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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16 views

Finding a branch of the complex logarithmic function $\log(1-z).$

I have a question that asks me to find the holomorphic branch $L(1 − z)$ of $\log(1 − z)$ valid in the cut-plane $z \in \mathbb{C}\setminus [1, ∞)$ and such that $L(1) = 0.$ We have defined the ...
0
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0answers
19 views

Convergance of the average of a convergant complex sequence

So this is in exercise 3.14 (Neat!) in Baby Rudin, which I have found quite a simple and obvious proof for, however when I checked the answers the proofs I found were quite complicated so now I am a ...
1
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0answers
35 views

Almost found the limit points of this set

I want to find all limit points of the given set, and I think I almost got it. $M=\left\{\frac{x}{2^y} \mid x, y\in ℕ , x \leq y\right\}$ also $M \subset ℝ$. We say that: $\forall \delta > 0,\...
1
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2answers
43 views

Partial fraction decomposition of $\pi\cdot \tan(\pi z)$

Evaluate the partial fraction decomposition of $\pi \tan(\pi z)$ $$2\pi \tan(\pi z)=\cot\left(\frac{\pi}{2}-\pi z\right)-\cot\left(\frac{\pi}{2}+\pi z\right)$$ $$=\frac{2}{1-2z}+\sum_{k=1}^\infty \...
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0answers
24 views

How do I describe the analytic completion to the algebraic closure of $\mathbb{F}_2$?

Is it possible? I'm using the algebra generated by the set $\{0, 1^r\}$ for all fractions $r$ as my representation of the algebraic closure to $\mathbb{F}_2$. I can't seem to find a metric on it. ...
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0answers
33 views

Does $\min\{x_1x_3+x_2^2 | x_1^2+x_2^2+x_3^4 = 4\}$ has an optimal solution?

Does $\min\{x_1x_3+x_2^2 | x_1^2+x_2^2+x_3^4 = 4\}$ has an optimal solution? I think continuous function over closed and bounded domain has an optimal solution but I am not sure. Can anyone give me ...
0
votes
2answers
43 views

Finding all limit points of a set

How can I find all the limit points of this set? $S=\left\{\frac{x}{2^y} \mid x, y\in ℕ , x \leq y\right\}$ with $S \subset ℝ$. Could this be proved if I showed that $∀δ > 0, \exists z ∈ S\text{...
0
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1answer
42 views

Urysohn's extension theorem

Currently I am working my way through Ernest Michael's first article on continuous selections. Here, Urysohn's extension theorem is stated as follows: For a $T_1$-space, the following properties ...
1
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1answer
30 views

How to find all limit points of this set?

How to find all limit points of this set? $S=\left\{\frac{x}{y} \mid x, y\in ℕ , x \leq y\right\}$ also $S \subset ℝ$. Is the way to proof this done same way as it is for sequences? I have never ...
-8
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0answers
157 views

Show that map is norm preserving and determine Eigenvalues [duplicate]

Can someone of you give me a solution for this? Let $N\in \mathbb N$. a) We define the map $\mathfrak F:(\mathbb C^N, ||\cdot||_2)\to(\mathbb C^N, ||\cdot||_2)$ by $$(\mathfrak F(x))_k := \frac{...
0
votes
2answers
36 views

Is $f_n=x^n$ weakly convergent in $(\mathscr C[0,1],\lVert\cdot\rVert_\infty)$?

This is part of an old preliminary exam in Analysis I am working through. For earlier parts of the problem I have already shown that $f_n$ does not converge in $(\mathscr C[0,1],\lVert\cdot\rVert_\...
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0answers
16 views

Alexandrov Maximum Principle and $W^{2}_p$ estimates

I'm reading an article of N. V. Krylov: About an example of N. N. Ural'tseva and weak uniqueness for elliptic operators, Nonlinear partial differential equations and related topics, 131–144. This ...
1
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1answer
28 views

Prove that exists a linear continuous functional satisfying…

Let $E$ be a normed space over the field of real numbers. I have to prove that given two convex sets $A$, $B$ in $E$, with positive distance between then, there exists a linear continuous functional ...
2
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1answer
50 views

Convergent + divergent $\to$ divergent

Given sequences $(x_n)$, convergent, but $(y_n)$ is divergent, then $(x_n + y_n)$ is divergent. I am confident that it is true, but having trouble getting the formalities correct. I have tried proof ...
2
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1answer
53 views

Is it possible to construct such a function in analytical form?

Suppose $f\left(f\left(x\right)\right)=\sin(x)$ Is it possible to find $f$ in closed form, or any other forms so as to visualize $f(x)$ on $x\in[-\pi,\pi]$? Is it possible to prove the existence and ...
0
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1answer
31 views

Does an unbounded gradient implies a non-vanishing hessian determinant

Let $\Phi:\mathbb{R^{n-1}\to R}$ be a vector function. Suppose that exists a set $S_1\subset \mathbb R^{n-1}$ on which $G=\nabla\Phi$, the gradient of the function is unbounded. Does that imply ...
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4answers
58 views

The continuity of function's restrictions implies the continuity of function.

Let be $X \subset F_1 \cup F_2$, where $F_1$ and $F_2$ are closed. If the function $f\colon X \longrightarrow \mathbb{R}$ is such that $f|_{X \cap F_1}$ and $f|_{X \cap F_2}$ are continuous, so prove ...
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0answers
48 views

Convergence of $\sum a_n b_n$

In Rudin P.M.A The partial sums $A_n$ of $\sum a_n$ form a bounded sequence; i.e. $b_0\ge b_1\ge b_2\ge\cdots\ge b_n$ so that $\lim\limits_{n\rightarrow\infty}b_n=0$. Then $\sum a_n b_n$ converges. ...
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2answers
54 views

Equivalence relations and commutative diagrams

Let $\sim$ and $\dot\sim$ be equivanlence relations on the sets X and Y respectively. Suppose $f \in Y^X$ is such that $x \sim y$ implies $f(x) \dot\sim f(y)$ for all $x,y \in X$. Prove that there is ...
0
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1answer
69 views

Show: $(X,d_X)$ is complete $\Leftrightarrow $ $f(X)$ is closed in $(Y, d_Y)$ ($f: X \to Y$ is an isometric embedding)

I have the following task: Show that a metric Space $(X,d_X)$ is complete if and only if for every isometric embedding $f: X \to Y$ in another metric Space $(Y, d_Y)$ it holds true that $f(X)$ is ...
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1answer
19 views

Bounding an exponential integral

I'm having trouble seeing this bound I've seen on a proof. Let $f$ be a polynomial, and $F$ the polynomial obtained from $f$ by replacing each coefficient by its absolute value. Then: $$\bigg{|}\...
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1answer
19 views

What is $(A_1 \times … \times A_n) \cup(B_1 \times … \times B_n)=?$ ,$A_i$'s are intervals

What is $(A_1 \times ... \times A_n) \cup(B_1 \times ... \times B_n)=?$ ,$A_i$'s are intervals $[a_{Ai},b_{Ai}]$ and $B_i$'s are $[a_{Bi},b_{Bi}]$ respectively. What I mean is can $(A_1 \times ... \...
1
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0answers
24 views

Prob. 4, Sec. 4.3 in Kreyszig's functional fnalysis text: Application of the Hahn Banach Theorem

Let $X$ be a real or complex vector space, and let $p \colon X \to \mathbb{R}$ be a real-valued function satisfying $$p(x+y) \leq p(x) + p(y) \ \mbox{ for all } \ x, y \in X$$ and $$p(\alpha x) = \...
1
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1answer
50 views

Difference of functions inequality

In one book on complex analysis I see the following: But $f$ is continuous at the point $z$. Hence, for each positive number $\varepsilon$, a positive number $\delta$ exists such that $$\lvert f(...
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0answers
40 views

Proof about Dedekind Theorem

Following two pics are proof regarding Dedekind Theorem My question is: after prove there is a rational number r greater than $\beta$, we arrived at contradiction, why?
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1answer
40 views

If $\langle f(x),g(x)\rangle = 0$, and $f$ has periodic orbit, then $g(x)$ has equilibrium point

Let $x'=f(x), x'=g(x)$ be two ODE, with $f(x),g(x):\mathbb R^2\rightarrow\mathbb R^2$, such that $\langle f(x),g(x)\rangle =0$ for all $x\in\mathbb R^2$. If $f$ has a periodic orbit then $g$ has ...
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3answers
90 views

Isometric isomorphism between $R^2$ and $R$

Can someone help me solving the following problems? $(\mathbb R^2,d_2)$ and $(\mathbb R, d_1)$, $d_2, d_1$ being the respective euclidean norms, are not isometric isomorphic, i.e. there is no ...
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0answers
27 views

A problem involving Poincaré-Bendixson Theorem

This is the problem: Given the ODE $x'=f(x)$, suppose that $f:\mathbb R^2\rightarrow\mathbb R^2$, and $f$ satisfies the hipoteses of Poincaré-Bendixson Theorem, if exists $V:\mathbb R^2\rightarrow\...
0
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0answers
57 views

Spivak problem 26-4 (Calculus 3rd edition)

I am having difficulty understanding a problem requirements from Spivak's for Chapter : "Complex Functions". The problem descriptions is as follows: In this problem we will consider only ...
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0answers
22 views

How to prove a continuous function is uniformly continuous on a compact set using BW theorem?

Question. Let S be a compact set in $\mathbb R^n$ and $f:S\rightarrow \mathbb R^m$ be a continuous function. Prove that $f$ is uniformly continuous on S. I want to prove it using Bolzano-Weierstrass ...
2
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1answer
53 views

Functional Inequality given $s\cdot f(t)+t\cdot f(s)\leq 2$

Question: For all continuous $f:\mathbb R\to \mathbb R\ $ and $\forall s,t \in [0,1]$ that satisfies: $$s\cdot f(t)+t\cdot f(s) \le 2$$ a) Prove that: $$\int_0^1 f(x) \,dx \le {\pi/2}.$$ b) How many ...
0
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1answer
19 views

$d_1, d_2$ are metrics on $X$; $(X,d_1)$ is complete. Let $i:(X,d_1)\to(X,d_2)$ be continuous and $i^{-1}$ unif. cont. Show $(X,d_2)$ is complete.

This is a problem on an old preliminary exam in Analysis I'm working through. The problem initially looked easy to me; my plan is to show that for any $\{x_n\}$ Cauchy in $(X,d_2)$, we have that $\{i^{...
-4
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1answer
51 views

Hausdorff metric, Ultra metric

Can anyone prove the following statement though it seems simple. Let $(X,d)$ be an ultrametric space and $A$ and $B$ are closed, bounded subsets of $X$. Then for each $a$ in $A$ and $\varepsilon > ...
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1answer
46 views

What is the moment of inertia tensor of a hollow spheroid?

I am looking for exact or even approximate formulas for the moment of inertia of a hollow spheroid (oblate and prolate.) I have find formulae for a hollow spherical shell and for a filled ellipsoid ...
0
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1answer
34 views

Finding a base representation for a differential form that's given through the determinant of a matrix

Let $U \subseteq \mathbb{R}^n$ be open. I first want to show/acknowledge that the mapping $$\omega_p(v_1, ..., v_{n-1}) := det(p, v_1, ..., v_{n-1}), p \in U, v_i \in \mathbb{R}^n$$ defines a ...
1
vote
1answer
44 views

Succession in Peano axioms

In "Analysis I" - Herbert Amann states: "The natural numbers consist of a set $ N$ , a distinguished element $0\in N$ and a function $v:N\to N^*$ with the following properties: ($N0$) $v$ is ...
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0answers
27 views

Taylor expansions for functions of several variables

I need help with this question. a) Determine the Taylor expansions at the origin up to the square Terms of $f(x, y, z) = \cosh(x) - \sin(yz) - xy(z - 1)^7$ and $g(x, y) = e^{-y}/(1-x^2)$. b) ...
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1answer
23 views

What can be written instead of the supremum axioms to get same structure in the set of real numbers

Question goes as follows : "What can be written instead of the supremum axioms, to get the same structure in $\mathbb{R}$." Now for the supremum axioms I have : If $K_a \neq Ø \Rightarrow \exists$ a ...
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2answers
80 views

Why $I = [0,1]$ is a $1$-manifold and $I^2$ not?

I am stuck in this, I have no idea why! $[0,1]$ is a manifold with boundary, how to justify? Which are the charts? And how about $[0,1]^2?$ Why it is not a manifold? My definition of topological ...
2
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4answers
59 views

periodic continous function $f$

I have a function $f: \mathbb{R} \longrightarrow \mathbb{R}$ that is continous with $f(x)=f(x+2)$ for all real numbers $x \in \mathbb{R}$. So, I have to show that an $a \in \mathbb{R}$ exists that $f(...
5
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1answer
84 views

Prove there is a unique $y:[0,1] \to \mathbb{R}$ solving $y(x) = e^x + \frac{y(x^2)}{2}$ for $x \in [0,1].$

The title is the problem statement, but to reiterate, Prove there is a unique $y:[0,1] \to \mathbb{R}$ solving $y(x) = e^x + \frac{y(x^2)}{2}$ for $x \in [0,1].$ Looking for hints/solutions, thanks ...
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1answer
44 views

For which values of $\gamma$ does this inequality hold?

Edited: Just realised my first post was somewhat misleading and not precise. Thanks to the two commetators that pointed it out. I am working on an article and ended up wondering for which values of $\...
0
votes
2answers
48 views

Solving the equation $y+xy^2-e^{xy}=0$ to form $y=f(x)$.

I need help by this question. Problem I don't know where to start by solving such a questions. Can the equation $y+xy^2-e^{xy}=0$ in a neighbourhood of $(x_0,y_0)$ with $x_0=0$ and suitable $y_0$ be ...
0
votes
1answer
20 views

Cauchy Condensation Test - Proof

I have the following proof for the Cauchy condensation test in my lecture notes. I can understand all that has been carried out except for the conclusion. Can you help explain why the sequences are ...
0
votes
1answer
42 views

Suppose $f_n\to f$ in $L^1([0,1],\lambda)$. Prove or disprove: $\exists \{f_{n_j}\}$ such that $f_{n_j}(x)\to f(x)$ for almost every $x\in[0,1]$. [duplicate]

This is part of an old preliminary exam in Analysis I am reviewing to prepare for my own prelim. $\lambda$ is the Lebesgue measure. $f_n\to f$ with respect to the $L^1-$norm. I know that there exists ...
1
vote
1answer
56 views

Triangle Inequality for $\|x\|_{\infty}$

I have to show the triangle inequality for $\|x\|_{\infty}$. I'm not sure, if estimate is correct. To show: $\|x+y\|_{\infty} \le \|x\|_{\infty}+\|y\|_{\infty}$ Let $x \in \mathbb{R}^n$ and $\|x\|_{\...
1
vote
1answer
68 views

To show $\sum_{n=1}^{\infty}a_n$ is convergent and $b_n$ is convergent with limit $b\neq0$ then $\sum_{n=1}^{\infty}a_nb_n$ is convergent [duplicate]

To show $\sum_{n=1}^{\infty}a_n$ is convergent and $b_n$ is convergent with limit $b\neq0$ then $\sum_{n=1}^{\infty}a_nb_n$ is convergent I have tried some stuff with making a convergent $a_n$ series ...
1
vote
1answer
53 views

$w(x,y)=\frac{x^2+3y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}} \ dx - \frac{3x^2+y^2+2xy}{3 (x^2+y^2)^\frac{4}{3}} \ dy$ , calculate $\int_{+\gamma} w $

$\gamma$ is the curve of this equation: $$\rho=e^{-\theta} \qquad \theta \in [0,+\infty)$$ It is oriented in the growing $\theta$ $$w(x)= \sum_{i=1}^n a_i(x) \ dx_i $$ $$\int_{+\gamma} w=\sum_{i=1}^...
0
votes
1answer
25 views

Derivative and uniform convergence

$f_n(x) = \dfrac{\arctan (n^{1/4} x^2)}{n^{3/2}}$ I need to calculate first derivative of it and then tell if first derivative is uniformly convergent. I calculated it but I got now idea how to bound ...
1
vote
1answer
28 views

Supremum and infimum of function of two variables

Consider $D = \left \{ x \in \mathbb{R} : x_1^2 + 44x_2^2 \leqslant 5 \right \}$ and function $f: D \rightarrow \mathbb{R}$, $f(x) = 13x_1 - 22x_2$. Find supremum and infimum of $f$. For both of them ...