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

Are there way of proving that polynomials are relatively prime using number theory or abstract algebra?

This question is inspired by question A5 from the Putnam Mathematical Competition: Let $$P_n(x) = 1 + 2x + 3x^2 + \cdots + nx^{n-1}.$$ Prove that polynomials $P_j(x)$ and $P_k(x)$ are ...
1
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0answers
47 views

How to calculate the $(3)$ and $(4)$?

In Gérald Tenenbaum's book "Introduction to Analytic and Probabilistic Number Theory" Cambridge University Press 1995, On the page of 97-98, I Can calculate the $(1)$ and $(2)$, but I do not know how ...
0
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0answers
21 views

Why doesn't this approach work for integral of $\log(\sin(x))$?

Evaluation of: $$I = \int_{0}^{\pi} \log(\sin(x)) dx$$ Over closed rectangular contour $ABCD$ complex analysis. Kind of like the contour here: Contour Answer complex analysis. BUT INSTEAD the ...
0
votes
1answer
27 views

$\int_{0}^{2\pi}\int_{0}^{\pi}\sin^3y \ e^{3\cos x\sin y+4\sin x\sin y}\,dy\,dx$

I am working on this double integral $\displaystyle\int_{0}^{2\pi}\int_{0}^{\pi}\sin^3y \ e^{3\cos x\sin y+4\sin x\sin y}\,dy\,dx$ so far, I don't know how to start. Can someone give a hint? Thanks ...
0
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2answers
41 views

(some errors are corrected)Can $\int_0^T{\frac{e^{at}}{(1+t)^b}} \leq \frac{Ce^{aT}}{(1+T)^b}$ where $a,b>0$? [on hold]

I'm sorry to make some misunderstandings I need C to be independent of T i.e. Could there be a constant C which is independent of $a$ and $b$ and $T$, such that $$\int ...
1
vote
1answer
23 views

Mean value theorem for line integral

I am wondering if there is a mean value theorem for line integral. For example, let $f(x):\mathbb{R}^n\rightarrow \mathbb{R}$ be a continuous (not necessarily monotonic) function defined on smooth ...
2
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0answers
31 views

$ C^p_0(B,\mathbb{X})$ is a Banach space with the norm of $C^p\!\!$-topology?

Let $\mathbb{X}$ a Banach space with norm denoted by $\|\;\cdot\;\|_{\mathbb{X}}$. Let $ B $ the open ball with center at $ 0$ and radius $ r> $ 0, i.e. $B(0,r)=\{x \in \mathbb{X}: \|x-0\|<r\}$. ...
7
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1answer
22 views

$A:=\left\{y\in\mathbb{R}:\mu\left(f^{-1}(\left\{y\right\})\right)>0\right\}$ is countable, if $\mu$ is a finite measure and $f$ has compact support

Let $E$ be a metric space $\mathcal{E}:=\mathcal{B}(E)$ be the Borel algebra on $E$ $\mu:\mathcal{E}\to [0,1]$ be a measure $f:E\to\mathbb{R}$ be measurable and have compact support Assume that ...
0
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1answer
20 views

Get Progress from Three Values

I have a starting weight, current weight and target weight. How can I get a percentage of progress? Thanks!
0
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2answers
34 views

How to convert a discrete function to a continuous function

I was wondering because of this: Trick to find if number is composite or prime Is there any formal method to convert a discrete function to a continuous function. For example take $n!$, how was the ...
1
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0answers
37 views

Is the condition of continuity for the differentable functions necessary in Looman-Menchoff theorem?

Looman-Menchoff theorem states that a continuous complex-valued function $f(z)=u(x,y)+iv(x,y)$ defined in an open set of the complex plane is holomorphic if and only if it satisfies the Cauchy–Riemann ...
2
votes
2answers
20 views

Property of distance and adherence

Please how to prove that in a metric space $(E,d),$ for $A,B\subseteq E$ that $\forall x\in E, d(x,A)=d(x,\overline{A})$ and that $d(A,B)=d(A,\overline{B})=d(\overline{A},\overline{B})$ and ...
1
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1answer
27 views

Lowest consecutive number that is the result of an addition of 2 different integers

I need to know what the lowest consecutive number would be that is possible by simply adding 2 numbers any times necessary. I came up with a simple formula for numbers with greatest common divisor 1: ...
0
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1answer
32 views

hard question on singularities

If every series converging to the singularity has a sub sequence such that limit of the function of the subsequence is zero what can the singularity be?
0
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3answers
21 views

Integration substitution: How do you find the derivative of the denominator?

I have to integrate: $$ \int_1^2 \frac{37 x}{x^2-6 x+10} \, dx $$ $$ =37 \int_1^2 \frac{x}{x^2-6 x+10} \, dx $$ Then Wolfram Alpha tells me to rewrite the integrand as $$ \frac{2 x-6}{2 ...
0
votes
1answer
12 views

Showing that $f(z)$ is differentiable throughout a region

I'm having trouble with an algebraic operation in a proof, which I will copy here: Specifically, I do not see the connection between steps 4.8 and 4.9. As best as I can tell, the equations in 4.8 ...
0
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1answer
74 views

Studying mathematical Analysis [on hold]

I want to study Mathematical analysis for my use in signal processing and physics I want to study : real and complex analysis in particular what topics in what subjects in mathematics Should I read ...
0
votes
1answer
47 views

$\int_{0}^{2\pi} f(x) \text{e}^{-ilx} \, \text{d}x =0$

Suppose $f,g \in \mathcal{L}^1(\mathbb{R} / 2 \pi)$ with $f(x)=g(mx),m \in \mathbb{Z}$. I want to show that $$ \forall \, l \in \mathbb{Z} \ \text{with} \ l \not\equiv 0 \ \text{mod} \ m \colon ...
1
vote
2answers
50 views

How to prove that $(x-a)^{2} \cdot f(x) $ is differentiable only at $a$, where $f(x)$ is the Dirichlet function?

Let's $f$ be a function, which is equal to 1 at irrational points and to 0 in rational. Let's denote the following: $g(x)=(x-a)^{2} \cdot f(x)$, how to prove that it's differentiable only at $x=a$? ...
1
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3answers
77 views

prove when $r \geq t $ then $ x^{r} \geq x^{t} $

when $ r,t \in \mathbb{Q} , x \geq 1 $ and $ r\geq t $, prove $ x^{r} \geq x^{t} $. i have tried so much but i can't prove it. :( My attempt : i can derive from it that this proposition is equal ...
-8
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0answers
94 views

What's wrong here? [on hold]

As everybody knows that $\frac{1}{-1}=\frac{-1}{1}$. Now taking $\sqrt{}$ on both sides we get, $\sqrt{\frac{1}{-1}}=\sqrt{\frac{-1}{1}} \implies ...
2
votes
0answers
19 views

multivariable calculus, existence of partials

Let $f(x,y)$ be a continuous funciton on $I=(a,b)\times (c,d)$ such that (i) $f_x$ exists and continuous on $I$. (ii) For some $x_0\in (a,b)$, $f'(x_0,y)$ exist. (iii) $f_{xy}(x,y)$ exist and ...
1
vote
1answer
16 views

Origin of min/max notation

Here I am referring to the notation $x \wedge y = \min \{ x,y \}$ and $x \vee y = \max \{ x,y \}$. These seem to reference the corresponding usages in logic, where $\wedge$ means "and" and $\vee$ ...
1
vote
1answer
43 views

Continuous extension on compact set in $\mathbb{R}^n$

I'm an undergrad student reading through Deimling's Nonlinear Functional Analysis and have come across the following proposition. Let $A\subset\mathbb{R}^n$ be compact and $f:A\to\mathbb{R}^n$ be a ...
4
votes
2answers
86 views

Inequality involving $\frac{\sin x}{x}$

Can anybody explain me, why the following inequality is true? $$\sum_{k=0}^{\infty} \int_{k \pi + \frac{\pi}{4}}^{(k+1)\pi-\frac{\pi}{4}} \left| \frac{\sin \xi}{\xi} \right| \, \text{d} \xi \geq ...
1
vote
1answer
25 views

Lipschitz continuity of a generalized Rayleigh quotient

I am thinking about the Lipschitz continuity of a generalized Rayleigh quotient: $f(x)=\frac{x^\top Ax}{x^\top Bx}$ with the constraint $||x||\geq c$, where both $A$ and $B$ are positive definite ...
0
votes
0answers
23 views

Geometric Intuition behind the Dual Norm?

What is the geometric intuition behind the dual norm, $\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$ Specifically, if possible in terms of hyperplanes defined by $x$ and $z$. My interest ...
1
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0answers
11 views

Equivalence of Theorems; the sphere on $\ell^2$ is finitely oscillation stable.

I just started reading the book Dynamics of Infinite-dimensional Groups, by Pestov, and right in the introduction the following theorem by Milman is cited: Let $\mathbb{S}^\infty$ denote the sphere in ...
4
votes
2answers
243 views

Meaning of the backslash operator on sets

I am self-studying analysis and ran across this: $\mathbb R \setminus \mathbb N$ is an open subset of $\mathbb R$ My best guess for interpretation was this: the set $\mathbb R \setminus \mathbb N$ ...
1
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1answer
19 views

The inner product of the Cartesian Product space

I want to know how can one define the inner product in the Cartesian product of spaces, i.e. let $A,B$ two hilbert spaces. Let $a_1, a_2 \in A$ and $b_1, b_2 \in B$, how can one express the inner ...
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0answers
14 views

Stopping Times and Directed Processes [on hold]

The book "Stopping Times and Directed Processes" uses the techniques of stopping times to convergence problems. Just wondering, what are the advantages of using this approach? Would it make the ...
3
votes
2answers
55 views

Demonstration of $\int_{-a}^a \frac{f(x)}{1+e^x} \,dx= \int_0^a f(x) \,dx$ [duplicate]

Good morning, Can you give me a help to demonstrate this proposition: $f$ is an even and continuous function on the interval $[-a,a], a>0$. Demonstrate: $$\int_{-a}^a \frac{f(x)}{1+e^x} \,dx= ...
0
votes
1answer
37 views

Ml inequality for $\log(z+i)$

I do not need a complete proof, just a hint. This is what the problem is: $$\int_{0}^{\infty} \frac{\log(1+x^2)}{1+x^2} dx$$ Over this contour: The radius is $R$ from the midpoint. I am trying ...
0
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0answers
33 views

Kuratowski Theorem on uniform convergence

There is a nice criterion of uniform convergence of the sequence of continuous functions due to Kuratowski: as far as I remember this involves checking whether $f_n(x_n) \to f(x)$ where $x_n \to x$ is ...
0
votes
1answer
27 views

Show directly from Cauchy definition [on hold]

I can't show directly from the definition that the following is Cauchy sequence, please help me : ( only directly from definition) $\left(1+\dfrac{1}{1!}+\cdots+\dfrac{1}{n!}\right)$
0
votes
2answers
33 views

Limit of a sequence problem

Suppose that $(x_n)$ is a sequence such that $\lim_{n\to\infty}\sum_{k=1}^n\frac{{x^4_k}}n=0.$ How do I show that $\lim_{n\to\infty}\sum_{k=1}^n\frac{{x_k}}n=0$?
2
votes
1answer
72 views

Evaluating $\int_0^{\infty} \frac{\xi x^{\alpha}}{ e^{x}-\xi} \:\mathrm{d}x$

I am supposed to integrate for $\alpha \ge 0$ $$\int_0^{\infty} \frac{x^{\alpha}}{ \xi^{-1}e^x - 1} \:\mathrm{d}x,$$ where $\xi e^{-x} < 1$ which means, I want to express this in terms of simple ...
4
votes
3answers
110 views

Rolle's Theorem: why do we need the premise $f(a) = f(b)$?

If a real-valued function $f$ is continuous on a closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then there exists at least one $c$ in the open interval ...
2
votes
3answers
267 views

A not complete metric space?

Please how to prove that the space $\mathbb{R}$ endowed with the metric $d(x,y)=|e^x-e^y|$ is not a complete space? I don't find a Cauchy sequence but not convergent Please Thank you.
0
votes
2answers
43 views

Question about distance, open and closed sets in a metric space

Let $(E,d)$ a metric space, and $\emptyset\neq A,B\subset E$ My first question, when $A\subset B$ we have that $d(x,A)\leq d(x,B)$ or the inverse ? My second question is how to prove that $\Omega$ ...
1
vote
1answer
46 views

Brouwer theorem

Is the Brouwer's fixed point theorem true for the topological space '+' sign(cross)? $$ + = \left( [-1,1] \times \{0\} \right) \cup \left( \{0\} \times [-1,1] \right) $$ I have tried using spencer's ...
0
votes
1answer
11 views

Distinguishing two cases for $\lim_L\delta_L^3 L^d$

Let $\lim_{L\to\infty}\delta_L=0$ and $\lim_{L\to\infty}L^{d/2}=\infty, d\geqslant 2~~~(*)$ Now two cases are distinguished. case I $\lim_{L\to\infty}\delta_L^3L^d=0$ caseII The ...
0
votes
1answer
30 views

for monotonic differentiable increasing functions, show $f'(x_0)\geq 0$

Let $X$ be a subset of $\Bbb R$, let $x_0$ be a limit point of $X$, and let $f:X\to \Bbb R$ be a function. If $f$ is monotone increasing and $f$ is differentiable at $x_0$, then $f'(x_0)\geq0$. ...
2
votes
2answers
84 views

One dimensional vector space and not Hausdorff

I read that all vector spaces that do not have the Hausdorff property and are one-dimensional need to have the trivial topology. I am not quite sure how to approach this problem, but I would like to ...
-1
votes
1answer
30 views

Sequence in $A$ converging to $\inf(A)$ or $\sup(A)$ [on hold]

Given a set $A$, how can I prove that there exist sequences $(x_n)$ and $(y_n)$ in $A$ such that $\lim_{n \to \infty}x_n = \inf(A)$ and $\lim_{n \to \infty}y_n = \sup(A)$?
0
votes
2answers
73 views

Convergence and continuity of $\sum\limits_{n=1}^\infty n^2x^n$ and $\sum\limits_{n=1}^\infty n^{-2}\sin(nx)$ [on hold]

I have two questions in particular that have been bothering me. Any help would be greatly appreciated: Consider the sum $$\sum_{n=1}^\infty n^{2}x^n$$ Determine the interval in which it is ...
1
vote
1answer
33 views

Construct a $C^1$ function from $\mathbb{R}^2$ to $\mathbb{R}^2$ with prescribed rank

Construct a $C^1$ function from $\mathbb{R}^2$ to $\mathbb{R}^2$ such that rank of $D(F)$ has rank $2$, except at the origin where it has rank $1$. I'm having trouble constructing such a ...
5
votes
3answers
128 views

Is the function $f(x) = \begin{cases} 1 & \text{$x\in\Bbb Q$} \\[2ex] 0 & \text{$x\notin\Bbb Q$} \end{cases}$ Riemann integrable?

$f(x) = \begin{cases} 1 & x\in\Bbb Q \\[2ex] 0 & x\notin\Bbb Q \end{cases}$ Is this function Riemann integrable on $[0,1]$? Since rational and irrational numbers are dense on $[0,1]$, no ...
0
votes
0answers
41 views
+50

Application of Fubini's theorem (in a proof of energy minimizing harmonic maps)

Let $u\in H^1(B_1,S^k)$, where $B_1$ is the open unit ball in $\mathbb{R}^n$ and $S^k$ is the unit sphere in $\mathbb{R}^{k+1}$. Suppose that $u$ is a minimizer for the Dirichlet energy functional $$ ...
3
votes
1answer
48 views

Mathematical Analysis: What is the Velocity of the Falling object through each point in its path?

I have been working through the book called "Mathematics" written by A.D. Aleksandrov, A.n. Kolmogorov and M.A. Lavrent'ev recently and have had some difficulty with understanding Examples given by ...