1
vote
1answer
154 views

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.
4
votes
0answers
104 views

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 ...
3
votes
1answer
162 views

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 ...
0
votes
1answer
47 views

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?
1
vote
2answers
35 views

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 ...
3
votes
1answer
182 views

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, $\mathbf{x}$ ...
2
votes
1answer
69 views

A couple of limit questions.

I'm working on my calculus homework right now and have become stumped on two questions about limits. $\lim\limits_{x \to 2^-} \frac{3x^2 + x - 7}{|x - 2|}$ $\lim\limits_{x \to 0^+} ...
0
votes
1answer
303 views

Fundamental group of this space

Based on this question: What is the homology groups of the torus with a sphere inside? I'm trying to find the fundamental group of this space using the Seifert–van Kampen theorem. If $U$ is the torus ...
2
votes
5answers
2k views

Proof via Induction for A Summation

I'm starting to understand how induction works (with the whole $k \to k+1$ thing), but I'm not exactly sure how summations play a role. I'm a bit confused by this question specifically: $$ ...
1
vote
0answers
86 views

Diophantine Equation: $f(x)f(y) = f(z^2)$ where $f$ is quadratic

In the study of the Diophantine Equation $f(x)f(y) = f(z^2)$ where $f$ is quadratic, the computational proofs I have seen (for specific $f$) rely on Pell's Equation. For example, if $f(t) = t^2+t+1$, ...
0
votes
1answer
90 views

Question about the properties of an ideal in the polynomial ring over a field

This is my homework question: Let $F$ be a field and $f(x), g(x) \in F[x]$ be polynomials. Show that $N = \{r(x)f(x) + s(x)g(x) \, | \, r(x), s(x) \in F[x] \}$ is an ideal in $F[x]$. Show that if ...
1
vote
0answers
36 views

Symmetric algebra and complex polynomial on Lie algebra

Let $G$ a Lie group and $\mathfrak{G}$ its Lie algebra. How can I identify the symmetric algebra on $\mathfrak{G}$ ($S(\mathfrak{G})$) with the algebra of complex polynomials on $\mathfrak{G}$, that I ...
2
votes
1answer
76 views

Directional differentiable property

I am stuck in constructing a function that is locally Lipschitz continuous at $x_0$ but it does not have directional differentiation at $x_0 $in any direction. Thank you for all help and comments.
1
vote
4answers
84 views

Analytic complex function

Show that $f(z) = \bar{z}$ is nowhere differentiable. I must use the definition in order to do this, the definition is: $$\frac{df}{dz}(z_0) = f'(z_0) = \lim_{\triangle z \rightarrow ...
4
votes
0answers
159 views

Representations of non-semisimple Lie algebras

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, and suppose $\mathfrak{g}$ is semisimple. An integral weight for $G$ is an element $\lambda \in \mathfrak{t}^*$ with ...
1
vote
1answer
306 views

Nondeterministic finite automaton proof

I am having a really hard time working the problem below out. I am not sure I am even on the right direction with this logic . Swapping the accept and reject states alone is not sufficient to accept ...
2
votes
2answers
70 views

Norm on normed algebra

I am reading a book " Abstract Linear Algebra " - M. L. Curtis Chapter V is about normed algebras, which is a title. Here normed algebra $X$ is a algebra with norm satisfying the following ...
0
votes
0answers
99 views

Sum of angles in a hyperbolic triangle with one ideal angle

I want to calculate the sum of the angles of the triangle formed in the hyperbolic plane from the points $(-1,1), (0,1)$, and $(1,1)$. This forms an angle at the origin which has an infinite slope for ...
3
votes
1answer
413 views

Proof of a norm

Prove that: $ \|\cdot\|_1:\mathbb{R}^n \rightarrow \mathbb{R};\vec{x} \mapsto \sum_{j=1}^n |x_j| $ is a norm defined on the vector space $\mathbb{R}^n$. 1) Zero vector: $\sum_{j=1}^n |x_j| = |x|_1 ...
8
votes
3answers
500 views

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 ...
0
votes
1answer
73 views

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 ...
1
vote
2answers
42 views

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 ...
0
votes
1answer
142 views

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.
1
vote
2answers
112 views

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 ...
3
votes
1answer
80 views

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 $a$ ...
-2
votes
4answers
108 views

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
0
votes
3answers
89 views

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 ...
3
votes
2answers
453 views

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 ...
2
votes
4answers
173 views

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 ...
3
votes
1answer
1k views

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 ...
3
votes
0answers
52 views

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 ...
0
votes
0answers
38 views

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 ...
1
vote
1answer
87 views

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 ...
5
votes
1answer
201 views

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 ...
3
votes
1answer
154 views

Order of operations of multiple Matrix Elementary Row Operations

I have two elementary row operation matrices (elimination matrices): $E_{31} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ ...
3
votes
1answer
1k views

$f$ bounded on $[a,b]$ with one or finite discontinuities implies $f$ Riemann-integrable.

I have two problems: Prove that if $f$ is bounded on $[a, b]$ and has exactly one discontinuity in $[a, b]$ then $f$ is Riemann-integrable on $[a, b]$. Prove that if $f$ is bounded on $[a, b]$ and ...
1
vote
2answers
189 views

Show that $\sum\limits_{ n \in \mathbb{Z} \setminus \{0\}}\frac{e^{2 \pi i x n}}{n}$ is a convergent series where $x$ is any real

Show that $\displaystyle \sum_{ n \in \mathbb{Z} \setminus \{0\}}^{} \frac{e^{2 \pi i xn}}{n}$ is a convergent series for all real x. I think that this could be done by breaking up the sum into two ...
1
vote
2answers
69 views

Value of Investment in the Past

An amount of $1000$ is to be accumulated at a compound rate of discount of $9$% per year. (a) Find the present value $3$ years before (b) Find the value of i corresponding to d. For a) i have done ...
1
vote
2answers
240 views

Exercise - Fourier Transform

I have got another question concerning the Fourier transform. I hope somebody can help me. Let $f \in L^1(\mathbb{R}^n)$. Prove that $\hat{f}$ is continous. (ok, I was able to show it) If ...
0
votes
0answers
79 views

Markov Chain and cryptanalysis

Where I will be able to found papers to read the state-art of the use that Markov chain in cryptanalysis. I founded this Canteaut, A. and Chabaud, F. (1998). A new algorithm for finding ...
1
vote
1answer
76 views

Disjoint Sets Seemingly Not Disjoint in Description of Lebesgue Measure

I am currently working through Knapp's Basic Real Analysis. I am currently working in Chapter 5 on "Lebesgue Measure and Abstract Measure Theory." The book states the following, giving Lebesgue ...
10
votes
1answer
450 views

Distance minimizers in $L^1$ and $L^{\infty}$

If $H$ is a Hilbert space, we have the Hilbert Projection Theorem, which tells us that given a nonempty, closed, convex subset $K \subset H$, and a point $x \in H$, there is a unique point $y \in K$ ...
3
votes
1answer
93 views

Function in $L^1([0,1])$ that is not locally in any $L^{\infty}$

Can we find a function such that $f\in L^1([0,1])$ and for any $0\leq a<b\leq 1$ we have that $||f||_{L^{\infty}([a,b])}=\infty$?
1
vote
1answer
109 views

Self-adjointness of $D=\frac{d^2}{dx^2}-1$ with boundary conditions $u'(0) = 0 = u'(a)$ on $[0,a]$.

Im trying to show that $$D=\frac{d^2}{dx^2}-1$$ is self adjoint on $[0,a]$ subject to $u'(0)=u'(a)=0$. I think I need to use integration by parts but I'm not sure how to do that.
1
vote
0answers
74 views

Inequality with Expectation and vectors

Let $x=(x_1,\ldots, x_n)\in R^n$. Let $r_i, i=1, \ldots, n$ be Rademacher i.i.d. random variables (i.e. $P(r_i=1)=P(r_i=-1)=1/2$). It is a well-known inequality that: $$ E\left(\left|\sum_{i=1}^n ...
2
votes
0answers
127 views

$l^1$ norm estimate for inverse of Vandermonde matrix

As title, I would like to know the known upper bound for the $l^1$ norm for inverse of Vandermonde matrix. A quick search gives this paper by Gautschi 40 years ago, but it deals with the infinity norm ...
0
votes
1answer
53 views

Number of sliding window in the string

I have to calculate number of sliding window in a string. If the sample string is "CHECKIT" and window size is 2 then CH HE EC CK KI IT Window size may vary for the string.
4
votes
2answers
125 views

Where does this equality about expectation of random variables come from?

In order to prove this Lemma in my course about Probability : Let $X=(X_1,\dots ,X_p)$ be a gaussian random variable such that $\mathbb{E}[X_j]=0$ for all $j=1,\dots,p$. Then ...
5
votes
2answers
409 views

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 ...
4
votes
3answers
2k views

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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