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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14 views

Rayleigh quotient strictly increases

Consider the Rayleigh quotient $$\lambda_{L} := \max_{u \in H^{1}_{0}([0, L])}\frac{-\int_{0}^{L}u'^{2}\, dx}{\int_{0}^{L}u^{2}\, dx}.$$ Is $\lambda_{L}$ strictly increasing in $L$? Fix an $L_{1}, ...
2
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1answer
17 views

Reference for Gradient expression of a function on matricies

I'm looking for a reference (I suppose the statement is correct) for the following formula: $$ \langle\nabla f(\rho)^\dagger,V\rangle=\left.\frac d{dt} f(\rho+tV)\right|_{t=0} $$ for any direction ...
2
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2answers
39 views

$\{a_n\} \to a$ iff $\limsup_{n \to \infty} \{a_n\} = \liminf_{n \to \infty} \{a_n\}$

It is clear that if $$\limsup_{n \to \infty} \{a_n\} = \liminf_{\to \infty} \{a_n\},$$ then $\{a_n\} \to a$, since we can just squeeze the terms in the middle. I understand that to prove the ...
3
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1answer
59 views

Real Analysis, Cauchy but not null.

I came across this question in a book on p-adic numbers and thought it looked interesting. However, I am having trouble getting started with it. Any hints/suggestions is much welcomed Let $(a_n)$ be ...
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1answer
27 views

Algorithm and top-points.

Problem: For an array $A[1],\dots,A[n]$, with $n\geq 3$, it holds that $$A[i+1]>\frac{A[i]+A[i+2]}{2},\qquad i\in \{1,2,\dots, n-2\}$$ That is, it holds that $$A[2]>\frac{A[1]+A[3]}{2},\dots, ...
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1answer
18 views

Weak convergence of bounded nets

Let $(x_\alpha)_{\alpha\in A}$ be a bounded net in $c_0$. For all $\alpha\in A$, let $x_\alpha = (x_\alpha^n)\in c_0$; if, for every $n\in\mathbb{N}$, $(x_\alpha^n)_{\alpha\in A}$ is a net that ...
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1answer
27 views

Understanding this proof about the intersection of compact subsets

The following proof is theorem 2.36 from Rudin's Principles of Mathematical Analysis: Theorem: If $\{K_\alpha\}$ is a collection of compact subsets of a metric space $X$ such that the intersection ...
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1answer
22 views

Justification for Interchange of integral and sum

Let $\mu$ be a probability measure and $t\in\mathbb{R}$. I would like to write this equality $$\int_{\mathbb{R}}e^{ixt}d\mu(x)=\sum_{n\geq0}\frac{(it)^{n}}{n!}\int_{\mathbb{R}}x^{n}d\mu(x).$$ This is ...
2
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1answer
12 views

Investigating uniform convergence of a sequence

I am trying to determine if the sequence $f_n$:= $\frac{x^{2n}}{1+x^{2n}}$ is uniformly convergent on $D_1:=[-q,q],0<q<1$, and $D_2:= (-\infty,r] \cup [r,\infty),r>1$. I have determined that ...
1
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1answer
30 views

Characteristic function of a measurable set.

Let $X=L^p[0,1]$ $(1\leq p<\infty)$ be the Lebesgue space of p-integrable real functions on $[0,1]$. Let $D\subseteq [0,1]$ be measurable subset. The characteristic functions $\chi_D$ is defined as ...
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1answer
29 views

If the limit of power series exists, it converges.

Let $f(x) = \sum a_n x^n$ converges on $(-R, R)$. Does $\sum a_n R^n$ converge if $\lim _{x \to R-} f(x)$ exists?
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1answer
66 views

positive term series$\displaystyle \sum_{n=1}^\infty a_n$ is convergent,Prove$\displaystyle \sum_{n=1}^\infty a_n^{1-\frac 1n}$ is convergent

When I do my homework ,I met this problem: If positive term series$\displaystyle \sum_{n=1}^\infty a_n$ is convergent,Prove:$ \displaystyle \sum_{n=1}^\infty a_n^{1-\frac 1n}$ is convergent. I ...
3
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1answer
23 views

All closed rational rays measurable implies $f$ measurable

Is the following proof correct? Let $f: X \to \mathbb{R}$ where $X$ is a measurable space. Suppose $\{x: f(x) \geq r\}$ is measurable for each $r \in \mathbb{Q}$. Then, $f$ is measurable. Proof: ...
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1answer
19 views

If $A^k$ consistently approximates $\nabla^2f(x^k)$ with $x^k\to x^*$ and $\nabla^2f(x^*)$ regular, then the $A^k$ are regular, too

Let's call $\left\{A^k\right\}\subseteq\mathbb R^{n\times n}$ a consistent approximation of $\left\{B^k\right\}\subseteq\mathbb R^{n\times n}$ iff ...
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0answers
53 views

rudin's definition of a compact set

Here are some definitions given in my book: Definition 2.31 By an open cover of a set $E$ in a metric space $X$ we mean a collection $\{G_\alpha \}$ of open subsets of $X$ such that $E \subset ...
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1answer
28 views

Give an example of a linearly independent sequence $[x_0,x_1,x_2,\dots]$ of vectors in $\ell^{\infty}$ such that $\sum_{n=1}^{\infty} (x_n) =0$

Give an example of a linearly independent sequence $[x_0,x_1,x_2,\dots]$ of vectors in $\ell^{\infty}$ such that $\sum_{n=1}^{\infty} (x_n) =0$. Solution: Let $X_n=[x_0,x_1,\dots]$; define ...
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2answers
37 views

Let $X$ and $Y$ be finite sets. Then $X \cup Y$ is finite and $| X \cup Y| \leq |X| + |Y|$.

Let $X$ and $Y$ be finite sets. Let us assume that they are distinct at least, for otherwise $X \cup Y = X$ and $X$ is finite. Also let us assume that $X$ has cardinality $n$ and $Y$ has cardinality ...
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1answer
32 views

Solution to the wave equation in $\mathbb{R}^{3}$ with certain initial data

Suppose $f$ is a smooth function satisfying $f(0) = f'(0) = 0$. The question I am working on is to determine the solution $u$ to $u_{tt} - \Delta u = 0$ in $\mathbb{R}^{3}$ with $u(x, 0) = f(|x|)/|x|$ ...
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0answers
27 views

Jacobian from $\mathbb{R}^n \to \mathbb{R}^m$

Consider a diffeomorphism $F: V \subset \mathbb{R}^N \to U \subset \mathbb{R}^n$, where $U$ and $V$ are open, $N>n$. Moreover, suppose that $f:=F|_M: M \subset V \to U' \subset U$ is also a ...
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17 views

Measure-preserving map between a function and its symmetric rearrangement

Let $f \, \colon \mathbb{R}^d \rightarrow[0, \infty)$ be a function such that the sets $ \{ y \: \colon f(y) > \lambda \}$ are of finite Lebesgue-measure, for every $\lambda \geq 0$. Then, we can ...
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1answer
20 views

Let $A\subset R$ and $sup\{f(x)^2:x\in A\}=M(f^2)$, $inf\{f(x)^2:x\in A\}=m(f^2)$. Show that $M(f^2)=(M(|f|))^2$ and $m(f^2)=(m(|f|))^2$.

Let $A\subset R$ and $sup\{f(x)^2:x\in A\}=M(f^2)$, $inf\{f(x)^2:x\in A\}=m(f^2)$. Show that $M(f^2)=(M(|f|))^2$ and $m(f^2)=(m(|f|))^2$. I'm having difficulty showing the above equalities. I ...
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0answers
12 views

An elementary inequality in the context of Strictly Convex Function

Suppose: whenever $\epsilon \gt 0$ , define: $\zeta (y, \epsilon) = \frac{\eta (y+\epsilon) - \eta (y)}{\epsilon} - \eta'_{+}(y)$ ; where: $\eta$ is STRICTLY CONVEX CONTINUOUS FUNCTION & ...
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1answer
8 views

When we say f(x)->l as x->c then how c becomes a limit point of the domain of defination of f.

I think that if c be an isolated point of the domain of f then continuity of c does not imply existence of the limit of f at c.Is it the only cause?
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17 views

What are the asymptotic considerations in the following?

The following is from this paper that discusses polynomials and classic number theory functions. The proof of theorem 1.3 has a final statement saying that $R$ must be null because we arrive at ...
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3answers
53 views

Why is 1 + 1 = 0 when we make the addition table for F = {0, 1} (F = field)

In Analysis, I learned that any number system satisfying all the axioms (commutativity, associativity, identity elements, invertibility, distributivity) is called a field. Then the professor mentioned ...
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1answer
39 views

Showing that $\sup_{(x,y)}f(x,y)=\sup_x\sup_yf(x,y)=\sup_y\sup_xf(x,y)$

Can anyone help me prove this: Let $X$ and $Y$ be nonempty sets and $f:X\times Y\to\Bbb R$ such that $f(X\times Y)$ is bounded. Prove the following statement: ...
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2answers
40 views

$y'=\frac{y^2}{2x(y-x)}$

I'm trying to solve the following differential equation: $$y'=\frac{y^2}{2x(y-x)}$$ It is supposed to have a relatively easy general solution, but I can't find it. I've tried several things, the ...
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1answer
51 views

Why is $\overline{\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}}=L^2(\mathbb{T})$?

I want to know, why $\{e_n\mid n\in\mathbb{Z}\}$ is an orthonormal basis of $L^2(\mathbb{T})$, where $\mathbb{T}=\{z\in\mathbb{C}\mid |z|=1\}$, $e_n(z)=z^n$, and $\int_{\mathbb{T}} ...
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2answers
40 views

Orthonormal basis of $L^2(T)$

Why is $\{e_n\mid n\in\mathbb{Z}\}$ an orthonormal basis of $L^2(T)$, where $T=\{z\in\mathbb{C}\mid |z|=1\}$, $e_n(z)=z^n$, and $\int_T f(z)\,dz:=\int_0^1f(e^{2\pi i t})\,dt$? My try: If $n=m$, ...
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1answer
22 views

Verification of some conditions and facts of the Laplacian on a Riemannian manifold

I came across the following introduction to a paper I was reading: "Let $L$ be a second-order, elliptic, differential operator, with smooth coefficients, on a Riemannian manifold $M$. Assume $-L$ is ...
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0answers
63 views

How to construct $\mathbb{R}^N$ where $N$ is a random variable?

How does one rigorously construct $\mathbb{R}^N$ where $N$ is a $\mathbb{Z}^{++}$-valued random variable on some Borel probability space $(\Omega,\mathcal{B},\mathbb{P})$? Would someone be so kind ...
2
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1answer
22 views

Definition of differentiability at the point in multivariable calculus.

I'm self-studying the analysis from Zorich and the next definition of differentiability is given: $f:E\to \mathbb{R}^n$ is differentiable at the point $x$, which is a limit point of $E\subset ...
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24 views

In the geometrical interpretation for integration how lower and upper rectangular approximation are functions of natural number?

I've attempted to prove this in the following manner. Let Q be a subset of $P[a,b]$ which contains partitions of each order exactly once. Now, if we consider mappings $F:N \to Q$ defined by $F(n)=p$ ...
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1answer
39 views

Reposting Question about Schroder-Bernstein

Assume there exists a $1$-$1$ function $f:X\to Y$ and another $1$-$1$ function $g:Y\to X$. Follow the steps to show that there exists a $1$-$1$, onto function $h:X\to Y$ and hence $X\sim Y$. a) The ...
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1answer
22 views

Set invariant under reflections is a ball?

Say that $A\subset \mathbb{R}^n$ is measurable and of positive, finite measure. I'm trying to see if the following is true. If $A$ is invariant under all orthogonal reflections across $(n-1)$ ...
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1answer
60 views

Solve complex integral with $\Gamma$-function

Let $s\in\mathbb C$ and $r\in\mathbb R$. In the integral $$\int_{-\infty}^\infty \frac{1}{z^{r+s}\overline{z}^s} dx$$ we have $z=x+iy$ where $y>0$ is fixed. I read that you can explicitly compute ...
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35 views

On the continuity of Li's numbers. [on hold]

Consider Li's numbers defined by $\lambda_n = \sum_{\rho} \left(1-\left(1-\dfrac{1}{\rho}\right)^n\right)$ where $n$ is a nonnegative integer and the $\rho$ are the nontrivial zeros of the Riemann ...
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2answers
41 views

Decide what is the number of roots of the equation

Decide what is the number of roots of the equation $2^x=100x$. I know I can draw a sketch and then check but maybe there is a better method to do that? It's an exam question, thus it must require ...
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2answers
40 views

Convergence of $\int_2^{\infty}f(x)\,dx$ with a given condition

Let , $f$ be continuous function on $[2,\infty)$ and $\displaystyle\lim_{x\to \infty}x(\log x)^pf(x)=A$ , where $A$ is a non-zero finite number.. Then $\displaystyle\int_2^{\infty}f(x)\,dx$ is (A) ...
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1answer
13 views

Find the distance such that the angle will be the gratest

Rectangle shaped screen in a cinema is 8m high. It is place on a wall in such a manner that the upper edge of the screen is 12m above the floor. Find the distance between the viewer and the wall where ...
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2answers
73 views

Taylor series expansion, $f(x)=\frac{1}{4-x^2}$

Find taylor series expansion of the following function: $f(x)=\frac{1}{4-x^2}$ In the neighbourhood of the point $x_0=1$ determine the radius of convergence of this series. I do not understand the ...
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0answers
16 views

Proof of Darboux's Theorem when the function has infinite derivatives at both endpoints.

I have a question about the statement in the NOTE above. It says that the Darboux's Theorem is also valid when one or both the one-sided derivatives are infinite. So say $f_{+}'(a)=-\infty, ...
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1answer
92 views
+300

Intuition for visualising dense monotonic discontinuous function

My question is about the function defined in Rudin 4.31, mentioned by this question: Remark 4.31 in Baby Rudin: How to verify these points? The function is defined as $$f(x) \colon= \sum_{x_n < ...
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3answers
71 views

Limit $\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\sin x^2+y^2}$ [on hold]

Find the limit of: $$\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\left(\sin x^2 \right)+y^2}$$ How to find this limit? What is the most straightforward method?
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20 views

Find minimum distance between the plane and the beginning of Cartesian plane.

Find minimum distance between the plane: $S=\{\left(x,y,z\right) \in \mathbb{R}^3: x+yz=2012 \}$ and the beginning of Cartesian plane $(0,0,0)$. I want to minimize this with use of lagrange's ...
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0answers
30 views

Prove that $g$ has a limit at $x_0$ if $f$ has a limit at $x_0$ and $\lim_{t\to x_0} f(t)=f(x_0)$

The problem: Suppose $f:[a,b]\to R$ and define $g: [a,b]\to R$ as follows: $g(x)=\sup \{f(t):a\le t\le x \}$ Prove that $g$ has a limit at $x_0$ if $f$ has a limit at $x_0$ and $\lim_{t\to ...
5
votes
1answer
80 views

A possible dumb question about derivative

I was solving some differentiation problems when I found the function $$g(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}.$$ So I thought: If I define the function $f:\mathbb{R_{x>0}}\to \mathbb{R}$ as ...
0
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2answers
31 views

On the Spivak's proof of the theorem 3-11 (calculus on manifolds)

In second paragraph of the case 1 within the proof: What is $U$ s.t $A\subset U$ and satisfies in the proof of the case 1 of theorem 3-11. $\psi_i$ is defined on $U_i$ and its support is not ...
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votes
0answers
19 views

Is the following derivative also differentiable with respect to $n$? [on hold]

Let $f(x)$ be the $n-th$ derivative with respect to $x$ of $x \exp (n-1) log (x-1)$ evaluated at $x=1$. Is $f(x)$ differentiable with respect to $n$ ?
3
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3answers
94 views

Is the $n$th derivative of a continuous function also continuous?

Consider a differentiable (and hence continuous) function of order $n-1$. Is the $n$th derivative of such a function always continuous? As an example, is the $n-th$ derivative of the function $f(s) = ...