4
votes
1answer
208 views

Question on Intersection Theory of Effective Divisors

I am reading Section 1.1C of Lazarsfeld "Positivity in Algebraic Geometry I" and I need help understanding one line. On Page 17, Remark 1.1.13(iii), he says the following: If $D_1,..., D_n$ are ...
1
vote
1answer
39 views

similar matrices have the same bandwidth?

If $A$ is symmetric with bandwidth $p$ then $A_+ = Q^{T} A Q$, where $Q$ is orthogonal, is orthogonally similar to $A$. How can we show/prove that $A_+$ also has bandwidth $p$ ?
1
vote
1answer
43 views

How do you find L of LU for this LU factorization?

$$ \begin{bmatrix} 3 & -7 & -2 \\ -3 & 5 & 1 \\ 6 & -4 & 0 \\ \end{bmatrix} $$ The method my book gave me of doing this is: divide col1 ...
3
votes
1answer
64 views

How prove $\binom{n}{m}\le\left(\frac{en}{m}\right)^m$ [duplicate]

Show that $$\binom{n}{m}\le\left(\dfrac{en}{m}\right)^m$$ where $0<m\le n,m,n\in N^{+}$ My idea: since ...
1
vote
2answers
103 views

Finding a particular solution to the non-homogenous system

I have the following problem $\vec{x}^{'}(t)=\begin{pmatrix} 2 & -5\\1 & -2 \end{pmatrix}\vec{x} + \begin{pmatrix} \csc t\\ \sec t \end{pmatrix}$ Step 1) Find the Eigenvalues ...
1
vote
1answer
62 views

Prove $p \times(1 - p)^n$ is a probability function

$p \times (1 - p)^n$ where $p$ is the probability an event will happen after $n$ number of failure attempts. I know I have to show that the sum of the probabilities of all possible events equals one. ...
1
vote
2answers
130 views

How does one arrive at the asymptotic expressions for the bessel functions?

It is known that Bessel functions for large arguments will behave as exp or cos/sin however I was wondering how does one arrive at those results. The motivation being that I would like to use these ...
87
votes
2answers
4k views

Some users are mind bogglingly skilled at integration. How did they get there?

Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this. Especially in a ...
0
votes
0answers
42 views

Concept for an “Order” of Exponents

Given a set $S$ of abstract mathematical objects, let's say that every element $A\in S$ has order 1, denoted $Ord(A)=1$. I then define that for any 2 elements $A$ and $B$ in $S$, $Ord(A\cdot B) = ...
0
votes
3answers
180 views

Showing quotient rings are isomorphic

Can anyone explain to me how to show two quotient rings are isomorphic? For my particular case. Both quotient rings are based off ideals in the ring $\mathbb Z_3[X]$: $$ \mathbb ...
0
votes
1answer
167 views

PDE from London's Equation with Cylindrical Symmetry

The question is from ISSP by Kittel and as follows: (a)Find a solution of the London equation that has cylindrical symmetry and applies outside a line core. In cylindrical polar coordinates, we want ...
0
votes
1answer
62 views

Can an entire, non-constant function map $\mathbb{C}$ to a proper subset of $\mathbb{C}$?

Can an entire, non-constant function map the complex plane to a proper subset of the complex plane? And, by what theorem?
1
vote
0answers
38 views

Simple Conditional Expectation Problem

$$Y := \begin{cases} \begin{matrix} 5 & \text{w.p. } \frac{1}{5} \\ 2 & \text{w.p. } \frac{1}{2} \\ 1 & \text{w.p. } \frac{3}{10} \end{matrix} \end{cases}$$ Note: $\text{w.p.}$ stands for ...
0
votes
2answers
29 views

Construct a sequence with certain property

Construct a sequence $(s_n)$ which satisfies the following property: $\forall x \in \mathbb{R}$ and $\epsilon > 0$, there exists some $N$ such that $|x−s_N| < \epsilon$
1
vote
1answer
53 views

Convolving two functions

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 ...
0
votes
1answer
1k views

Moments of inertia of a torus

So, I can show that the moment of inertia of a torus about its axis of symmetry is $I_z = 4\pi^2\rho r^5\left[ \frac{3a}{8b} + \frac{a^3}{2b^3}\right]$ where $a$ is the distance from the axis to the ...
0
votes
1answer
73 views

Proving that these two sets are denumerable.

(a) $S_k=\{A\subset\mathbb{N}: |A|=k\}$ for $k\in\mathbb{N}$ (b) $S = \bigcup_{k=1}^\infty S_k$ Work: For (a), I am not too sure about what approach I should use. I think finding a bijective ...
0
votes
1answer
46 views

Polar graph question

Can you only graph periodic functions using polar graphing? I'm not really understanding this I guess. It you are to get all of the x and y values on a finite graph, then the original must be ...
2
votes
3answers
189 views

Manifolds and their dimension

Why are the following two sets manifolds and what are their dimensions? The set of all 2 by 2 matrices with determinant 1. The set of all inner products on $\mathbb R^3$. I have difficulty in ...
0
votes
1answer
48 views

Lebesgue measure unique on semiring?

in our lecture it was stated that the Lebesgue measure can be uniquely extended from a semiring to a sigma algebra by Caratheodory's theorem. Unfortunately, we did not show that it is unique on the ...
1
vote
3answers
37 views

Expected value for games where you can replay?

Lets just say there's a game where you roll one fair die. If you roll a 1 or 2, you pay 1. If you roll a 3 or 4, you win 2. If you roll a 5 or 6 you roll again until you get a 1, 2, 3, or 4. How ...
0
votes
1answer
98 views

Lagrange condition and second-order conditions

Given a function to minimize or maximize with equality and/or inequality constraints, I can use Lagrange multiplier and/or KKT to solve such problems. So I understand how it works. My problem is ...
3
votes
1answer
107 views

Find all integer solutions of $x^4 + 2x^3 + 2x^2 + 2x +5 = y^2$

Find all integer solutions to $x^4 + 2x^3 + 2x^2 + 2x +5 = y^2$. I'm in a dead end. I've transformed the expression in the following state: $(x^2+1)(x+1)^2 = y^2 -4$ I couldn't see anyway in ...
4
votes
2answers
2k views

Is it possible to diagonalize a singular matrix?

I have not seen anywhere written that it is impossible, but it seems impossible, so I want to check if I missed something. According to a theorem, an nxn matrix is diagonalizable if it has n ...
1
vote
1answer
214 views

How do i show that if every continuous function on $X$ is bounded, then $X$ is compact? [duplicate]

Let $(X,d)$ be a metric space. Assume every continuous function on $X$ is bounded. Prove that $X$ is compact. Well, i don't know which continuous function should i fix to start an ...
3
votes
1answer
92 views

A problem in Combinatorial Analysis

It's a question of a exercise list... Let A be a set with n points on the plane such that for each point P of A there are at least k points in A equidistant to P. Prove that $$k < \frac{1}{2} + ...
2
votes
1answer
1k views

Finding Tangent line for a Graph with the Natural Log

I'm really confused on how my professor did this problem. Any in depth explanation would be awesome. Thanks for your time.
0
votes
1answer
96 views

Calculating confidence interval - formula

I have the following problem that I get the feeling I'm mixing formulas. ...
0
votes
1answer
31 views

Let S be the set of all reals where every other decimal place, starting with the first one, contains a 1.

(so, for instance, S contains 23.101816191... but not 0.123419...) (a) Show that the cardinality of positive integers is less than or equal to the cardinality of the set S (b) Is the cardinality of ...
0
votes
1answer
59 views

How to prove maxSpec $\mathbb{C}[x,y]/(x^2)$ is homeomorphic to $\mathbb{A}^1$ as topological spaces?

I am new to algebraic geometry, and am reading the book An Invitation to Algebraic Geometry. Exercise 2.6.4 in this book asks you to prove the topological space ...
1
vote
1answer
34 views

How to prove that the product of two divergent limits is divergent.

I'm trying to prove that, if $a_n\rightarrow\infty$ and if $b_n\rightarrow\infty$, then $a_nb_n\rightarrow \infty$. Here's my proof: But what happens when $0<K<1,$ in which case $K>K^2$? ...
1
vote
2answers
332 views

How to use Fermat's little theorem to find $50^{50}\pmod{13}$?

I don't understand how to use Fermat's little theorem to find remainders e.g if we are asked to find remainder of $50^{50}$ on division by $13$, what is a and what is $p$ in the formula? Also I ...
2
votes
1answer
86 views

Estimate for weak $L^{1}$ norm

Let the weak $L^{1}$ norm on $f$ be defined by $\|f\|_{\mathrm{WL}^{1}} = \sup_{t > 0}t D_{f}(t)$ where $D_{f}(t) = \mu(\{x \in \mathbb{R}: |f(x)| > t\})$ and $\mu$ is the standard Lebesgue ...
0
votes
1answer
33 views

Possible Quadratic extensions of $\mathbb{Q}_{2}$ are 1-1 with $\mathbb{Q}_{2}^{*}/(\mathbb{Q}_{2}^{*})^{2}$

Why do they have to be units? $\mathbb{Q}_{2}[\sqrt{2}]$ is a quadratic extension but $|2|_{2}=\frac{1}{2}\neq 1$. Where do I need them to be units below? ...
0
votes
1answer
65 views

Limit of division by zero problem.

Find the limit as $x\rightarrow0$ of $1/x$. 1) infinity. 2) 1. 3) 0. 4) The limit doesn't exist. So I tried an experiment by plugging some values and I found that as I put small values, 1/that value ...
0
votes
0answers
39 views

Why an element of an order of a number field K is always an algebraic integer of K?

Let $K=\mathbb Q(\sqrt{N})$ be a number field, $\mathcal O$ be an order of $K$ (i.e. $\mathcal O$ is a subring of $K$ and $\mathcal O$ is a free $\mathbb Z$-module of rank 2). In the begining of ...
3
votes
6answers
1k views

How do I begin proving this binomial coefficient identity: ${n\choose 0} - {n\choose 1} + {n\choose 2} - {n\choose 3} + \dots = 0$

This is a homework question. I'm asked to prove the identity: $${n\choose 0} - {n\choose 1} + {n\choose 2} - {n\choose 3} + \dots = 0$$ (The sum ends with ${n\choose n} = 1$, with the sign of the ...
4
votes
2answers
195 views

Successively longer sums of consecutive Fibonacci numbers: pattern?

Consider the following: $$\begin{align} F_{n-1}+F_{n-2}&=F_n\\ F_{n-1}+F_{n-2}+F_{n-3}&=F_{n-1}+F_{n-1}\\ &=2F_{n-1}\\ F_{n-1}+F_{n-2}+F_{n-3}+F_{n-4}&=F_n+F_{n-2}\\ &=L_{n-1}\\ ...
3
votes
5answers
129 views

$\sqrt[3]{2}$ is not the root of a quadratic with rational coefficients?

How can one show that $\sqrt[3]{2}$ is not a root of a quadratic with rational coefficients? It is clear that if $\sqrt[3]{2}$ is the root of such a quadratic, then it is also the root of a quadratic ...
3
votes
2answers
119 views

Almost perfect numbers

A positive integer $n$ is called almost perfect if the sum of its divisors smaller than $n$ is $n-1$. What are all almost perfect numbers $n$ such that some power $n^k$ is also almost perfect for at ...
0
votes
1answer
57 views

Find the Axis of rotation of rotation matrix $K$ after solving $(K-I)v=0$

$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$ Find the axis of rotation for the rotation matrix $K$. This is from my previous thread click here ...
1
vote
1answer
148 views

Interior of the set {1/n}

What is the interior of the set {1/n} where n is a natural number? For me by considering the definition I think the interior is an empty set.Am I right here?
2
votes
0answers
23 views

Finding expression that runs through zeros of a function

A list plot of the zeros of the imaginary part of $2^s\pi^{s-1}\sin\bigg(\frac{\pi s}{2}\bigg)\Gamma(1-s)$ for $s=\frac{1}{2}+it$ looks like this: How would I find the expression for the line ...
1
vote
1answer
29 views

Find a vector $t \in \{x,y,z\}$ with base $\{u, v, w\}$

I don't know how to find a vector $\vec t$ that will suffice the condition: $\vec t \in \{x,y,z\}$ with bases $\{u, v, w\}$ the given vectors are: $$ \begin{array}{rcrrrrrl} u &=& [ & ...
3
votes
0answers
83 views

Unitary Farey Sequence Matrices

Take the Farey sequence $\mathcal{F}_n$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\frac1{\sqrt{|\mathcal{F}_n|}}\biggr(\exp(2\pi i k a_m)\biggr)_m $$ The dimension of ...
0
votes
1answer
38 views

Which is true about this NFA?

Let $M$ be an NFA with alphabet {0,1} that accepts every binary string. Which of the following is true? a) Every state of $M$ must be an accept state b) $M$ does not have any accept state c) The ...
0
votes
1answer
167 views

Why divide by N (length of input sequence) during IDFT?

During DFT of a input sequence of length N, we find X(k). We find inner product with a basis vector to get the coefficient: X(k) = <x[n], e[k, n]>    |  k = 0, 1, 2, ... ...
0
votes
1answer
239 views

Backus-Naur Form with automata

Parsers & compilers usually utilize deterministic finite automata to parse input. It's very easy to implement a generic DFA tool, that simulates any DFA table for example to validate input. ...
0
votes
2answers
57 views

Dihedralize Twice - dihedralize a dihedral group $D_n$

A simplest dihedral group $$D_4=C_2 \ltimes C_4$$ can be regarded as dihedralizing a $C_4$ by a semi-direct product. Q: Can one dihedralize the group $D_4$ a second time by defining $$C_2 \ltimes ...

15 30 50 per page