1
vote
2answers
783 views

Is this language and its complement is Context free Language?

The language $\mathcal{L}=\{a^n b^n c^n\;|\;n={1,2,3,4}\}$ is not context free from my point of view. Because no. of a's pushed in to stack is totally compensated by no of b's. popped. Please help me. ...
0
votes
2answers
244 views

If a matrix power has a limit which is positive, then there exists some power of the matrix such that the matrix is positive for this power.

Let $A$ be a matrix of order $ n \times n$ and let $ \lim_{ k \rightarrow \infty} A^k = L > 0.$ I want to show that there exists some power $m$ such that $A^m>0$ and $ A^{m+i} > 0$ for any $ ...
1
vote
1answer
56 views

How to prove this expression?

It's from a programming language. "%' here is a modulo operation. (a * 2^32 + b) % c = (((a % c) * (2^32 % c)) + b) % c
0
votes
1answer
179 views

Finding area of surface of revolution

how do you find the area of the surface obtained by rotating the curve about the x-axis? Given hint in the question: write $y$ in terms of $x$. $3y^2 = x(1-x)^2,\ 0\leq x\leq 1$ I got ...
0
votes
1answer
627 views

What's the proper way for calculating first quartile?

How should I calculate quartiles? In a book I read (by Amir D. Aczel) it is said that: First quartile is th 25th percentile, that means a value for which $1/4$ of oberervation's results are ...
1
vote
1answer
233 views

Sum of reciprocals, reciprocals of the sum

I have two vectors $a$ and $b$. I have the two following quantities, $\sum_i a_i \frac{1}{b_i}$ and $\sum_i a_i \frac{1}{\sum_j b_j}$. I know that for every $i$, $0\leq a_i \leq b_i \leq 1$. Which ...
1
vote
1answer
588 views

Surface Area Problem with Specific Formulas

I noticed this question in a Math Problem-Set book. These were the only formulas allowed: $1. Area(quadrant) = \frac{1}{4}\pi r^2$ $2. Area(square) = (side)^2$$3.Area(semicircle) = \frac{1}{2} \pi ...
0
votes
1answer
212 views

Finding the arc length

How do you find the arc length of $y = \sin^{-1}x + \sqrt{1-x^2}$? I got $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sqrt{2-2\sin t}dt$ and became stuck. Any hints?
4
votes
0answers
260 views

Jacobi theta function

This is a question from Stein & Shakarchi's complex analysis book. Show that if $\rho$ is fixed with $Im(\rho)>0$, then the Jacobi theta function $$\theta(z|\rho)=\sum_{n=-\infty}^\infty ...
1
vote
1answer
62 views

First order differential equation

Im clueless on how to solve the following question... $xe^y\frac {dy}{dx} = e^y +1$ What i've done is... $\frac {dy}{dx} = \frac 1x + \frac {1}{xe^e}; \frac {dy}{dx} - \frac {1}{xe^e} = \frac 1x $ ...
2
votes
3answers
187 views

Do these two sequences converge to same limit?

$(x_n)_{n\geqslant1}$ and $(y_n)_{n\geqslant1}$ are two real sequences such that $x_1 > 0$, $y_1 > 0 $, $x_{n+1} = \frac{1}{2}(x_n + y_n)$ and $\dfrac{2}{y_{n+1}} = \dfrac{1}{x_n} + ...
2
votes
3answers
194 views

Are there five complex numbers satisfying the following equalities?

Can anyone help on the following question? Are there five complex numbers $z_{1}$, $z_{2}$ , $z_{3}$ , $z_{4}$ and $z_{5}$ with ...
1
vote
3answers
903 views

Trouble understanding equivalence relations and equivalence classes…anyone care to explain?

What exactly are equivalence relations and equivalence classes? The latter is giving me the most trouble; I've tried to read multiple sources online but it just keeps going over my head. Example ...
3
votes
0answers
157 views

Hazard Rate Probability HW Question

I am working on a homework problem below from Pitman's Probability book: Suppose the failure rate is $\lambda$($t$) = $at$ $+$ $b$ for $t$ $\geq 0$. The problem asks to find the formula for the ...
6
votes
2answers
298 views

Equating sums of square roots

I solved the following equation the hard way: $$\sqrt{x+1} +\sqrt{x+33}=\sqrt{x+6} +\sqrt{x+22}$$ The only solution is $x=3$. I am wondering if there is some easy observation that solves the equation ...
2
votes
2answers
99 views

Parameter values that make function values side lengths of a triangle

I have been trying to solve the following problem for more than a week without any success. Given the function: $$f(x)=\frac{x^2+mx+4}{x^2+x+4}$$ Find all possible values of the parameter $m$ such ...
16
votes
7answers
4k views

How to write \mathcal letters by hand? [closed]

I have a formula in a paper that I want to write out by hand, and it contains two "D"s, a normal D, $D$ in latex, and a 'mathcal', caligraphic D, $\mathcal{D}$ in latex. What are some ...
9
votes
2answers
185 views

Can an algebraic variety be described as a category, in the same way as a group?

Can an algebraic variety be described as a category, in the same way as a group? A group can be considered a category with one object, with elements of the group the morphisms on the object.
1
vote
1answer
424 views

The proof of Brower Fixed Point Theorem for 2-dimensional case

Note the followin theorem Theorem : any continuous map from a unit two dimensional disk $E^2$ into itself has a fixed point. To prove this theorem, Harper and Greenberg's book use the following ...
2
votes
1answer
228 views

properties of the poset of self adjoint elements in a $C^*$ algebra

It is well known that the set of self-adjoint elements in a $C^*$ algebra naturally forms a poset. I assume that the general properties of this poset are known. Can anybody point me to a reference ...
3
votes
2answers
208 views

how can I evaluate this integral?

how to evaluate this integral: $$l(y)=\int\limits_\beta^\infty \theta\exp(-y\theta)\alpha\exp(-\alpha\theta) \, d\theta$$ where $\alpha,\beta,\theta,y>0.$ Because I find it infinity! Can anyone ...
3
votes
0answers
119 views

smooth morphism on schemes

Imagine a projective, noetherian and flat family of curves $C \rightarrow S$ (i.e. every geometric fiber is a curve, this is a integral, non singular, proper scheme of dimension 1 over an ...
10
votes
2answers
1k views

Is the Nash Embedding Theorem a special case of the Whitney Embedding Theorem?

The Whitney Embedding Theorem states that every smooth manifold can be embedded in Euclidean space. The Nash Embedding Theorem states that every Riemannian manifold can be embedded in Euclidean ...
2
votes
1answer
70 views

Endomorphism of Algebraic Closure

This is a homework problem and I have some questions regarding it (not looking for solution): Let $\tau$ be an endomorphism of a field $F$ that is algebraically closed. i.e. $\tau: F\rightarrow ...
0
votes
1answer
35 views

If $\tau\in\mathscr{L}(V)$, why does $\tau^{-1}=p(\tau)$ for some polynomial $p$?

I'm working on an exercise from Steven Roman's Advanced Linear Algebra. He asks Let $\dim(V)<\infty$. If $\tau,\sigma\in\mathscr{L}(V)$, prove that $\sigma\tau=\mathrm{id}$ implies $\tau$ and ...
2
votes
3answers
260 views

Show that: $\mu\left(\bigcup_N \bigcap_{n=N}^{\infty} A_n \right) \leq \lim \inf \mu(A_n)$

Let $(X,\mathcal{M},\mu)$ be a measure space. Let $A_1, A_2, \ldots \in \mathcal{M}$. Then, I want to show that: $$\mu\left(\bigcup_N \bigcap_{n=N}^{\infty} A_n \right) \leq \lim \inf \mu(A_n)$$ ...
0
votes
1answer
53 views

Isomorphy of vector bundles over a variety $X$

As in my other question, let $X$ be a variety over a field $k$, and let $\pi:F\to X$, $\psi:G\to X$ be vector bundles of rank $r$ over $X$ defined on the same open cover $\{U_i\}$. That is, we have ...
1
vote
2answers
112 views

Question about maximal ideal.

I am trying to solve a problem in the book Introduction to commutative algebra from Atiyah. Page 11, exercise 5 (iv). Let $A[\![x]\!]$ be the ring of all formal power series of the form ...
1
vote
2answers
81 views

Question on a proof of a sequence

I have some questions 1) In the forward direction of the proof, it employs the inequality $|x_{k,i} - a_i| \leq (\sum_{j=1}^{n} |x_{k,j} - a_j|^2)^{\frac{1}{2}}$. What exactly is this inequality? ...
0
votes
1answer
19 views

Need a reference for techniques for the evaluation of limits of functions

Is anyone familiar with a comprehensive source (a book or a web-site) for techniques used in evaluating the limits of functions ?
3
votes
1answer
201 views

What are the morphisms $\mathcal{O}_X\to k_p$ on a variety $X$?

Let $X$ be a variety over a field $k$ (we always assume $k=\bar k$, but I think this doesn't matter here), and let $k_p$ denote the skyscraper sheaf on $X$ w.r.t. the point $p\in X$. I want to find ...
1
vote
1answer
71 views

Finding all differentiable functions with a certain property

Find all differentiable functions $f \colon (0,\infty) \to \mathbb R$ for which there is a positive real number $k$ such that: $$ f(x) \cdot f'(k/x) = x, \qquad\text{for all }x > 0. $$
1
vote
1answer
556 views

Nonabelian order 28 group whose Sylow 2-subgroups are cyclic

Prove that if such a group exists it is unique (up to isomorphism). Also, determine the numbers of elements of each order, and the class equation of the group. I don't really know how to do the ...
1
vote
1answer
68 views

Isomorphisms of linear representations of finite groups

Let $G$ be a finite group with representations $\rho_1, \rho_2:G\rightarrow GL(V)$. According to the definition of representation isomorphisms, $\rho_1$ and $\rho_2$ are isomorphic if there exists a ...
0
votes
1answer
71 views

A question on groups actions of permutation groups

This is the final step into completing a problem and I am a bit stuck. I need to show that: Consider the action of $S_{n-1}$ on $S_n/S_{n-1}$ by left multiplication. Does this action have exactly two ...
2
votes
2answers
82 views

Diagnostic Tests and Expected Values

I have this question from my textbook I'm not sure how to answer. I got the first part but the second part is a bit confusing. It goes something like this: "Two percent of the population has a ...
0
votes
1answer
240 views

7 card stud counting problem

Consider a standard deck of 52 cards. A game called 7 card stud is played where each player is dealt 7 cards. How many possible: 7 card hands are there? --my answer:133,784,560 hands from (1) ...
1
vote
1answer
66 views

For any odd integer $x,y$, $(x^2+2) \nmid (y^2+4)$ [duplicate]

Possible Duplicate: Quadratics and divisibility Prove that for any odd integers $x$ and $y$, we have $(x^2+2) \nmid (y^2+4)$
1
vote
1answer
66 views

Interpolation of a function

Given the function $$f (x) = x\bigg(x − {1\over4}\bigg)\bigg(x − {1\over2}\bigg)$$ How can I interpolate $f(x)$ with $p(x) = a_0T_0(x) + a_1T_1(x) + a_2T_2(x) + a_3T_3(x)$ to show that $$a_0 = ...
0
votes
1answer
53 views

Absolute convergence of series…a hint?

The question: Show that if $\sum_{n=1}^\infty a_n$ converges absolutely, then $\sum_{n=1}^\infty a_n^2$ converges absolutely. I'm stuck, not sure what path to take in solving this. Can anyone ...
0
votes
2answers
103 views

Stuck: Counting Problem..Prove that among any 3 integers…

Prove that among any 3 integers there are always two whose difference is divisible by 2.
2
votes
3answers
380 views

How to prove $r^2=2$ ? (Dedekind's cut)

Let a (Dedekind) cut $r=\{p \in \mathbb{Q} :p^2<2 \text{ or } p<0\}$ and a cut $2^*=\{t\in \mathbb{Q} : t<2\}$. I want to prove $r^2=2^*$. I could show that $r^2 \subset 2^*$ easily, but I ...
2
votes
2answers
2k views

Runge-Kutta algorithm for a given ODE system

consider the system given by: $$x'_{1}=9x_{1}+24x_{2}+5\cos t-\dfrac{1}{3}\sin t$$ $$x'_{2}=-24x_{1}-51x_{2}-9\cos t+\dfrac{1}{3}\sin t$$ with initial values $$x_{1}(0)=\dfrac{4}{3}$$ and ...
1
vote
3answers
396 views

Discrete Math - Counting

Prove that among any 100 integers there are always two whose difference is divisible by 99. How can I prove this?
4
votes
1answer
74 views

Functional Interpretation of Variety?

I am simultaneously taking courses in functional analysis and commutative algebra. In doing so, I found that there is, at least heuristically, some similarity between the notion of an algebraic ...
0
votes
1answer
78 views

Ordered set of integers

$\{x_i\}_{i = 1}^7$ is a set of 7 integers that satisfy $1≤ x_i ≤ 8$. How many such ordered sets of $7$ integers are there, such that $$x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 - x_1x_2x_3x_4x_5x_6x_7 ...
2
votes
1answer
126 views

Expectations Homework Questions Clarification

This is another one of the questions I didn't get a chance to ask the TA at my school today. I thought I had a pretty good grasp on Expectations but apparently I could still use some clarification. ...
0
votes
1answer
45 views

Help with the proof that $A^*$ has the UMP of the free monoid

In the following proof $A^*$ here is a Kleele closure and $*$ is a product of $a$'s or "concatanation": Proposition 1.9. $A^*$ has the UMP of the free monoid on A. Proof. Given $f:A\to|N|$, ...
0
votes
1answer
214 views

How to find the coefficient of this Fourier sine series?

From $$1=\sum_{k\geq 1} a_k \sin((k\pi+\frac{\pi}{2})x),$$ I want to find $a_k.$ My unsuccessful approach is first multiplying both side by $\cos((k\pi+\frac{\pi}{2})x)$. That is, ...
1
vote
3answers
161 views

Intuitive understanding of relations and their basic properties

Can anyone explain relations and the four basic properties of them (reflexive, symmetric, antisymmetric, transitive)- or direct me to a source that does. Particularly, are these statements ...

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