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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20 views

find out the number of students as required

in a survey of 900 student in a school, it was found that 600 students liked tea, 500 liked coeffee and 125 did not liked both the drinks. find the numberb of students who didnt liked tea only.
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1answer
41 views

Sign fluctuation in the harmonic series (sum)

Lets begin with a few simple rules that we know. -$\sum_{k=1}^\infty \frac{1}{n}=\infty$ -$\sum_{k=1}^\infty \frac{\color{red}{(}-1\color{red}{)}^{n-1}}{n}= \ln2 $ Going off of this knowledge, we ...
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3answers
45 views

Prove that if $B = \{x-y : x,y \in A\}$, where $A$ is a Borel measurable subset of $R$ with positive measure

Suppose that $m$ is Lebesgue measure, and $A$ is a Borel measurable subset of $R$ with $m(A) > 0$. Prove that if $B = \{x - y : x,y \in A\}$, then $B$ contains a non-empty open interval centered at ...
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0answers
18 views

Gauss divergence theorem, $div(F) = 0$?

I'm trying to solve the following problem using the Gauss divergence theorem. I have to calculate the Flux of $$ f(x,y,z) = (\sin(yz),y+\sqrt{x^2 + z^2}, 1-z) $$ through the surface $$ \Omega = ...
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1answer
32 views

If $d_1(x,y)$ and $d_2(x,y)$ are metrics, prove that $d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$ is a metric.

$$d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$$ The first three properties are trivially proven. The triangle inequality, not so much. I tried using the triangle inequalities that apply to $d_1$ and $d_2$, ...
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1answer
26 views

Define a metric using scalar product and prove that it is indeed a metric

So this is how I went about this: $\langle\,\cdot\,,\,\cdot\,\rangle: X \times X \to \mathbb{R}$ such that (by definition I list the properties of scalar product) and I can east prove the first three ...
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4answers
56 views

Secondary solving method of polynomial

$$x+1+\frac{1}{x}=0$$ This is a fairly trivial and possibly bland equation to solve. But for the sake of the question I will display them here: $$x\left(x+1+\frac{1}{x}\right)=x(0)$$ $$x^2+x+1=0$$ ...
3
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0answers
47 views

Understanding tensors

Locally in a chart, a tensor field looks like $$T= T^{i_1,...i_n}_{j_1,...,j_m} dx^{j_1} \otimes...\otimes dx^{j_m} \otimes \partial_{i_1} \otimes ... \otimes \partial_{i_n},$$ where ...
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1answer
56 views

Why is this statement true? [on hold]

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be the function $f(x,y)=(y-x^2)(y-2x^2).$ Why is this statement true: $t\mapsto f(t\xi)$ has in $t=0$ a local minimum for every $\xi\in\mathbb{R}^n$
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1answer
24 views

Heat equation with fourier transformation

I want to understand a solution from an exercise where we should find a solution of the heat equation: $$\frac{\partial u(x,t)}{\partial t}=\sum_{j=1}^{n}\frac{\partial^2 u(x,t)}{\partial x_j^2} $$ ...
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0answers
11 views

Primitive-ability of a function

Prove that the function $f:{R}\to{R}$ is primitive-able(does this term exist in English?) and find one of its primitives. $$f(x)=\left\{\begin{array}{cc} 1-x & x<1 \\ x^2-2x+1 & x \geq 1 ...
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1answer
39 views

Is every continuous 1-1 function onto? [on hold]

Is it true that every continuous 1-1 function on the interval [0,1] to [0,1] onto?
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0answers
15 views

Convergence of Laplace transform and its inverse

There is a sequence of functions $F^{\epsilon}(\lambda)$ which converges to 0 as $\epsilon \rightarrow 0$. Assume that each $F^{\epsilon}(\lambda)$ has a inverse Laplace transform f(s) such that ...
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1answer
46 views

$f$ is continuous on $E$ if and only if its graph is compact.

This question may be asked before under different formulation, the original problem is Chapter 4, Exercise 7 of Rudin's text: The Principles of Mathematical Analysis: Problem: If $f$ is defined on ...
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2answers
31 views

Proving Lower bounds on an Approximately Linear Function

We are looking for a lower bound on the function, $\frac{1.31}{e^{\frac{1.31}{x+1}} - 1}$ for $x \geq 2$. This function seems to behave linearly. We believe that the following statement holds: ...
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1answer
35 views

Closed form defines locally a function?

In my particular example, I have a volume form on a one-dimensional manifold, so i.e. this volume form $dx$ is closed. Now, I was wondering, does the integral function $f(y):=\int_{x_0}^{y} dx$ define ...
3
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1answer
44 views

Covariant and partial derivative commute?

I know that we have for a function $\Gamma: (-\varepsilon,\varepsilon)^2 \rightarrow M$ we have (at least I think I know that this is true) $$\nabla_{\frac{\partial \Gamma}{\partial s}} ...
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0answers
32 views

Jacobi fields and variations

I want to show that every Jacobi field is a variation of geodesics, i.e. let $Y: I \rightarrow TM$ be a Jacobi field along a geodesic $\gamma$, then I want to show that $Y$ can be written as a ...
4
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2answers
52 views

Every neighborhood $N_r(x)$ in $\mathbb{R}^n$ is connected

I am working on an exercise in baby Rudin (Ex 2.20 in particular) and as part of that I am trying to show that any neighborhood in a metric space is connected. I've seen several differing definitions ...
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2answers
21 views

If I have let's say a finite number of points in a metric space, and lets say that space has a couple of different metrics.

If I have let's say a finite number of points in a metric space, and lets say that space has a couple of different metrics defined. Is the diameter of a subset unique with respect to the two most ...
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1answer
25 views

Why do we study multi-valued(set valued) mappings?

I am working on a problem that has to do with multi-valued mappings. Precisely, Iterative methods for Fixed points for multivalued mappings. However, I have no clear motivation for studying such ...
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0answers
21 views

If $\partial E$ has Jordan outer measure zero, then $E$ is measurable.

I am going through Tao's measure theory book, and have to prove If $\partial E$ has Jordan outer measure zero, then $E$ is measurable. where $\partial E$ denotes the boundary of the set $E$. I ...
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0answers
30 views

Convergence of a sequence of integration

I am considering one problem and I am stuck in this step. The problem is that What conditions on function $f(u,\epsilon)$ are required to satisfy $$ \int_0^\epsilon f(u,\epsilon)\,du \rightarrow 0 ...
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0answers
43 views

Inner Product Properties And Applications

In every calculus or analysis class we are told that the concept of inner product is very important, and that its applications are vast, diverse, and extremely useful. I don't think there is a single ...
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0answers
12 views

About a property of the upper triangular projection of a matrix

I need a hand checking that a property about the upper triangle projection of an infinite matrix holds. $\bullet$ Let A be an infinite matrix $A=(a_{ij})_{i\geq 1\;j\geq 1}$. We define its upper ...
2
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1answer
15 views

$1$-form of a antiholomorphic function, Cauchy-Goursat Theorem

Let be $f:U\to \Bbb C$ antiholomorphic function. Show that the 1-form $f(z)d\overline{z}$ is closed. We have that $\overline{f}$ is a holomorphic function, so by Cauchy-Goursat Theorem the ...
4
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1answer
46 views

Schwarz Lemma of Complex Analysis

Let be $f:B(0,1)\to B(0,1)$ holomorphic function such that $$f(0)=f'(0)=\cdots=f^{(n-1)}(0)=0$$ but $f^{(n)}(0)\neq 0.$ Show that $|f(z)|\le |z|^n,$ for every $z\in B(0,1)$ and $|f^{(n)}(0)|\le ...
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0answers
12 views

Inverse Laplace operator $\Delta^{-1}$ in $H^1_0, \ H^{−1}$.

Let open domain $\Omega \subset \mathbb R^n$, $u : \Omega \rightarrow \mathbb R$, $u \in H^1_0 (\Omega)$ and $f \in H^{−1}(\Omega)$. I'm looking for some known expressions of the inverse Laplace ...
2
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3answers
79 views

Show that $\frac{n}{n^2-3}$ converges

Hi I need help with this epsilon delta proof. The subtraction in the denominator as well as being left with $n$ in multiple places is causing problems.
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0answers
24 views

Operator Norm and Submultiplicativity against the Spectral Norm

Consider $\mathcal{A}:\mathbb{R}^{n\times m}\to \mathbb{R}^{p\times q}$ to be a linear operator. I know that by considering the trace norm and using the submultiplicativity of the operator norm we ...
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0answers
35 views

local inverse functions

consider $f(x,y)=(x\sin y,x\cos y),\; (x,y)\in (0,\infty)\times (0,3\pi)=U$. f is locally invertible at every point in U, because $\det(Df(x,y))\not= 0$ for all $(x,y)\in U$. I want to know : What are ...
3
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1answer
41 views

Jacobi field strange condition.

I am currently reading a textbook (Kuehnel) saying that if $V,W \in T_pM$ are such that $\langle V,W \rangle =0$ and $\|V\|=\|W\|=1,$ then $Y(t):=D \exp(tV)(tW)$ is a Jacobi field. The thing is, I ...
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5answers
125 views

Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$”

If $C$ and $A$ are the circumference and area of a circle, then $$ \frac{C^2}{A}=4 \pi\; . $$ I'm looking for a reference to an elementary but rigorous proof of it. I'm particularly interested in ...
2
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1answer
48 views

Assumptions on functions so that integral is zero

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two arbitrary functions. Assume $g\in L^2(\mathbb{R})$. I'm looking to find out the minimal set of assumptions on $f$ and $g$ such ...
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1answer
43 views

convolution and integral limits

Let $\xi$ be an increasing function , and $f$ be a continuous function on the interval $[0,1]$. Take $\phi$ a smooth function such that $\int_0^1 \phi(s)\, ds= 1 $ and consider an approximation of ...
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1answer
27 views

Local Lipschitz continuity

In some proof I have seen the author use that if $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and bounded, then it is locally Lipschitz continuous. I have never seen that before and I don't find ...
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0answers
76 views

Show a function defined by summation is increasing, another is decreasing

Problem: For real numbers $x\ge1$ and $k>0$, let $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as follows. $f(x) = -\frac{1}{x}+\sum_{n=1}^{\infty}\frac{1}{(nk+x)^2}$ , ...
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1answer
60 views

Zeros of an analytic function [duplicate]

How to prove zeros of a real analytic function (non-zero function) is always countable?
2
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1answer
61 views

Rigorously proving $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx= \frac{\pi}{2}$ [duplicate]

I want to prove the famous formula: $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx = \frac{\pi}{2}.$ There are many ways to do it, for example, by some Fourier analysis. But how about a simple ...
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1answer
25 views

Show that the Lebesgue Stieltjes measure corresponding to $\alpha(x) = \mu((0,x])$ is $\mu$.

This is exercise 4.1 from Bass: Let $\mu$ be a measure on the Borel $\sigma$-algebra fo $R$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0,x])$ if $x \ge 0$ and ...
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1answer
41 views

Is $C^{\infty}$ dense in $W^{k,p}$?

The $C_c^{\infty}$ are certainly not sense in an arbitrary $W^{k,p}$ space. Despite, I started wondering whether at least $C^{\infty}$ is dense? Now, this can certainly not be true for general ...
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0answers
9 views

Calculate factor for FWHM in a sech(x)-function

I have a $sech\left(\frac{\pi}{2}a\cdot x\right)$-function, and I want to calculate $a$ such that the FWHM of the function meets a specific width $\Delta x$. So I started with ...
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0answers
41 views

Research areas lying at the confluence of Analysis and Geometry [on hold]

I wanted to get expert opinion on what are the areas of active research lying at the confluence of Analysis and Geometry. Two areas that I have heard about are : (1)Geometric Analysis and ...
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1answer
26 views

ODE, Picard approximation of a second order equation: How do I make sure that this is correct.

I have the following problem: $$\ddot{x} + \dot{x}^2-2x=0$$ and I.V are: $x(0)=1 \qquad$ $\dot{x}(0) = 0$. and I need to find two first "Picard" approximations. I first arranged it in the form ...
2
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1answer
68 views

Is the space of continuous functions with bounded variation separable?

Let $$BVC([0,1]) = \{f:\mathbb [0,1] \to \mathbb R,\, f\in BV([0,1])\cap C([0,1]),\, f(0)=0\},$$ and $$\|f\|_{BVC}=\mathrm{Var}(f).$$ Is $BVC([0,1])$ separable?
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1answer
24 views

Nest of intervals explanation

I am currently reading Konrad Knopp book about infinite series, I just don't get the part where he mentions that the nest of intervals would determine or define as he said a rational number s if it ...
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0answers
24 views

Methods of Real Analysis solution book [on hold]

Where can I find a solutions book for Methods of Real Analysis by RIchard Goldberg?
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2answers
38 views

When is a continuous function piecewise monotone?

Given a continuous function $f:[a,b]\mapsto \mathbb{R}$, are there known additional conditions that ensure $f$ is piecewise monotone? Like this question, my motivation is to decompose the interval ...
0
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1answer
15 views

For every $n \ge 2 $ , existence of uncountably many mutually disjoint closed balls , complement of whose union is path connected

For every $n \ge 2 $ , does there exist an uncountable family $\{D_\lambda: \lambda \in \Gamma\}$ of mutually disjoint closed balls in $\mathbb R^n$ , such that $\mathbb R^n \setminus \cup_{\lambda ...
3
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2answers
37 views

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ?

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ? Or , does every infinite dimensional normed linear space has ...