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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0
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1answer
34 views

Local extremes of $f(x) = (x-2)^{\frac{1}{5}}(x-7)^{\frac{1}{9}}$

The task is to find local extremes of $f: \mathbb R \to \mathbb R$, $f(x) = (x-2)^{\frac{1}{5}}(x-7)^{\frac{1}{9}}$ There is theorem that if $x_{0}$ is local extreme of $f(x)$ then $f'(x_0) = 0$ So ...
1
vote
1answer
40 views

Difficulties of convergence of the partial sums of the Fourier inversion formula

Define the Fourier inversion formula over $\mathbb{R}^n$ by $$ f(x)=\int_{\mathbb{R}^n}e^{2\pi ix\cdot\xi}\hat{f}d\xi $$ where $\hat{f}$ is the Fourier transform over $\mathbb{R}^n$ $$ ...
3
votes
2answers
122 views

Is $f(x) = x^3 \sin \frac{1}{x} $ uniformly continuous on $(0, \infty)$?

Since the derivitive of $f$ is bounded on a neighborhood of $0$, $f$ is uniformly continuous on $(0, M)$ where $M$ is any positive number. I'd like to prove that $f$ is uniformly continuous on a ...
0
votes
1answer
24 views

Series of positive-definite kernels

Suppose I have a positive definite, shift invariant kernel $k_1(x-y)=k_1(\delta)$. I want to know whether the sum (where $a_n\geq 0$) $$ k(\delta) = \sum_{n=1}^{\infty} a_n k_1(n\delta)\tag{*} $$ is ...
2
votes
1answer
57 views

Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.

Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.(Take into consideration metric $d_2...$) I was ...
7
votes
2answers
173 views

Does anyone have a proof that the intersection and union of two compact sets is compact.

I have my take on it. It is quite informal and don;t know where it would be evaluated correctly on an exam. Since the sets are compact that means for every open cover there is a finite cover. When ...
1
vote
1answer
37 views

Examine the uniform convergence of the series $\sum^{\infty}_{n=1}\frac{1}{\sqrt{x+n}}$ if $x \in [0, \infty]$

Examine the uniform convergence of the series $$\sum^{\infty}_{n=1}\frac{1}{\sqrt{x+n}}$$ if $x \in [0, \infty)$ Which series should I choose in Weierstrass M-test to show that is divergent? ...
4
votes
1answer
31 views

What are the maps of these closed sets in $\mathbb R^3 \mapsto \mathbb R$

What is the map of an elipsoid(closed) if $H(x,y,z)=x+y+z$ This is kind of an tricky question, because I am not sure precisely what the answer is. I think when it says closed elipsoid it can be ...
1
vote
0answers
30 views

Proof of Gruss inequality

I've been reading articles that use the Gruss inequality for some time now, but I can't seem to find a proof of it anywhere. The only source I could find that actually has the proof is the original ...
0
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0answers
15 views

For what does the formula $(\prod_{t=1}^d[\begin{array}{c}-\frac 12&1&-\frac 12\end{array}]_{l_t,i_t})f$ stand for?

Let $f:\mathbb R\to\mathbb R$ and $$a_{l,i}:=f(x_{l,i})-\frac{f(x_{l-1,(i-1)/2}+f(x_{l-1,(i+1)/2})}2$$ for some $x_{l,i}$. I've read, that we can write $a_{l,i}$ in the following "operator form": ...
2
votes
1answer
57 views

What Does the Term “Regularity” Mean?

When I was an undergraduate, I took a course on regularity theory for nonlinear elliptic systems. This included topics such as the direct method of calculus of variations, mollifiers, integration over ...
1
vote
1answer
35 views

If a compact subset is contained in an open subset in $\mathbb{R}^n$, is a small cylinder of this compact subset also contained in the open set?

Let $O\subseteq \mathbb{R}^n$ be an open set, $K\subseteq \mathbb{R}^{n-1}$ a compact set and $a\in \mathbb{R}$, such that $$\{a\}\times K\subseteq O$$ holds. Does there exist an $\epsilon>0$, ...
-1
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0answers
33 views

Prove the norm axioms

Prove the norm axioms for example 7 . Thank you. :)
0
votes
1answer
14 views

Prove that $\left|\frac{x^p-1}{p}\right|\leq x+|\ln(x)|$ for all $x\in(0,\infty)$ and for all $p\in(0,1)$

So far I have shown that $$\displaystyle \lim\limits_{p\to 0^+}\frac{x^p-1}{p}=\lim\limits_{p\to 0^+}\frac{e^{p\ln(x)}-1}{p}=\ln(x)\lim\limits_{p\to 0^+}\frac{e^{p\ln(x)}-1}{p\ln(x)} =$$ (L'Hopital) ...
2
votes
2answers
63 views

$\mu(A \cap I) \le a \mu(I)$ implies $\mu(A) = 0$?

Let $\mu$ be lebesgue measure on $\mathbb{R}$, $0<a<1$. If $\mu(A \cap I) \le a \mu(I)$ holds for any interval $I$, can I say $\mu(A)=0$? I tried to construct a counterexample by considering ...
18
votes
4answers
176 views

How to Prove : $\frac{2}{(n+2)!}\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^{n+2}=\frac{n(3n+1)}{12}$

While I calculate an integral $$ \int\limits_{[0,1]^n}\cdots\int(x_1+\cdots+x_n)^2\mathrm dx_1\cdots\mathrm dx_n $$ I used two different methods and got two answers. I am sure it's equivalent, but ...
-1
votes
1answer
29 views

Prove that in norm linear space [closed]

Prove that in norm linear space the condition $||x||=1$ and $||x-y|| < \epsilon < 1$ imply that $||x-(y/||y||)||<2\epsilon$
1
vote
2answers
52 views

show that $F''$ is strictly increasing.

If $f$ is continuous and always positive in $[0,\infty)$ and $$F(x)=\frac{1}{2}\int_0^x(x-t)^2f(t)\,dt$$then show that $F''$ is strictly increasing. I found that the integrand is continuous and ...
-2
votes
0answers
25 views

prove the norm axioms for Lebesgue integral functions [closed]

prove the norm axioms for this example. ||x||=0 iff x=0. thank you
0
votes
1answer
26 views

Do convergence a.e. + limit function being in $L^p$ imply $L^p$ convergence?

Suppose $f_n\in L^p$ such that $f_n \to f$ almost everywhere. If we further know $f \in L^p$, can we say that $f_n \to f$ in $L^p$ norm?
0
votes
0answers
75 views

Can you explain to me why we need Limits at all? [closed]

I'm just studying about limits and I don't understand it so well because I don't get the idea of it. 1. How come there is negative infinity and also positive infinity? How can infinity help in math at ...
0
votes
0answers
28 views

De Rham Theorem for other coefficients

Are there versions of De Rham's theorem for which the coefficients are not just $\mathbb{R}$? In particular, is there a vector-valued version of De Rham's theorem?
1
vote
2answers
27 views

Relations between upper and lower Riemann sum

Let , $f$ , $g$ , $h$ be bounded functions on the closed interval $[a,b]$ such that $f(x)\le g(x)\le h(x)$ for all $x\in [a,b]$. Let , $P=\{a=a_0<a_1<a_2<\cdots <a_n=b\}$ be a partition ...
8
votes
0answers
64 views

Question on complete metric spaces and whether the following is a complete metric space:

Let $ S \subset C^2([0,1])$(set of all two-times differentiable functions on $[0,1]$), which satisfy $$f(0)+f(\frac{1}{2})+f(1)=0.$$ Question :Is $ (S,d)$ is a complete metric space, where $d$ is ...
4
votes
1answer
102 views

Changing the values of an integrable function $f:[a,b] \to \mathbb R$ countably infinitely many points not a dense subset of $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a function Riemann integrable over $[a,b]$ . It is known that if we change its values at finitely many points of $[a,b]$ , then the changed function still remains ...
2
votes
1answer
61 views

Changing the values of a function $f:[a,b] \to \mathbb R$ of bounded variation for countably infinitely many points not a dense subset of $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a function of bounded variation. It is known that if we change its values at finitely many points of $[a,b]$, then the changed function still remains of bounded ...
3
votes
0answers
58 views

A integral inequality

Let $g\in C_0^\infty((-1,1))$.Prove $\forall t\in (-1,1)$,$${g^4}\left( t \right) \le 16\int_{ - 1}^1 {\left( {{{\left| {g'\left( s \right)} \right|}^2} - \frac{{{g^2}\left( s \right)}}{{4{{\left( ...
1
vote
1answer
37 views

The behavior of $f(x)=\alpha x+x^2\operatorname{sin}1/x$ for $x\neq 0$ near $0$, where $\alpha \ge 1$.

Consider $\alpha \ge 1$. Let $f(x)=\alpha x+x^2\operatorname{sin}1/x$ for $x\neq 0$ and let $f(0)=0.$ In order to find the sign of $f'(x)$ when $\alpha \ge 1$ it is necessary to decide if ...
0
votes
2answers
105 views

If $f:X \to Y $ is continuous then $f^{-1}(\emptyset)= \emptyset?$

If $f:X \to Y $ is continuous then $f^{-1}(\emptyset)= \emptyset?$ (I know the inverse of a closed set of a continuous function is closed, but is this a must?) And does the following apply to all ...
1
vote
1answer
51 views

Whether a real number is a dyadic rational iff its binary expansion terminates?

In self-studying a textbook on computability theory, I found that many of the exercises depend on the following factlet: A dyadic rational is a rational number whose denominator is a power of two, ...
4
votes
1answer
94 views

Find a formula for $f''$ in terms of $f$, where $f\gt 0$ and $(f')^2=f-\frac{1}{f^2}.$

Problem: Suppose that a function $f \gt 0$ has the property $$ (f')^2=f-\frac{1}{f^2} $$ Find a formula for $f''$ in terms of $f$. Hint: Use Theorem 7. Theorem 7: Suppose that $f$ is ...
-1
votes
1answer
22 views

$D\det_A$ exists and equals $D\det_A (H)=\det (A) \operatorname{trace} (A^{-1}H) $? [closed]

Consider the determinant function $\det : M_n(\mathbb R ) \to \mathbb R$ , the is it true that $D\det _{A}$ exists ? Does it exist if $A$ is assumed to be invertible also and at $H \in M_n(\mathbb R)$ ...
-1
votes
2answers
43 views

Evaluation on a basis of gaussian integral

Knowing that $$\int_{-\infty}^\infty e^{-x^2} dx= \pi^{\frac{1}{2}}$$ Find: $$\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{\frac{-x^2}{2}} dx$$ And my question is how does this help if have the ...
1
vote
1answer
26 views

How to show that this works (complex numbers)

So if I have a set of complex numbers: $A= \{z\in\mathbb{C} |\ \text{Re}\,(z) > 0, |z|<1\}$ So I have a problem showing this: For any $z\in A$ exists $w\in A$ such,that this works: ...
1
vote
1answer
36 views

Boundaries change in double integral

Calculate: $$\int_0^1 \int_0^{x^3} e^\frac{y}{x} dydx$$ Obviously i need to change it to $dxdy$ thus i need to change the boundaries of the second integral but how to do that in this case?
1
vote
2answers
23 views

Double integral on a compact subset

Calculate: $$\int \int _D \left(6x+2y^2 \right) dxdy$$ where D is a compact subset of $\mathbb{R}^2$ enclosed by a parabola $y=x^2$ and a line $x+y=2$. How to find that, how to find the limits of ...
1
vote
0answers
19 views

Generalized Hyperbolic and Circular Functions

I have recently posted a couple of questions in regards to Generalized Hyperbolic and Circular Functions and I was hoping to find a couple more papers available on the particular subject. The papers ...
2
votes
0answers
34 views

Rudin's Rank theorem

Rudin states the following: 9.32 Theorem: Suppose $m,n,$ are nonnegative integers, $m\geq r,n\geq r$, $F$ is a $C^1$ mapping of an open set $E\subset R^n$ into $R^m$, and $F'(x)$ has rank $r$ for ...
2
votes
3answers
285 views

Double integral problem: $\int_0^\pi\int_x^\pi \frac{\sin y}{y} dy\, dx$

Calculate: $$\int_0^\pi \int_x^\pi \frac{\sin y}{y} dydx$$ How to calculate that? This x is terribly confusing for me. I do not know how to deal with it properly.
3
votes
2answers
56 views

find smallest $x>0$ such that $\frac{A}{cx}e^{-cx^2}\le \varepsilon$

I was estimating some error and I got $$\varepsilon(x)\le\frac{A}{cx}e^{-cx^2}$$ $A,c$ are known and positive, $x$ is also positive. The bigger the $x$ smaller the error. But I need to find the ...
2
votes
0answers
41 views

Inequality problem involving log function

Given $|f(x+y)-f(x)-f(y)| \leq x+y$ for all $x > y > 0$, prove that real valued function $f$ satisfies the inequality $|\frac{f(x)}{x} - \frac{f(y)}{y}| \leq M(1+\log_2\frac{x}{y})$ where M is ...
4
votes
0answers
62 views

Assumption in PDE theory

I have an exercise in PDE theory. Let $w \in C^2(U)\cap C(\overline{U})$ where $U$ is open, bounded and connected and $c \in C(\overline{U},\mathbb{R})$ with $c(x) \le 0$ everywhere. Moreover, ...
0
votes
1answer
36 views

Bijective Conformal Mapping onto the Open Unit Disc $\mathbb{D}$

What is the explicit bijective conformal mapping $f(z):G_n\to\mathbb{D}$, $z\in\mathbb{C}$ for the following domain transformations: $G_1=\{x+iy~|~x>1/2,y>0\}$ is the open region of the first ...
1
vote
0answers
18 views

Range of values of $\frac {\epsilon}{\sup \delta}$ for the modified step function

Let the modified step function be defined on $[0,1]$ by : $f(x) = \begin{cases} \bigg( \dfrac {2^n+1}{2}\bigg )x - \dfrac {2^n-1}{2^n} ; & n \in \mathbb N~~ , \dfrac {2} {2^n+1} ...
1
vote
2answers
38 views

Polynomial must be monotone between its extrema

Suppose that the polynomial function $f(x)=x^n+a_{n-1}x^{n-1}+\cdots +a_0$ has $k_1$ local maximum points and $k_2$ local minimum points. Show that $k_2=k_1+1$ if $n$ is even, and $k_2=k_1$ if $n$ is ...
3
votes
2answers
47 views

What's the relationship between a measure space and a metric space?

Definition of Measurable Space: An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$. Definition of Measure: Let $(\Omega, F)$ ...
0
votes
1answer
21 views

Equation of the plane tangent to the given surface

Find the equation of a plane tangent to the surface given by $$xyz+x^2-3y^2+z^3=14$$ at $$P=\left( 5,-2,3 \right)$$ In my opinion answer is: $$4x+27y+25z-41=0$$ If not please tell me what am i doing ...
-7
votes
0answers
66 views

engineering math (curl) [closed]

A ferris wheel with motorised cabins,measuring 550 ft in height and 520 ft in diameter. a. What purpose is served by the cabin motors with respect to the orientation of the cabins and their ...
2
votes
2answers
29 views

Use the lagrange's multipliers method to find a points on an ellipse

Question: Using the Lagrange's Multipliers method, find the points on the ellipse $x^2+2y^2=1$, that are situated in the longest and shortest distance from the line $x+y=2$. I know how to use ...
0
votes
0answers
15 views

proof coordinate functions of integrable function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ integrable

If $$f(x)=f_1(x_1)\cdots f_n(x_n)$$ and $f$ is an integrable function from $\mathbb{R}^n$ to $\mathbb{R}$. Proof that $f_i(x_i)$, $i = 1, \ldots, n$ are integrable. With the Fubini theorem?