# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...