Questions tagged [analysis]

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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1answer
12 views

Preimage of multivariable function is convex

Let $\phi:\mathbb R^n\to \mathbb R\in C^2(\mathbb{R^n})$, such that $D^2\phi$ is positive semi-definite and let $c \in\mathbb{R}$ be arbitrary. Show that $\phi^{-1}((-\infty,c])$ is convex. Hint: ...
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0answers
35 views

Mathematical analysis question [closed]

Could someone help solve this problem please Solve $f(x,p) < g(y,p)$ where $$ \begin{split} f(x,p) &= 2 \exp\left(-\frac{xp}{2}\right)\\ g(y,p) &=\left[\frac{yp +2}{2}\right] \cdot \exp(...
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0answers
22 views

Union of images under arbitrary linear maps with boundness assumptions

Let $\mathcal{A}\subset B(\mathbb{R}^n,\mathbb{R}^n)$ be a bounded family of linear maps (matrices). Now let $X_0 = \{ x\in\mathbb{R}^n ~\vert~ \lVert x \rVert \leq 1 \}$, where $\lVert \cdot \rVert$ ...
1
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1answer
27 views

Weak convergence for $L^2$ on the torus

It is well-known that if we consider, for example, $L^2(\mathbb{R})$, we can pick any function $f\in L^2(\mathbb{R})$ with norm $\Vert f\Vert_{L^2}=1$, and then the sequence $$ f_n:=f(x-n) $$ will ...
1
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1answer
19 views

Find the completion of the metric space.

Let $\ell^1(\mathbb{N})=$ $\{(a_n): a_n ∈ \mathbb{R} , \sum |a_n| \lt \infty \}$, and define the distance function $d, d_* : \ell^1(\mathbb{N})\times\ell^1(\mathbb{N}) → [0, \infty)$ by $d(a,b)=\...
1
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1answer
28 views

Set of orthogonal matrices is a compact manifold

The following is a problem from Munkres's Analysis in Manifolds. Problem: Let $\mathcal O(3)$ denote the set of all orthogonal 3 by 3 matrices, considered as a subspace of $\mathbf R^9$. (a) ...
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0answers
39 views

Conneceted metric space

First of all my english is not the best, so sorry for that. I have math task: For a connected metric space $(M,d)$ there is no steady, non constant function $f:M \to \{ 0,1 \}$, when $\{ 0,1 \}$ the ...
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0answers
18 views

Total order in quotient set of Cauchy sequence?

I have been studying the construction of real numbers through Cauchy sequences in my analysis class and it defines $\Bbb R :[(x_{n})]/\sim$ where $(x_{n})_{n\in \Bbb N}$ is a $\Bbb Q $-Cauchy sequence ...
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2answers
50 views

Show that $f$ is not differentiable at $(0,0)$, despite being differentiable in every direction $v\in\textbf{R}^{2}$ at $(0,0)$.

Let $f:\textbf{R}^{2}\to\textbf{R}$ be the function defined by $f(x,y) := \frac{x^{3}}{x^{2}+y^{2}}$ when $(x,y)\neq(0,0)$, and $f(0,0) = 0$. Show that $f$ is not differentiable at $(0,0)$, despite ...
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1answer
62 views

Theorem of residue [closed]

Use the residue theorem to calculate $$\int_\gamma\frac{2z^2-5z+2}{\sin z}\,\mathrm{d}z$$ where $\gamma$ is the unit circle When i calculate residue in 0, i get the case 0/0. I made something wrong?
2
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2answers
38 views

Show that this function is decreasing

I have to show that the function $f(x)=(\ln x)^2(1-\ e^{-\frac{t}{x}}), t>3$ is decreasing on $[\max(e^4,2t),\infty[$ and deduce that $f(x)\leq \max(16,(\log2t)^2)$, $x\geq 1$. The exercise ...
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0answers
27 views

Cauchy sequences of Cauchy sequences are Cauchy sequences

In 'Analysis' by Elliott H. Lieb and Michael Loss, Theorem 2.13 is the Mazur's lemma for $L^{p}(\Omega)$ ($1<p< \infty$). Let $f^{j}$ be a sequence in $L^{p}(\Omega)$ that converges weakly to $F$...
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1answer
49 views

Lower bound on ODE (Gronwall)

Consider a well-defined first-order ODE $$ \frac{du(t)}{dt} = F(t, u) + G(t), $$ which has a unique continuous solution on $t\in[0,\infty)$ Suppose that The solution $u(t)>0$, and $u(0)>0$. $...
2
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4answers
79 views

Why are picewise continuous functions on $[a,b]$ bounded?

Consider a picewise continuous function $f:[a,b] \to \mathbb{R}$, i.e there are $a=t_0<\dots <t_n=b$ such that $f$ is continuous on each open interval $(t_i,t_{i+1})$ and the limits $\lim\...
0
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1answer
41 views

How to prove inequality $(2n-1)x^{-1} + (2n+1)x^{2n-1} > \frac{(2n+1)^2}{2n}, \quad n > 2$ and $ 0 < x < 1$

How to prove the following inequality for $n > 2$ and $ 0 < x < 1$: $$(2n-1)x^{-1} + (2n+1)x^{2n-1} > \frac{(2n+1)^2}{2n}, \quad n \in \mathbb{N}, x \in \mathbb{R}$$ I tried using ...
2
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2answers
63 views

Solid torus is a 3-manifold.

I'm working on a problem from Munkres' Analysis on Manifolds, where I must show that a solid torus is a 3-manifold, and that the boundary of this manifold is the torus $T$. Letting $g$ be the ...
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1answer
50 views

How calculate ${\lim}_{n→∞}\int_0^{\pi/2}{7^{-2^n\cos^nx}dx}$ [closed]

How calculate it? $$\lim_\limits{n\to\infty}\int_0^{\pi/2}{7^{-2^n\cos^nx}dx}$$
1
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1answer
25 views

Unique Solution to 1st Order Autonomous ODE

Take the ODE $y'=F(y)$. Show it has a unique solution with initial condition $y(t_0) = y_0$ in a neighborhood of $t_0$ provided $F$ in continuous and $F(y_0) \neq 0$. I am trying to use the inverse ...
2
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3answers
51 views

Continuous and Integration [closed]

Question: Suppose $f,g\colon [a,b]\to\mathbb{R}$ are continuous and $\displaystyle\int_a^bf(x)dx=\displaystyle\int_a^bg(x)dx$. Prove that there is $c\in[a,b]$ such that $f(c)=g(c)$. If we let $T(x)=f(...
0
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0answers
18 views

Rationals, irrational and dyadic numbers with archimedean propierty

i need help to test the following: “prove that every open interval in real numbers contains rational, irrational and dyadic numbers.” I had tried defining an isometry on a set where the property ...
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0answers
18 views

Existences partial and directional derivatives

If $f:\mathbb{R}^n \to \mathbb{R}^m$, there is a partial derivative of $f$ in $a$ ($D_if_a$) if and only if there is a directional derivative in direction $e_i$, and also $f^{'}_{e_i}(a)=\frac{\...
0
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1answer
15 views

Best approximation of a function among closed linear manifolds

Let $H$ be an infinite-dimensional Hilbert space and consider a $n-dimensional$ closed linear manifold generated by a subset of orthonormal basis, say, $M = span(\{u_1,u_2,\cdots,u_n\})$. Of course, ...
0
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1answer
30 views

if $ \limsup_n x_n\leq y $then there exists a subsequence $\{x_{n_i}\}_i$ such that, for all $i\in\mathbb{N}$: $ x_{n_i}\leq y $? [closed]

Let $\{x_n\}_n$ be a real sequence and $y\in\mathbb{R}$ such that: $$ \limsup_n x_n\leq y $$ can we say that there exists a subsequence $\{x_{n_i}\}_i$ such that, for all $i\in\mathbb{N}$: $$ x_{n_i}\...
-1
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1answer
17 views

Accumulation points of a sequence in a metric space

I am working with sequences in metric spaces and I think the following may happen. Let $(X,d)$ a complete and separable metric space, if $x \in X'$, we know that exist a sequence $(x_n)_{n \in \...
0
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1answer
33 views

Can we say that: $f(\beta)=\alpha$?

Let $X$ be a Hausdorff topological vector space and $f:X\to \mathbb{R}$ be an affine, sequentially continuous function. Let $\{x_n\}\subset X $ be a sequence such that: $$ \lim_{k}\frac{1}{k}\sum_{n=...
1
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1answer
44 views

Proving solution is defined and $C^{\infty}$

Consider the harmonic pendulum problem with $l=1$, $\ddot{x}=-g \sin(x)$ with $x(t_0) = x_0$; $x'(t_0)=v_0$. Show that the solution $\varphi(t,t_0,x_0,v_0,g)$ is defined on all $\mathbb{R}^5$ and is ...
0
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1answer
34 views

Prove that $\int_{B^2(0,1)}||D(\Phi)||=\int_{\Phi(B^2(0,1))}||D(\Phi^{-1})||$

Let $\Phi:\Bbb R^2 \to \Phi(\Bbb R^2)\subseteq \Bbb R^2$ be a diffeomorphism. Prove that $\int_{B^2(0,1)}||D(\Phi)||=\int_{\Phi(B^2(0,1))}||D(\Phi^{-1})||$, where $||A||=(\sum_{i,j=1}^2a_{ij}^2)^{1/...
1
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0answers
48 views

Unique Solution for Square-root function on Matrices

Prove that for every n by n matrix M sufficiently close to the identity matrix there exists a square-root matrix (solution of $A^2 = M$) and the solution is unique if $A$ is required to be ...
2
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0answers
41 views

Harmonic Pendulum

Let the equation of the harmonic pendulum given by $$x''=-g\sin{(x)} \qquad x(t_0)=x_0 \qquad x'(t_0)=v_0$$ Is the solution $\varphi(t,t_0,x_0,v_0,g)$ is defined in all $\mathbb{R}^5$? If so, how ...
0
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1answer
27 views

Locally Lipschitz? Globally Lipschitz

Does this function $$f(x,y) = -\frac{2xy}{(\frac{3}{2}\sqrt{|y|}+1+x^2)} $$ Satisfy the local Lipshitz condition $$|f(x,y_2)-f(x,y_1)| \leq M|y_2-y_1| $$ for $x,y \in \mathbb{R}$ I have no idea how ...
0
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1answer
55 views

$\int_0^1 \exp(f(x))\,dx \geq \exp(\int_0^1 f(x)\,dx$

As in the title with a bit more detail, assume $f$ is Riemann Integrable from $[0,1] \rightarrow \mathbb R$, prove that $\int_0^1 \exp(f(x))dx \geq \exp(\int_0^1 f(x)\,dx$, I did a bit of research ...
0
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0answers
29 views

The boundary of a disc has zero measure

Consider that a subset $A$ of $\mathbb{R}^{n}$ has measure $0$ if for every $\epsilon > 0$, there is a cover $\{U_{1}, U_{2}, \cdots \}$ of $A$ by closed (or open) rectangles such that $$\sum_{i = ...
0
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2answers
26 views

How do I calculate the roots of this complex polynomial? [closed]

$$z^4-5z^3+3z^2+19z-30=0$$ I'm not sure how I should proceed.
1
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1answer
32 views

Roots of trancendental equations and their relation to Dirichlet series and Mellin transforms

Consider a function of the form $F(x)=x^{\alpha}f(\ln(x))$, with $0<\alpha<1$ and $c_1<f(\ln(x))<c_2$ for some positive constants $c_1,c_2$, such that $F(x)$ is strictly increasing. ...
0
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2answers
23 views

Prove that if $f$ is differentiable and $f'$ is discontinuous at $c$, $\Rightarrow \nexists \lim_{x \to c^-} f' \lor \nexists \lim_{x \to c^+} f'$

I'm studying how I can limit the possible discontinuities of the derivative of a function and at a point I want to prove that if $I$ is an interval and $c \in int(I)$: $f : I \rightarrow \mathbb{R} \...
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0answers
27 views

Lebesgue measurable set for a function

Let $E⊂[0,∞).$ Show the following for the function $f(x)=x^{2}$: The requirement and sufficient condition for $m^*(E)=0$ is $m^*(f(E))=0.$ (Guiding: $E_n=E∩(0,n)$ use the set of sets.) Depicts the ...
0
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1answer
30 views

Bounded analytic in a region is constant

Let $f:G\to\mathbb{C}$ is an analytic function where $G$ is a region in $\mathbb{C}$ such that $|f(z)|\leq|f(\alpha)|$ for some $\alpha\in G$. Then $f$ is a constant function. Can somebody help how ...
0
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0answers
5 views

A integrable relation related to the same order function

When I read book I meet the following statement: As $\frac{\phi(t)}{2 \sin\frac{t}{2}}\sim\frac{\phi(t)}{t}$, so since the $\frac{\phi(t)}{t}$ is integrable or absolutely integrable, the $\frac{\...
0
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2answers
34 views

Can the set of natural numbers have a injection with a finite set? [closed]

My intuition says that is not possible, but i am not achieving a great formal proof
2
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0answers
28 views

Finding adjoint operator

Let $X, Y$ be compact Hausdorff spaces. Let $\tau:Y\to X$ be continuous and $A \in \mathcal B(C(X),C(Y))$ the operator given by $Af:=f\circ \tau$. Show that the adjoint operator $A' \in \mathcal B(...
1
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3answers
26 views

Does $[−2, 3]\subset \operatorname{Im} f'$ for the defined function?

I'm trying to prove that if a function $$ f : [−1, 1] \rightarrow \mathbb{R}$$ is continuous in $[−1, 1],\phantom{2}$ differentiable in $(−1, 1)$ and verifies $$ f(−1) = 1,\phantom{1} f(0) = −1, \...
0
votes
1answer
21 views

Differentiation of geometric mean

I've just encountered the following problem. Define the following geometric mean $$Y_t = \left( \prod_{k \neq i} X^k_t \right)^{1/n} $$ where $i = 1,2,...,n$ and $t$ means time. My question is how to ...
1
vote
1answer
66 views

Uniqueness of O.D.E

Prove that the o.d.e. $(\frac{3}{2}\sqrt{|y|}+1+x^2)\frac{dy}{dx}+2xy=0$ has unique local solutions with $y(x_0) = y_0$ for any $x_0$ and $y_0$. Does the existence and uniqueness theorem for o.d.e's ...
3
votes
1answer
63 views

Evaluate the limit for $p<1$

I found this limit calculation problem in a book. For a real number $p\geq 0$ we have $$\lim_{n\rightarrow \infty}\frac{\left (1^{1^p}2^{2^p}\dots n^{n^p}\right )^{\frac{1}{n^{p+1}}}}{n^{\frac{1}{p+1}}...
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votes
1answer
27 views

Prove that there exists a unique $x^* \in X$ such that $T(x^* ) = x^*$ . [closed]

Suppose $(X, \rho)$ is a complete metric space, and suppose the function $T : (X, ρ) \rightarrow (X, ρ)$ is such that $T_n = T ◦ T ◦ · · · ◦ T$ (n times) is a contraction map for some $n \ge 2$. Prove ...
2
votes
2answers
37 views

Using definition of derivative at an inequality

The question is really simple but I'm not sure how can I prove it. Let $f : \mathbb{R} \rightarrow \mathbb{R} \phantom{2}$ a function that verifies : $\exists\, K \in \mathbb{R^+}, \phantom{1}\...
0
votes
0answers
9 views

What the summation of ordered pairs? [closed]

I am sorry if this sounds completely stupid to you, but I am having hard time conceptualize the summation of ordered pairs. Could anybody please tell me how to calculate the summation of ordered ...
0
votes
1answer
63 views

$\lim_{x\to\infty} (\sqrt{x+1} + \sqrt{x-1}-\sqrt{2x})$ using little o function

Problem has to be solved specifically using little o function. I was going to transform $\sqrt{x+1}$ into $1+\frac{1}{2}x+o(x)$ and $\sqrt{2x}$ into $1+\frac{1}{2}t+o(t)=1+\frac{1}{2}(2x-1)+o(2x)$ but ...
0
votes
3answers
30 views

How to prove the Riemann Zeta fuction tends to infinity when $x$ tends to $1$

The Riemann Zeta Function is convergent over the interval $(1,\infty)$, and $\sum_{n=1}^{\infty}\frac{1}{n^x}$ tends to infinity when $x\rightarrow 1^ {+} $, it seems one can feel it is right because ...
0
votes
1answer
41 views

Does there exist a continuous function from $[a,b]\to (-\infty,c]\cap[d,\infty)$ [closed]

Does there exist a continuous function from $[a,b]\to (-\infty,c]\cap[d,\infty)$, or $\mathbb{R}\to (-\infty,c]\cap[d,\infty)$ where $a,b,c,d\in\mathbb{R}$ and $c<d$? If it doesn't exist, how can ...

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