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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unboundness of an infinite series $f(t)\cos(tx)\sim t^{-1}\cos(tx)$

If $\lim_{t\to \infty} f(t)t=1$, i.e., $f(t)\sim t^{-1}$, then \begin{equation} {\text{ess}\sup}_{x\in [-\pi,\pi]}\sum_{t=1}^{\infty} f(t)\cos(tx)=\infty ? \end{equation} Here ${\text{ess}\sup}$ is ...
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2answers
48 views

How to prove if $\int^{\infty}_{0}f(x)dx$ a converges, then there is increasing sequence $x_n$, $\lim_{n \to \infty}f(x_n)=0$

I tried to prove it directly, but examples like $\sin(x^{2})$ makes it impossible to find the proper subsequence $x_{n}$; I also tried proving by contraposition, but the converse negative statement ...
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4 views

Metric space of non empty closed bounded parts of $R$ with the Hausdorff metric

Consider the metric space of non empty closed bounded parts of $R$ with the Hausdorff metric. For n $\in N_{0}$ and $F_{n} = \{0,1/n,2/n,3/n, ..., 1\}$ i am wondering if $(F_{n})_{n}$ is convergent? ...
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36 views

Analysis and algebra [closed]

I'd like to know if there exist a field of the theoretical math that really combines analysis and algebra. Some people say that Model theory combines those two subjects but I personally want ...
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1answer
21 views

Controlling the size of a function

Consider the function $$f(\delta,r)=\frac{2e^{-\delta r} }{r}\sinh\left(r/2\right)$$ with $\delta >0$ Show that $\exists r >0 \text{ such that }f(\delta,r)<1$ My attempt: \begin{align*} ...
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1answer
49 views

$f(x)=\frac{x}{x}$ is continuous at $0$?

$x$ is divided by $x$. Thus, $f(x)=1$ when $x \neq 0$. However, at $0$ can we consider $f(x)$ as $1$? More specifically, do we have to define a rational function as a reduced form?
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1answer
45 views

Any Video Lectures Of An MIT, Harvard, Stanford, UC Berkeley, Yale, or Princeton Analysis Course Based On Baby Rudin?

I'm learning analysis from the book Principles of Mathematical Analysis by Walter Rudin, third edition. This book, popularly known as Baby Rudin, is being used for analysis courses at such elite ...
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23 views

Showing that $f$ is convex given that $(\nabla f(x) - \nabla f(y)) \cdot (x-y) \geq c(x-y)\cdot(x-y), \;\;\forall x,y \in \mathbb{R}^n$

Assume that $f$ is continuously differentiable and that for some constant $c > 0,$ the gradient $\nabla f$ satisfies, \begin{equation} (\nabla f(x) - \nabla f(y)) \cdot (x-y) \geq c(x-y)\cdot(x-y), ...
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18 views

Estimate the number of candidates who obtained fewer than 70 scores.

In an examination, the number of candidates who obtained scores between certain limits are as follows: Scores $0—19$, $20—39$, $40—59$, $60—79$, $80—99$, Number of candidates $41$, $62$, $65$, ...
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1answer
73 views

Show that the limit $\displaystyle \lim_{n\to \infty }\frac{a_{n}}{n}$ exists.

So $\left \{ a_{n} \right \}_{n\geq 1}$ is a sequence of real numbers and $C>0$ is a fixed constant. We assume that $a_{n+m}\leq a_{n}+a_{m}+C, \forall n, m\geq 1$. What is a good way to prove ...
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11 views

Limit over general chain

We all know classic definition of limit of a sequence. There is also definition of limit of a function. Now consider general chain, i.e. linearly ordered set $(\mathcal{I}, \le)$ with specified values ...
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0answers
35 views

The space of sequences which are eventually zero in $l^2$ is not a Hilbert space.

Define $V$ to be the space of sequences which are eventually zero, i.e. $$V=\bigcup_{N=1}^{\infty}\{(x_n)_{n=1}^{\infty}\in l^2: x_n=0 \; \text{for}\; n\ge N\}.$$ Is $V$ a Hilbert space with ...
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2answers
39 views

Prove a doubly periodic entire analytic function in complex plane is a constant [duplicate]

I got stuck on this problem. So I really appreciate if anyone can give me some hint to move on. Thanks a lot. Prove that an entire analytic function $f:\mathbb{C} \rightarrow \mathbb{C}$ is a ...
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1answer
30 views

Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$.

Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$. I'm thinking about a contradiction proof. Assuming ...
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0answers
18 views

Aproximation in $W^{1,p}(U)$ with U disconnected.

Consider $U=(-1,0)\cup(0,1)$. Define $$v(x)=\left\{\begin{array}{rc} 0,&\mbox{se}\quad -1<x<0,\\1, &\mbox{se}\quad 0<x<1. \end{array}\right. $$ Clearly $v\in W^{1,p}(U)$ for each ...
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1answer
22 views

$\sup\limits_{t>0}[\frac{g(t)}{\sup\limits_{t<u<\infty}g(u)}]\leq 1$

I can't prove that the following inequality true or not: $$ \sup\limits_{t>0}[\frac{g(t)}{\sup\limits_{t<u<\infty}g(u)}]\leq 1 \tag{*}, $$ where $g$ is a positive function. I think it is true ...
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2answers
44 views

Theorem 3.19 in Baby Rudin: The upper and lower limits of a majorised sequence cannot exceed those of the majorising one

Here is Theorem 3.19 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Let $\{s_n \}$ and $\{t_n \}$ be sequences of real numbers. If $s_n \leq t_n$ for $n \geq N$, ...
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0answers
21 views

Theorem 3.17 in Baby Rudin: Infinite Limits and Upper and Lower Limits of Real Sequences

Here're Definitions 3.15 and 3.16 and Theorem 3.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Definition 3.15: Let $\{s_n \}$ be a sequence of real numbers ...
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65 views

I have a answer to a question about trace. Is there an easier answer to this question?

Let $A\in M_n(\mathbb{C})$. Show that $$tr\left(\frac{A+A^*}{2}\right)\leq tr((A^*A)^{1/2}).$$ My answer: It is easy to see that $$tr\left(\frac{A+A^*}{2}\right)=\text{Re}(tr(A))\qquad and\qquad ...
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16 views

Compute the characteristic strip and conoid solution of a geometric PDE

This is a problem from Fritz John's Partial Differential Equations, which I'm working through for self-study. Given a family of spheres of radius $1$ with centers in the $xy$-plane ...
2
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1answer
26 views

Let the sequence of functions $f_n(x)$ equal $1$ if $x ∈ [n, n + 1)$ and $0$ otherwise. Why doesn't $f_n$ converge uniformly?

Let the sequence of functions $f_n(x)$ equal $1$ if $x ∈ [n, n + 1)$ and $0$ otherwise. How can I use the definition of uniform convergence to show that $f_n$ does not converges uniformly? If a ...
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1answer
28 views

Prove the integral is always imaginary

Show that if f is analytic on D and γ is a closed curve in the region then the integral $$\int \overline{f(z)}f'(z)$$ is purely imaginary. I think this problem would use some extension of cauchy ...
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1answer
44 views

Continuity of the operator “inverse of” in the space of linear, bounded and bijective operators

Let $(X,\|\ \|_X),(Y,\|\ \|_Y)$ be two Banach spaces over $K$. Definition: $\qquad\qquad\qquad\quad I(X,Y):=\big\{A\in \mathscr B(X,Y):A\text{ is invertible and }A^{-1}\in\mathscr ...
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2answers
15 views

Calculating Taylor series of complex function

I'm going through a past exam paper and found a question I can't do. The question is to write down the Taylor expansion of $\frac{z^2}{z-2}, z \in C$ \ {2}, on the disc $|z| < 2$ I've been ...
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3answers
27 views

Proving continuity of the following function

Let $X,Y$ be compact sets in $\mathbb{R}^n$ (with the usual topology) and let $f:X\times Y \rightarrow \mathbb{R }$ be a continues function function moreover let $P(Y)$ be the space of all ...
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22 views

Proof on a statement

Unfortunately, I cannot seem to prove this question. The integral of a regulated function does not depend on the approximating sequence of step functions. Is it possible to give a guided description ...
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3answers
48 views

How to write the following polynomial in $(1-\frac{x}{a}) (1-\frac{x}{b}) (1-\frac{x}{c}) (1-\frac{x}{d})$?

I was given the following problem: Write the polynomial $f(x) = \frac{1}{24} \displaystyle \prod_{i \mathop = 1}^4 (x-i)$ in the form $(1-\frac{x}{a}) (1-\frac{x}{b}) (1-\frac{x}{c}) (1-\frac{x}{d})$ ...
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1answer
32 views

Proving that a function is real-analytic

I try to solve the following exercise: Let $f:\mathbb{R}\to\mathbb{R}$ with $f(x):=\frac{1}{1+x^4}$. Prove that $f(x)$ is real analytic and compute the radius of convergence of it's Taylor series at ...
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1answer
30 views

Is there any flaw in this proof relating derivative and monotonicity?

I came across this lemma in my notes: Suppose that $f:(a,b)\rightarrow \mathbb{R}$ and $f'(c)$ exists for some $c \in (a,b)$. If $f'(c) > 0$, then there exists $\delta > 0$ such that $$f(x) ...
2
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1answer
32 views

Calculating the flux of $\langle x,y,x\rangle$ over $z=1-x-y$

Find the flux of $\bar F = \langle x, y, x\rangle$ over $z = 1 - x - y$ in the first octant use the upward unit normal ($\bar n$) flux = $\int\int_S \bar F \cdot \bar n dA$ $dS = \sqrt 3 dA$ $\bar ...
2
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1answer
45 views

A problem about tangents and areas.

Let $f(x) = x^3 - 4x^2 + 4x$, its graph denoted $C$. If for any $x_1 \not = \frac 43$, the tangent of $f(x)$ at the point $P_1(x_1, f(x_1))$ intersects $C$ at another point $P_2(x_2, f(x_2))$, and ...
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1answer
49 views

Show that $d(z,z_1)$ is a metric.

I am relatively new to proofs so I am struggling to understand what really counts as "proved" and what doesn't. Consider the vector space $\mathbb M $ of real numbers $\mathbb R$. Show that for ...
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34 views

Holomorphic, entire functions and the Cauchy-Riemann equations

Where is $f\colon\mathbb C\to\mathbb C,~z\mapsto |z|^2$ $\mathbb C$-differentiable? Is there a restriction of $f$ that is holomorphic? Is there an entire function $f$ with ...
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2answers
26 views

Let $C$ be a convex closed nonempty subset of a Hilbert space $H$. Show that there is a unique element in $C$ with the minimum norm.

Let $H$ be a Hilbert space and $C \subset H $ be a convex, closed and nonempty subset of $H$. Prove that there exists a unique $x_0\in C$ with minimum norm among the elements of $C$. I don't know ...
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0answers
7 views

Question conserning the existence and continuity of derivatives of function's shperical mean

I heard a rumor that the claim beneath is true and I'm trying to prove it (or find a counterexample). Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$, $f\in C^k(\mathbb{R}^n)$. Fix $\varepsilon > 0$ ...
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10 views

Power-Series strictly convexity

Watch the power-series $B(\beta):=\sum_{i=0}^{\infty}b_{j}e^{\beta\cdot j}$ with $b_{j}\geq 0$ for $0<\beta<r$ where $r$ is the radius of convergence. At least one $b_{j}$ for $j\geq 2$ is non ...
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1answer
22 views

Closed, convex and balanced subset of a vector space

Let $(X,\|\ \|)$ be a vector space over $K$ and let $B\subseteq X$ be closed, convex and balanced. I want to prove the following: If $x_0\in X\setminus B\Rightarrow\exists\;f\in X^*$ s.t. ...
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3answers
27 views

evaluate the following limit else prove that limit does not exist

The function/sequence of interest is as follows: $(\frac{n!}{n!+2})^{n!}$ I have a feeling the limit does exist, as if we divide the numerator and denominator by $n!$ we get ...
3
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1answer
34 views

Laplace transforms of powers of cosine

During the past several hours, while studying the Laplace transform, I've started wondering what \begin{equation} \mathcal{L} \{ \cos^n(at)\}(s) \end{equation} would look like – since it won't ...
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15 views

Let $E$ be the set of the $n\times n$ symmetrics and $U$ the subset of those are positive definite.

Then $f(X) = X^2$ is a diffeomorphism of $U$ onto itself. Ok, first of all, I am having troubles to show that $f(U) = U.$ How can I show that the image of this function is the set $U$? For the ...
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2answers
18 views

How to show that the inverse function is $C^r, ~ r \ge 1$

Let $g : \mathbb{R}^n \to \mathbb{R}^n \in C^r(\mathbb{R}^n), ~ r\ge 1.$ Suppose that $\|Dg(x)\| < 1$ for all $x \in \mathbb{R}^n.$ Define $f(x) = x + g(x).$ I can easily show that $f(x)$ ...
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1answer
53 views

Taylor series has zero convergence radius?

Let $$f(x):=\sum_{n=0}^{\infty} \frac{f^{n}(0)x^n}{n!}$$ where the $$|f^{n}(0)| \le C\frac{\Gamma(\frac{n+1}{\alpha})}{\alpha^{\frac{n+1}{\alpha}+1}}$$ for a constant $C>0$ and $\alpha>0$. Does ...
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0answers
35 views

rational function cancellation [duplicate]

This is probably a trivial question however i cannot find the correct information online. When simplifying mappings from $\mathbb{R}$ to $\mathbb{R}$ such as: $$\frac{x(x-1)}{(x-1)}$$ Why is it ...
0
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1answer
49 views

Show that $e^{\varepsilon |x|^{\varepsilon}}$ grows faster than $\sum_{k=0}^{\infty} {|x|^{2k}}/{(k!)^2}$

I am wondering whether we have for $$f(x):=\sum_{k=0}^{\infty} \frac{|x|^{2k}}{(k!)^2} $$ that $$\lim_{x \rightarrow \infty} \frac{e^{\varepsilon |x|^{\varepsilon}}}{f(x)} = \infty$$ for any ...
2
votes
2answers
32 views

A “special” inclusion of $W^{1,p}_0(\Omega)$ in $L^{\infty}(\Omega)$

I'm in trouble with this stuff. If $\Omega$ is a bounded open set in $\mathbb{R}^n$ of class $C^1$ and $f\in W^{1,p}_0(\Omega)$, is it true or not that $f\in L^{\infty}(\Omega)$? I think so, but I ...
1
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1answer
55 views

Real Analysis, Folland Problem 6.2.17 Dual of $L^p$

Theorem 6.14 - Let $p$ and $q$ be conjugate exponents. Suppose that $g$ is a measurable function on $X$ such that $fg\in L^1$ for all $f$ in the space $\sum$ of simple functions that vanish outside a ...
2
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0answers
22 views

Vector Calculus: solution to Poisson equation

This is problem 8.4.17. from Marsden Vector Calculus book. Let $\rho$ be a continuous function which vanishes outside a 3D region $W$. Define ...
3
votes
1answer
44 views

Real Analysis, Folland Problem 6.1.16 $L^p$ spaces

Problem 6.1.16 - If $0 < p < 1$, the formula $\rho(f,g) = \int |f-g|^p$ defines a metric on $L^p$ that makes $L^p$ into a complete topological vector space. Attempted proof - Suppose $a,b ...
0
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1answer
21 views

About $L^{p}$ norms and the Hilbert transform

When we are proving $L^{p}$ estimates for the Hilbert transform, we can proceed in the following way: Step 1 Prove that $H$ maps $L^{2}$ to $L^{2}$. Step 2 Prove that $H$ maps $L^{1}$ to ...
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2answers
33 views

Proof of $ax\le bx, x>0 \Rightarrow a \le b$ using only field axioms

I'm reading ELEMENTARY CLASSICAL ANALYSIS (2nd, Marsden), 1.1.2 Proposition. How can I prove $ax\le bx \land x>0 \implies a \le b$ using only sixteen field axioms? My proof was Prove $\forall ...