Prime of the form $4x+1$ within two bounds November 30    2

Let's have the number $5^{2^{n-1}}$ where $n$ any non-zero natural number. I conjecturally say that between the following two bounds we will always obtain $n$ primes of the form $4x+1$. $[(5^{2^{n-1}})^{1/n}]e^{1/n}<...>[(5^{2^{n-1}})^{1/n}]e^{-1/n}

3D coordinates rotation — new direction for Z axis November 30    1

I need to rotate 3D coordinate system so Z axis points in new direction.So, I have a direction defined by spherical coordinates ($\theta$, $\phi$), where $\theta$ (in $[0, \pi]$ range) is polar and $\phi$ (in $[0, 2\pi]$ range) is azimuthal angles. I

A basic doubt on a problem on sequence November 30    2

Suppose a sequence of "distinct" elements $\{x_n\}$ converges to $x$ and $x$ is not in the sequence. Then given an $N$ can we find an $\epsilon >0$ s.t. there exists $M >N$ with the property that $\forall n \geq M$ $|x_n-x| <\epsilon$ a

About unbounded linear functional November 30    1

About unbounded linear functional
I'm reading the chapterfrom a optimization book and cannot understand the example listed below:" " Actually i have seen its linear. My question is why f is unbounded? Let $e_n = \{ ...

A confusion regarding an infinite sequence. November 30    2

Sorry this question might be unclear/coherent, but it baffles me logically. Say we have an infinite sequence $\{x_i\}\forall i\in\Bbb{N}$. Can we choose every point in the sequence? Say we start from $x_1$, and choose every point after that; i.e. $x_

Simple Proof about Sequences and Unbounded Functions November 30    1

Let $X$ be a compact interval, let $V$ be a normed vector space and suppose that a map $f:X \rightarrow V$ is unbounded. I'm trying to see why there must exists a sequence $(x_i)$ in $X$ such that $|f(x_i)| \geq i \; \; \forall i \in \mathbb{N}$.My a

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