5
votes
1answer
224 views

My proof: Frobenius Map generates $\mbox{Gal}(\mathbb F_{p^n}/\mathbb F_p)$

I would like to ask, whether anyone can confirm or correct the following version of the proof that the Frobenius map generates the Galois Group of a finite field. Proof First note that ...
1
vote
1answer
206 views

Factorial of $(7/2)!$ [duplicate]

It's been many years since I studied maths, and I'm trying to figure out the half factorials $(7/2)!$ without a calculator. I did $(7/2) \times (5/2) \times (3/2) \times (1/2) = (105/16) ^ \pi = ...
1
vote
1answer
52 views

Finding the types of singularities of $f(z)=\frac{1}{z\cdot (e^z -1 )}$

I am getting trouble to find the types of singularities of $$f(z)=\frac{1}{z\cdot (e^z -1 )}$$ What I tried to do is: $z=0$ $z=2\pi k i$ for $z=2\pi k i$ its in order 1, but for the first one I ...
1
vote
2answers
189 views

A question about tableau method for first-order logic

I have a doubt about tableau method for f-o logic. In Smullyan's book (First-Order Logic, 1968, Dover reprint) the method is defined (pag.53) for formulae but - if I'm not wrong - all examples that ...
9
votes
3answers
326 views

How to show $\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a})$? ($a\ge0$)

$$\int^{\infty}_{-\infty}\frac{\sin(ax)}{x(x^2+1)}dx=\pi(1-e^{-a}), \ a\ge0$$ I tried to solve but came up with $\pi(2-e^{-a}) $. Could you tell me where did I do the mistake? if $x=z$ then ...
4
votes
1answer
139 views

Gödel Incompletness theorem

I am not familiar with model theory. As a matter of fact, I only had my first Logic and Set theory courses last semester. But still, there is a question that bothers me, and It could be nice if ...
1
vote
2answers
118 views

Computing distance from line to point in geodetic environment

Supposing to be in a cartesian plan and that I have the following point: $$A(x_{1},y_{1}), B(x_{2},y_{2}), C(x_{3},y_{3}), D(x_{4},y_{4})$$ $$P(x_{0},y_{0})$$ Now immagine two lines, the fist one ...
0
votes
1answer
465 views

How to solve this fraction within minute? (or trick)

I want to solve this question within minute but bcz of fraction it take more than a minute. does any one know a trick to solve this type question.plz share thanks
4
votes
2answers
259 views

Find the number of irreducible factors of $x^{63} - 1$

I have to find the number of irreducible factors of $x^{63} - 1$ over $\mathbb{F}_2$ using the $2$-cyclotomic cosets modulo $63$. Is there a way to see how many the cyclotomic cosets are and what is ...
3
votes
2answers
119 views

Are these two 2-manifolds homeomorphic?

I have a 2-Sphere with a finite number $k$ of points removed (at least 3), and I want to equip it with a Riemannian metric of constant negative curvature. My first thought was to take a free ...
-1
votes
1answer
87 views

Determine whether the given binary relation is equivalence relation

$D$ is the binary relation defined on $R$ as follows: For all $x,y\in R,xDy\Leftrightarrow xy >0$. Determine whether the given binary relation is equivalence relation and if it is, give the ...
1
vote
4answers
100 views

How to understand that the $\lim_{x\to 0}\frac1x\cos(\frac1x)$ does not exist.

I'm studying the $\lim_{x\to 0}\frac1x\cos(\frac1x)$. Can I understand that the limit does not exist simply splitting it into two parts? That is, $$\lim_{x\to 0}\frac1x\cos(\frac1x)=\lim_{x\to 0}\frac ...
1
vote
0answers
67 views

Continuity in a physical context

I'm currently trying to solve an exercise for my quantum mechanics class and have run into a bit of a jam: Suppose we have the following potential : $V(x) = 0$ if $x > |a/2|$ but $V(x) = V_0$ if ...
1
vote
1answer
167 views

Biholomorphic map between the upper half plane with a slit and the unit disk

Let $\mathbb{H} = \{z\in \mathbb{C}| \ Im(z) > 0 \}$. I want to find a biholomorphic mapping between $\Omega_{} = \mathbb{H}-\{it \ | \ t \leq 1 \}$ and $D(0,1)$. Any hint ?
1
vote
0answers
156 views

How to use Chebyshev's inequality or the law of large numbers to a probability question

Let x be a random bit string that takes values $\{1,0\}^n$. Let r be the value of the most significant (MSB) bit of x (and r is a r.v. 1 or 0 that are equally likely). Let g be our guess for the MSB ...
0
votes
2answers
165 views

Prove that set contains least element.

Let $A\not=\emptyset ,A\subset \mathbb{Z}$ and if $(\exists d\in \mathbb{Z})(\forall a\in A):d\le a$ then set A contains least element. How do I prove this? I understand I can use WOP principle. What ...
1
vote
1answer
137 views

Composition of orthogonal projection

Given $\gamma: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (rotation around $o$) and $\sigma: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (reflection in one of the lines through the origin), I have to show that ...
0
votes
1answer
718 views

Summation of logarithmic series

I am solving a recurrence relation and it requires me to sum the following series upto $\log{n}$ terms - $1/\log(n) + 1/\log(n/2) + 1/\log(n/4)$..... The base in each term is $2$. Any help on ...
3
votes
1answer
64 views

What is a codomain of diagonal functor?

I'm reading a "Graph Transformations. An Introduction to the Categorical Approach" by H.J.Schneider. In a example 6.3.3 Graph Category constructed as a comma category of a identity functor $id_{Set} : ...
1
vote
0answers
73 views

Resolvable spaces

a space $X$ is called a resolvable space if it is expressible as a union of two disjoint dense subsets. I want to find a resolvable but not lindelof space? Is there any example such a space?
3
votes
0answers
118 views

$\mathfrak b \leq \mathfrak r$. Is this proof correct?

I am trying to prove that $\mathfrak b \leq \mathfrak r$ where $\mathfrak r$ is the minimal cardinality of a reaping family of sets. A family $\mathcal R \subseteq [\omega]^\omega$ of sets $\mathcal ...
2
votes
1answer
75 views

A problem of urns.

Suppose there are $N$ balls of different colors and $K$ urns. For each ball $i=1,...,N$ it is extracted a flat integer random number $k_i$ between $1$ and $K$ and the ball $i$ is randomly assigned ...
0
votes
2answers
401 views

Can't read integral method [closed]

I type this : fun = @(x) exp(-x.^2).*log(x).^2; q = integral(fun,2,4); q; when I run the above code, I get an error message Undefined function or method ...
2
votes
1answer
87 views

Philosophical side of MATH. knowing the path then walk it. [closed]

Can I find a book that gives me the purpose of theorems and definitions without going deep into proofs. It's just like knowing the path then walk it. That's will me the understanding reach the next ...
2
votes
4answers
243 views

how to evaluate $\int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin x}}\text{d}x$

I was solving a physics problem and eventually the problem boiled down to solving the following integral: $$\int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{\sin x}}\text{d}x$$ I have already tried ...
2
votes
1answer
62 views

Continuous extension of positive functions on a C-embedded set.

If $A$ is a discrete, closed and $C$-embedded subset of a completely regular Hausdorff space $X$. Then how can we prove that for every continuous function $f:A\rightarrow (0,\infty)$, there exists a ...
8
votes
5answers
191 views

The Limit of $\frac{\cos x } {x e^{x}}- \frac{1}{x}$ as $x \to 0$

I need to evaluate $$\lim_{x \to 0} (\frac{\cos x } {x e^{x}}- \frac{1}{x})$$ Using neither L'Hôspitale rule, nor Taylor series... My try: $$\frac{\cos x}{xe^x}-\frac{1}{x}=\frac{\cos x - ...
1
vote
1answer
98 views

How much time will the pipe take?

There are four outlet pipes of the same capacity fixed one above the other to a water tank. The first pipe is at the bottom level and the fourth pipe is at three-fourths of the height of the tank. The ...
0
votes
1answer
83 views

Show that $(\Bbb{N}, |)$ is a distributive lattice.

Show that the set of Natural numbers with divisibility form a distributive Lattice where for any $x, y\in\mathbb{N}$ we have $x$ meet $y = \operatorname{gcd}(x,y)$ and $x$ joint ...
0
votes
1answer
44 views

Find the constants in a 2D flow (incompressible, newtonian)

$$u_1=x_1^2x_2$$ $$u_2=A+Bx_1x_2^2$$ $$p=cosx_1$$ The fluid is an incompressible Newtonian fluid $\implies$ $u_i,i=0$ and $tor_ij=-p\delta_ij+\mu u_ij$ Fluid bounded by a stationary rigid plate at ...
1
vote
3answers
76 views

Representing Complex Exponentials with Real and Imaginary Parts

My confusion lies with this : http://www.wolframalpha.com/input/?i=modulus+%28cos%282+pi+r_1%29%2Bcos%282+pi+r_2%29%2Bi+%28sin%282+pi+r_1%29%2Bsin%282+pi+r_2%29%29%29+squared I was looking at ...
2
votes
1answer
65 views

If two powers of permutations are equal and have no common symbols, they're the identity. - Mulholland p. 44 Proof to Theorem 4.2

Theorem 4.2 (Order of a Permutation): The order of a permutation written in disjoint cycle form is the least common multiple of the lengths of the cycles. Proof: One cycle: As we noted above, a ...
0
votes
1answer
45 views

Mean of absolute difference series of two random series, uniformy distributed?

Suppose we have two series of 100 (or more) random numbers between 0 and 1. Naturally, the average of series 1 is close to 1/2, and the average of series 2 is also close to 1/2. (screenshot attached.) ...
0
votes
2answers
46 views

Questions regarding bound variables

$$x\in(\cap F)\cap(\cap G)=[\forall A\in F(x\in A)]\land[\forall A\in G(x\in A)]$$ Since the variable $A$ is bounded by universal quantifier, it is regarded as bounded variable, according to the ...
1
vote
1answer
101 views

One can integrate every monotonic function

I have a question related to the proof of "One can integrate every monotonic fucktion $f: [a,b] \to \mathbb R$." that I have as assignment. We are referring to Riemann integrals here. The idea I came ...
4
votes
2answers
178 views

Set theory aspects of category theory

I have never learnt axiomatic set theory, but have studied it from Munkres's Topology book first chapter. So I do not understand the difference between a class and a set except in some vague sense. ...
11
votes
4answers
160 views

Why is $\operatorname{Hom}(M,N)$ not necessarily an $R$ module?

Let $R$ be a ring, and $M,N$ be left $R-$modules. Then is it not true that $Hom_R(M,N)$ has the structure of an $R$-module? I was reading the preface of the Homological Algebra book by Rotman and ...
2
votes
5answers
142 views

I am going to learn these Mathematics Topics. I need advice and suggestions please .

I am really horrible when it comes to maths since I never had any maths background in my High school. I am fairly good at programming ( C++ and Java) but without mathematics I cant advance in any ...
1
vote
1answer
35 views

Difference equation formula $\sum a^t = \frac{a^t}{a-1}$.

As I explain below, this question was originally posted by user YYG, but then deleted. I am reposting the question (from memory) and I will answer it myself below. Question: In Difference Equations ...
0
votes
2answers
61 views

Second isomorphism theorem for subspaces

just like I did some days ago, I now have to show that $T/T\cap U \cong (U+T)/U $. Therefore I tried finding a surjective homomorphism and then, by using the first isomorphism theorem, I should be ...
0
votes
3answers
142 views

Ellipse to Standard Form

This is the equation to the ellipse, $9x^2+4y^2-72x+40y+208 = 0$, and I need it in standard form. I can't figure this one out. Could this be the answer? $\frac{(x-4)^2}{9} + \frac{(y+5)^2}{4} = 1$ ...
4
votes
2answers
664 views

Irrational number and Baire space

How to show that the set of irrational numbers is a Baire space ?
0
votes
1answer
61 views

Uniform Convergence of a Sequence of Summations

Given: $f_1(x)=x$ if $x\le1/2$ $f_1(x)=1-x$ if $1/2\le x\le1$ $f_1(x+1)=f_1(x)$ $\forall n\ge2,f_n(x)=(1/2)*f_{n-1}(2x)$ Let $S_m(x)=\sum_{n=1}^m f_n(x)$ $S_m$ is a continuous function on ...
8
votes
1answer
755 views

Uniform Convergence verification for Sequence of functions - NBHM

Following is a list of problems from an exam for admission into Ph.D program. I have just compiled all previous questions on uniform convergence of sequence of functions and i tried to work out . I ...
0
votes
2answers
52 views

Arithmetic Order of Operation

What is the square root of -4 + 13 x 8 - (7 + 2(3 + 20/5)) The answer seems to be 3 but I wanted help trying to get to the answer!
4
votes
3answers
189 views

Geometry of the dual numbers

A dual number is a number of the form $a+b\varepsilon$, where $a,b \in \mathbb{R}$ and $\varepsilon$ is a nonreal number with the property $\varepsilon^2=0$. Dual numbers are in some ways similar to ...
1
vote
1answer
75 views

Stopping time and filtrations

I have a definition problem. I know that a filtration on a probability space is an increasing sequence of $\sigma$-algebras. I was now thinking on the fact that constant times are stopping times. I've ...
1
vote
2answers
262 views

Prove that :- If K is a compact subset of R with non empty interior then it is of the form [a,b] or [a,b] - U{In}

The question is :- Let $K$ be a compact subset of $\mathbb R$ with non empty interior. Show that K is of the form $[a,b]$ or $[a,b] \setminus \bigcup I_n$ , where { $I_n$} is a countale disjoint ...
2
votes
1answer
195 views

Representation of integers as powers of the golden ratio

How to prove that any integer $n$ can be represented in the form of $$n= \phi^{z_1}+\phi^{z_2}+\phi^{z_3}+...+\phi^{z_m}$$ For $z_1$, $z_2$... $z_m$ $\in$ $\mathbb Z $ and $\phi =\frac{ \sqrt ...
1
vote
1answer
83 views

Help solving an optimization problem involving inverse square roots

Does anyone know if the following optimization problem has an elegant solution? Let $A=\{a_1, a_2, \ldots, a_n\}$ be positive real numbers. Let $B=\{b_1, b_2, \ldots, b_n\}$ be unknown positive real ...

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