Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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62 views

$f$ unbounded in $\mathbb{R}$ implies it cannot be in $L^2(\mathbb{R})$.

Intuitively , I feel this must be true. I'm looking for a rigorous proof. So,I would like to confirm that there are no counter examples to begin with.
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1answer
42 views

Questionable Intervals

Find the numbers $X_1 , X_2 , \ldots , X_{10} $ such: $X_1$ is in the interval $[0,1]$. If we divide the interval $[0,1]$ in halves,each half consists of only one of $X_1$ or $X_2$. If we divide ...
5
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1answer
63 views

$\int_0^1f(x)dx = 2, \int_0^1g(x)dx = 1, \text{and} \int_0^1[f(x)]^2 dx ≤ C$ for some constant $C > 4.$

Suppose $f$ and $g$ are nonnegative measurable functions on the interval $[0,1],$ with the properties $$\int_0^1 f(x)\,dx = 2, \int_0^1g(x)\,dx = 1, \text{ and }\int_0^1[f(x)]^2 dx \le C$$ for some ...
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0answers
52 views

sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$

Let $x_n$ be a sequence of real numbers such that $x_n\in(0,1).$ Find the sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$ My answer is $\frac{(x_n)'}{(1-x_n)^2}.$ But the term $(x_n)'$ make me ...
0
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1answer
36 views

existence of a number in (0,1) for ln(1+1/n)

Show that for each $n\in \mathbb{N}$ there is a numer $\theta_{n}$ where $0<\theta_{n}<1$ such that $log(1+ \frac{1}{n}) = \frac{1}{n} - \frac{\theta_{n}}{2n^{2}}$
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1answer
124 views

if $\mid f(x)\mid<1$ then $\mid f'(x)\mid<1$

I am wondering, if $\mid f(x)\mid<1$ then $\mid f'(x)\mid<1$. Is this true? I can not find counter example.
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1answer
47 views

Taylor's Theorem Question

Assuming Taylor's Theorem holds: Given $$F(x)=f(b)-f(x)-\frac{f'(x)}{1!}(b-x)-\cdots-\frac{f^{(n-1)}(x)}{(n-1)!}(b-x)^{n-1}$$ Show that: $$F'(x)=\frac{-f^{(n)}(x)}{n!}(b-x)^{n-1}$$
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4answers
38 views

Increasing, continuous, concave downward function normalised between 0.5 and 1

What would be a good function which is increasing, continuous, concave downward with $$\lim_{x \to 0} f(x) = 0.5$$ and $$\lim_{x \to \infty} f(x) = 1.$$ It should be concave downward whose ...
1
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0answers
51 views

If $f_{n}\rightharpoonup \bar{f}$ and $f_{n}(x) \rightarrow f(x)$ pointwise a.e., then is $\bar{f} = f$ a.e.? [duplicate]

Suppose $f_{n}$ is a sequence of functions in $L^{p}(\mathbb{R}^{d})$ such that $\|f_{n}\|_{L^{p}} \leq 1$ for all $n$ and $f_{n}(x) \rightarrow f(x)$ pointwise almost everywhere as $n \rightarrow ...
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3answers
100 views

Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$

Let $E ⊂ \mathbb{R}$ be a compact (i.e., closed bounded) set of real numbers. Suppose $\{f_n\}$ is a sequence of real-valued continuous functions which converges pointwise on $E$ to a function $f$ ...
1
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1answer
38 views

Finding rational points at rational distance in the plane

Take any point $p$ in the real plane. Does there always exist a rational point at a rational distance from $p$? (A rational point is a point $(q,r)$ where $q$ and $r$ are rational.)
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2answers
26 views

Show that $\int_X gdν=\int_X gfdμ$ for all $g∈L_1(ν).$

Let $μ$ and $ν$ be finite (positive) measures on a measurable space $(X, M),$ and suppose that $ν(E)=\int_E fdμ$, for all $E∈M,$ $E$ where $f$ is some function in $L_1(μ).$ Show that $\int_X ...
0
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1answer
24 views

Lipschitz function proof

Statement Let $F(t,X)=A(t)X+b(t)$ with $A(t) \in \mathbb R^{n\times n}$ and $b(t) \in \mathbb R^n$. If the coefficients $a_{ij}(t)$ and $b_i(t)$ are continuous functions of the variable $t$ in a ...
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0answers
12 views

Estimate of Projection Operator on two-torus

Let $\Lambda$ be a lattice, $\mathbb{T}=\mathbb{R}^2/\Lambda$ be a flat torus and $\Delta$ be the Laplace-Beltrami operator. There is any reference where the norm of the projection operator ...
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3answers
36 views

Computation of surfaces areas of some objects

I want to calculate the surface area of the following objects: 1) A cylinder with height $h$ and radius $r$ 2) A cone $C=\{(x,y,z) \in \mathbb R^3 : x^2+y^2=z^2, 0<z<4\}$ 3) A torus At first ...
2
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1answer
58 views

Show solvability of ODE without explicitly calculating solution

Show that $$ u + u^{(4)} - u^{(2)} = f $$ has a solution $u \in H^4(\mathbb R)$ (without explicitly calculuting it) for every $f \in L^2(\mathbb R)$! What criteria for solvability for such ODE's ...
4
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1answer
118 views

How to integrate these integrals? $\int{\frac{1}{(x^x-x^{-x})}} dx$ [closed]

Problem : $$\int{\frac{1}{(x^x-x^{-x})}} dx$$ I need answer about the following problem. Please help . I will be grateful to you. Thanks.
3
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1answer
25 views

Triangle inequality and homomorphisms

Here is my situation: I have two homomorphisms $f$ and $g$ from a group $A$ into the complex numbers $\mathbb{C}$. I know that they are 'close' on a subset $B \subseteq A$. More formally there is an ...
4
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1answer
33 views

Locally integrable function with a uniform bound…

I'm a bit lost... I have a measure space $(\Omega,\mathcal{B}(\Omega),\mu)$ where $\mathcal{B}(\Omega)$ is a Borel set. Let $f$ be a real-valued measurable function on $\Omega$ and $\mathcal{K}$ be ...
1
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1answer
24 views

Difference in Definitions of Quasiconvexity

So I've seen a few different definitions of quasiconvexity of a function and I cannot, after a bit of working, figure out how all of them are related: Let $X$ be a convex subset of a real vector ...
1
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1answer
37 views

nth term derivative with f(0) plugged in

Question: compute the sixth derivative of $\frac{\cos{(5x^2)}-1}{x^2}$ and plug in zero to the derivative. What is the answer? I keep concluding that this question should be 892857 when I plug in 0 ...
4
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1answer
69 views

Riemann Sums as in Königsberger Analysis 1

Intro: I must take a small detour here which is only relevant if you do not know the book itself and care about my background. I am working with Königsberger Analysis I (can be found on Springerlink). ...
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1answer
34 views

Clarification about a basic proposition about measurable functions

I am making my way through "Linear Functional Analysis" by Bryan P.Rynne and Martin A.Youngson (second edition). Given a measure space $(X,\Sigma ,\mu )$ we define a function $f$ to be measurable if ...
2
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2answers
159 views

How to calculate $\int_{-\infty}^\infty\frac{x^2+2x}{x^4+x^2+1}dx$?

I want to calculate the following integral: $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\frac{x^2+2x}{x^4+x^2+1}}_{=:f(x)}dx$$ Of course, I could try to determine $\int f(x)\;dx$ in terms of ...
5
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1answer
70 views

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

It is well-known that, for $f,g \in L^{1}(\mathbb R).$ Then, by Fubini's theorem, one can derive, $\widehat{f\ast g} = \hat{f} \cdot \hat{g},$ (that is, Fourier transform takes, convolution to point ...
1
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1answer
32 views

Weakly * continuous definition

What does it mean that $$t\to u(t,\cdot)$$ is waakly* continuous from $[0,T]$ to $L^\infty(\mathbb{R}^d)$? I guess that I have to see $L^\infty(\mathbb{R}^d)$ as the dual of $L^1(\mathbb{R}^d)$ and ...
2
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1answer
16 views

lower bound of modified Besselfunction

i'm looking for an lower bound for the modified Bessel function of the first kind $I_\nu(x)$ of a +ive real argument. There should be one of the form $$ce^{x^\alpha} \le I_\nu(x)$$.
2
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1answer
70 views

Positive and négative Parts

we denote by $u^+=\max(u,0)$ and $u^-=\max(-u,0)$ the positive and the negative parts of $u$ we have that $u=u^+-u^-$ my question is : what is $u'$ using $u^+$ and $u^-$ ? and what is ...
0
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1answer
25 views

Formula for $L^{q}$ norm using $C_{c}^{\infty}$ functions

We put, $L^{p}=L^{p}(\mathbb R), L^{q}=L^{q}(\mathbb R);$ $\frac{1}{p}+\frac{1}{q}=1;$ ($p$ and $q$ are conjugate exponents); and $<f,g> =\int_{\mathbb R} f(x)g(x) dx.$ Fix $g\in L^{q}, ...
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0answers
26 views

Conjunctive Normal Form representation/ First Order Logic.

in my research problem, I need to represent three types of three types of relationships between the variables x,y as the following:: " y Cooperates with x" relationship: means if there is two ...
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1answer
39 views

Converge of an inversion to a mirrorring

I want to ask something about a mirroring and a inversion in $\mathbb{R}^n$. An inversion in a sphere with center $m$ and radius $\rho$ can be written as $$ v \ \longmapsto \ ...
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0answers
29 views

How to establish the equivalence of these two statements about integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a, b]$. Then we have $$ \int_{ka}^{kb} s\left(\frac{x}{k}\right) \ dx = k \int_a^b s(x) \ dx $$ for every $k ...
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1answer
31 views

How to establish this equivalence for integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a,b]$. Then we have $$\int_{a}^{b} s(x) \ dx = \int_{a+c}^{b+c} s(x-c) \ dx.$$ Second Statement: Let $s$ be ...
0
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1answer
40 views

Example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable?

Can anyone come up with an example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable? Thanks.
3
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2answers
64 views

integration of a series in $x\in(0,1)$

Prove that if $p>0$, $$ \int_0^1 \frac{x^{p-1}}{1-x}\log\left(\frac{1}{x}\right)dx=\frac{1}{p^2}+\frac{1}{(p+1)^2}+\dots$$
1
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0answers
52 views

Question on integral

I need a confirmation and answer about the following problem: If we have $ g(x)=\ln x+{ x }^{ -1/2 }{ 1 }_{ x\le 1 } $ I'm trying to determine $ \int _{ 0 }^{ +\infty }{ \ln x+{ x }^{ -1/2 }{ 1 }_{ ...
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1answer
20 views

Question about convergence in $H^1_0$

Please how to prove that if $u_n\rightarrow u$ on $H^1_0$ we have that $||u_n||\rightarrow ||u||$ ? Please i need your help Thank you
1
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1answer
36 views

$\|g\|_{L^{1}(\mathbb R)}=\sup \{ {|\int_{\mathbb R} fg|: f\in C_{c}^{\infty}(\mathbb R), \|f\|_{L^{\infty}(\mathbb R)}=1\}} ?$

I learn the following from the book: Fact: If $g\in L^{1}(\mathbb R),$ then $$\|g\|_{L^{1}(\mathbb R)}=\sup \{ {|\int_{\mathbb R} fg|: f\in L^{\infty}(\mathbb R), \|f\|_{L^{\infty}}}=1\}.$$ We put ...
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1answer
55 views

Density and convergence

I have a small question: Is it true that if the basis of a space $A$ is dense in a space $B$ ($B\subset A$) then if $u_n\rightarrow u$ in $A$ we have that $u_n\rightarrow u$ in $B$ ?
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1answer
33 views

Small question about convergence

I have a small question: if i have that $$\int_0^{+\infty}p(t)|u'_n(t)-u'(t)|^2dt\rightarrow 0$$ is it true that $$\int_0^{+\infty} p(t)|u'_n(t)|^2 dt\rightarrow \int_0^{+\infty} p(t)|u'(t)|^2 dt $$ ...
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0answers
50 views

Proof that a set is open.

Let $(\Lambda_i)_{i\in I}$ a collection of linear operators from $X$ (Banach space) to $Y$ (Normed space). Let $\alpha : X \rightarrow [0,\infty]$ be the function $\alpha(x):=\sup_{i \in I} ...
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2answers
48 views

Can any measure be made into a bounded measure?

Is it possible to derive a bounded measure from any measure on a measure space? For example can the Lebesgue measure be made into a probability measure?
3
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2answers
69 views

How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields ...
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0answers
23 views

Question about convergence

If i have that $$\int_0^{+\infty} a(t)|u_n(t)-u(t)|^2 dt \rightarrow 0 $$ how we can deduce that $$\int_0^{+\infty} a(t)|~|u_n(t)|-|u(t)|~|^2 dt \rightarrow 0 $$ where $a>0, a\in ...
2
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1answer
86 views

Uniform convex space

Please I want to know if this space $$H^1_{0,p}([0,+\infty))=\lbrace u, u\in AC([0,+\infty)), u(0)=u(+\infty)=0,\sqrt{p}u'\in L^2\rbrace$$ where $p>0$, $p\in L^1((0,+\infty))$ ...
0
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1answer
43 views

Definition of weak divergence [closed]

Can anyone give me the definition of the divergence of a vector field in the distributional sense?
2
votes
1answer
29 views

How to proof the limit is convergent for arbitrary initial state?

$\{a_n\},\{b_n\},\{c_n\},\{d_n\}$ is series. And $d_n=c_{n-1},c_n=b_{n-1},b_n=a_{n-1},a_n=b_{n-1}+c_{n-1}$ how to proof for any $a_0,b_0,c_0,d_0$ belong to $Z^+$, $\lim\limits_{n\rightarrow \infty} ...
0
votes
0answers
21 views

Fuel efficiency word problem

I am writting a paper for school and came across this fact, A single Boeing 747 spends an avergae of 32 minutes per flight taxiing, departing, and landing. During this time, it produces 87 kilograms ...
1
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1answer
39 views

Find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$

How can I find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$. I've tried derivating it but didn't reach any result.
3
votes
0answers
41 views

extending the convergence of measures

Let $F$ the real vector space of all applications $\phi: X \times Y \rightarrow \mathbb{R}$ where $(X,\mathcal{B}_1, \mu)$, $(Y,\mathcal{B}_2 )$ measurable spaces with $X$ and $Y$ are compact metric ...