# All Questions

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### Is there any normal subgroup of $S_n$? [duplicate]

Possible Duplicate: Normal subgroups of $S_N$ I wonder if there is any normal proper subgroup of $S_n$? If yes, give an example.
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### What is the smallest integer $n$ for which $\theta(n) > n$?

What is the smallest integer $n$ for which $\theta(n) > n$? Here $\theta(x) = \sum_{p \leq x} \log p$. I googled around, checked some likely textbooks, and ran a program for $n \leq 10^7$, but ...
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### Questions on: $G$ a generic ultrafilter on $B$ if and only if $G$ is a generic filter on $(B+,<)$?

Here are the definitions, and then I shall explain which parts of the implication I understand, and which parts I don't, which are the questions. The definitions are from Jech, as well as the ...
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### Why are the conditions of the IFT not necessary

The Inverse Function Theorem states sufficient conditions for a function to have a continuous inverse. When, if it all, are these conditions necessary conditions? Is there a nice counterexample?
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### Differential Equation $y' = \frac{ty(4-y)}{(1+y)}$

$y' = \frac{ty(4-y)}{(1+y)}$ given $y_0 = 2$ At what time $t$ when the solution will first be 3.9 I tried solving this but it didnt work out so well. What I did was: separate dy/dx and move ...
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### Maximum principle for a control with mixed constraints

Consider the dynamical system $\mathbf{x}' = f(\mathbf{x,u},t)$ where $\mathbf{x} \in \mathbb{R}^2, \mathbf{u} \in \mathbb{R}^3$ and $f$ has no explicit time dependence. As conventional, $\mathbf{x}$ ...
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### Hilbert Hotel: what if countably many buses each with countably many guests arrived?

Situation: There's a hotel owner David Hilbert who owns a hotel with countably many (infinity that can be mapped by natural number surjectively) rooms, and there are countable guests who lived inside ...
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### Discontinuity of a semialgebraic function

Let $f : \mathbb R \to \mathbb R$ be semialgebraic. Is it possible that for some $x \in \mathbb R$ the limits $f(x-)$ or $f(x+)$ does not exist? In other words can it have a discontinuity of the ...
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### probability of distribution

In two independent throws of a die, how do you know how many ways that you are going to have or not for certain number ?? For example, if the number was 4, there are number of ways to have none of ...
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### How can I prove that “If $M$ is contractible differentiable manifold, then $M$ is orientable?”

If $M$ is a contractible differentiable manifold, then $M$ is orientable.
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### Confused about a Lipschitz problem

If $f$ is Lipschitz of order 1 at $x$, is it differentiable at $x$? A function $f$ is Lipschitz of order $\beta$ at $x$ if there is a constant D such that $$|f(x)-f(y)|\le D \,|x-y|^\beta$$ for all ...
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### Proving that a $T_{1\frac{1}{3}}$-space is $T_1$-space

Call a space $T_{1\frac{1}{3}}$ if every sequence in it has at most one limit. A $T_1$ space is a space in which for two distinct points $a$ and $b$, there are open sets $U$ and $V$ for which $a$ ...
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### what is answers of these two integrals

what is difference between: $$\int \sqrt {(a^2-x^2)}dx,$$ and $$\int_{x_1}^{x_2} \sqrt {(a^2-x^2)}dx$$ with mathematical solution
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### Equality of two probabilities

I would like to know what should I verifiy in order to show that two probabilities are equal. Here is the exercice : Let $F_0$ be an algebra of sets over $\Omega$ and $P$, $P'$ two probabilities ...
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### is this set a regular surface?

I'm reading "Differential Geometry of Curves and Surfaces of Manfredo Docarmo" I'm doing the exercises of the chapter 2. Here is the definition of regular surface that we are following: I have ...
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### Show $\frac{d}{dx} \int\limits^{a(x)}_{0} f(x,y)dy = \int\limits^{a}_{0} \frac{\partial f}{\partial x}dy + a'(x)f(x,a)$

I am trying to show that $$\frac{d}{dx} \int\limits^{a(x)}_{0} f(x,y)dy = \int\limits^{a}_{0} \frac{\partial f}{\partial x}dy + a'(x)f(x,a)$$ I know this has something to do with the fundamental ...
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### Lim Sup/Inf for real valued functions

To understand the notion of, say, limit superior for a sequence, is not difficult. Simply consider the set of all upper buonds for the set of all limit points of the sequence, and then simply pick the ...
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### Is there a construction known for associating a K3 surface to a curve or cover of curves

Let $X$ be a curve of genus at least two. Then one can associate an abelian variety to $X$; this is the Jacobian. Let $X\to Y$ be a double cover of curves. Then we can associate an abelian variety to ...
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### images stable under precomposing with epis

Suppose that $C$ is a category admitting images. Given an arrow $f:a\to b$ and an epi $e:a'\to a$ is there a common name for a category where (any) of the following properties hold(s)? 1) any image ...
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### Finding the general solution of the PDE $xu_x-xyu_y -u=0$ using a side condition

Find the general solution of the PDE: $$xu_x-xyu_y -u=0$$ I have found it to be: $u(x,y)=-xf(ye^x)$ This PDE has the property that $u(0,y)=0$. Therefore, $u(0,y)$ cannot be arbitrarily ...
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### Faithfully flat morphisms with all fibers complete

Prove or disprove: if $f: X \to Y$ is faithfully flat and each fiber is complete, then $f$ is proper. (I'd especially like to see a counterexample with a morphism of finite type between varieties over ...
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### Vertices and edges of a cube are assigned natural numbers in a particular way; can the sum of the vertices equal the sum of the edges?

At the vertices of a cube are written 8 different natural numbers, and on each of its edges is written the greatest common divisor of the numbers at the endpoints of the edge. Can the sum of ...
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### What is this angle in a right triangle with sides of length 5, 12, and 13?

How do I find the missing adjacent angle to leg b in a right triangle with the following side lengths: leg a = 5, leg b = 12, and hypotenuse = 13. Thanks

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