on $[0,1]$, and such $f(0)=f(1)$,and if $a$ is irrational number. show that $$\lim_{n\to+\infty}\dfrac{1}{n}\sum_{k=1}^{n}f(\{ka\})=\int_{0}^{1}f(x)dx$$ where ...
2
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1answer
71 views

Convergence and Constant sequence?

Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence $x_0 x_1 x_2...$ given by $x_n = g(x_{n-1})$. converges ...
0
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1answer
40 views

Can you help me with a limit?

Can you help me with this limit? What do I have to do ? I tried to multiply by the conjugate expression but it didn't work $$ \lim_{x\to 2} \frac{x - \sqrt{3 x - 2}}{(x^2 - 4)}. $$ Thanks
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1answer
361 views

Prove that $ \ln[e(2/e)] $ is a fast way to calculate $ \ln2 $

Consider formula $ (*) \ln(x) = \sum_{k=1}^{\infty} (-1)^{k-1}\cdot \frac{(x-1)^k}{k}$. If you calculate $ \ln2 $ with error less then $ \frac{1}{2} \cdot 10^{-6} $ we need more than two milion ...
2
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1answer
207 views

How prove this $f(a)\le f(b)$

Suppose $f(x)$ is continuous on $[a,b]$, and that for any $x\in (a,b)$, the limit $$\lim_{h\to 0}\dfrac{f(x+h)-f(x-h)}{h}$$ exists and $$\lim_{h\to 0}\dfrac{f(x+h)-f(x-h)}{h}\ge 0.$$ Show that ...
0
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1answer
51 views

A function satisfy $x\frac{\partial f(x,y)}{\partial x}+y\frac{\partial f(x,y)}{\partial y}=0$ in a convex domain implies it is a constant

Assume $D \subset \mathbf{R}^2$ is a convex domain which contain original point. $f \in C^{(1)}(D)$,if $$x\frac{\partial f(x,y)}{\partial x}+y\frac{\partial f(x,y)}{\partial y}=0, \left((x,y)\in ...
0
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2answers
75 views

$\frac{\partial f(x,y)}{\partial x}=\frac{\partial f(x,y)}{\partial y}$ implies $f(x,y)=g(x+y)$

Assume $f(x,y) \in C^{(1)}(\Bbb{R}^2)$,If $$\frac{\partial f(x,y)}{\partial x}=\frac{\partial f(x,y)}{\partial y}$$ for all $(x,y) \in \Bbb{R}^2$. Show that there exists a function $g(t)$,such that ...
0
votes
1answer
46 views

Determining whether a subspace of the metric space of real sequences is separable

Let $$X=\left\{(a_n)_{n \in \mathbb N} \in \mathbb R^N : \exists n_0 \in \mathbb N\, \forall n\ge n_0 \big(a_n\le \sqrt{n}\big)\right\}$$ with the metric $$d\big((a_n)_{n \in \mathbb N},(b_n)_{n \in ...
2
votes
2answers
138 views

Isometric identification of $c_0^*$ and $ \ell^1$

Let $\{x_n\}_{n=1}^{\infty}\subset \ell_1$ be a sequence in $\ell_1$ with $x_n = (x_n(1),x_n(2), x_n(3),\ldots )$ I want to show that $$\lim_{n\to\infty}\sum_{j=1}^{\infty} x_n(j)y(j) = 0 $$ for all ...
1
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0answers
26 views

Singular Points in Different Areas of Mathematics

What is the relationship between singular points of algebraic curves (as described here or here), singular points of ode's (as described here or here) and singular points in complex analysis (as ...
1
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0answers
114 views

Find $\lim_{n\rightarrow\infty} na_n$ given that $0<a_0 <1$ and $a_{n+1}=a_n-a_n^2$ for $n\geq 0$.

I understand that no matter what value $a_0$ takes on between $0$ and $1$ that $a_1\leq \frac{1}{2}$. This has lead me to believe that for all $n\geq 1$, $b_n=(\frac{1}{n}-\frac{1}{n^2})<a_n ...
0
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0answers
128 views

Proving a metric space is separable

Let $X$ be a metric space which satisfies the following condition:$$$$If $\{ F_\alpha \}_{\alpha \in \Gamma}$ is a family of closed sets such that $\bigcap_{i \in \mathbb N} F_{\alpha_i} \neq ...
1
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1answer
276 views

Continuity of distance function and its generalization

The starting is an easy undergraduate problem. The distance function $d: X \times X \rightarrow \mathbb{R}$ in a metric space $(X,d)$ is continuous. Please check if my proof is correct. If it is wrong ...