3
votes
1answer
123 views

Wealth indicator function for bidder agent logic

I want to create a wealth indicator function used by the logic of a bidder agent, that tells the agent if he's rich (in comparison to others). Given: Total number of competitors $n$ Amount of all ...
3
votes
2answers
377 views

Inverse image of prime ideal in noncommutative ring

We say an ideal $P\neq R$ in a ring $R$ is prime, if for any two ideals $I,J$ of $R$ the following implication holds: if $IJ\subseteq P$ then $I\subseteq P$ or $J\subseteq P$. If $f\colon R \to S$ ...
2
votes
1answer
111 views

Proving that Bombieri's Theorem implies Linnik's theorem

I'm stuck on a line in the proof of Bombieri implies Linnik, where Bombieri: For primitive $\chi$ mod $q$ with $q \leq T$ we define $$N(\alpha, T; \chi)=\#\{\rho=\beta+i\gamma \;:\; ...
5
votes
2answers
335 views

Examples of non-cyclic group with a cyclic automorphism group

In introduction to algebra we got the exercise: Let $G$ be a group. Show that when $\operatorname{Aut}(G)$ is cyclic $G$ is abelian. This doesn't make that much trouble. Denote the center (all ...
2
votes
3answers
11k views

Generate all possible combinations of 3 digits without repetition

It's possible to generate all possible combinations of 3 digits by counting up from 000 to 999, but this produces some combinations of digits that contain duplicates of the same digit (for example, ...
4
votes
3answers
175 views

Convergence of $\sum \frac{a_n}{(a_1+\ldots+a_n)^2}$

Assume that $0 < a_n \leq 1$ and that $\sum a_n=\infty$. Is it true that $$ \sum_{n \geq 1} \frac{a_n}{(a_1+\ldots+a_n)^2} < \infty $$ ? I think it is but I can't prove it. Of course if $a_n ...
1
vote
2answers
175 views

Using mathematical induction to prove an identity related to combinatorics

Using Mathematical induction on $k$, prove that for any integer $k\geq 1$, $$(1-x)^{-k}=\sum_{n\geq 0}\binom{n+k-1}{k-1}x^n$$ How should I proceed? The tutorial teacher attempted this question and ...
1
vote
1answer
123 views

Find the Cramer-Rao bound for an unbiased estimator of $b^2$

$X$ is a RV with pdf $f(x,b) = \frac{x}{b^2} \exp \{-\frac{x^2}{2b^2} \}$ I've got two different estimates: $\hat{b^2} =\frac{2}{\pi} (\frac{1}{n} \sum_{i=1}^n X_i)^2 $ using MME, and $\hat{b^2} = ...
1
vote
1answer
69 views

How are these inequalities simplified?

How does this: (a > b && a > c && b <= c) || (a > b && a <= c && b < c) simplify down to this: ...
0
votes
2answers
59 views

Find the relation between the dimension of the nullspace of $A$ and $A^t$

Let $A$ be a $n \times n$ matrix, what is the relation between the dimension of the nullspace of the homogeneous system of $A$ and the one of $A^t$?
3
votes
2answers
516 views

Analytically determine that $\arctan x$ is an odd function

Without producing the maclaurin series for $\arctan x$, how would determine whether it was odd or even?
2
votes
0answers
63 views

How to compress a linear operator and have the lossless composition property.

Consider a linear operator on $\mathbf{R}^n$ represented by a square matrix of size $n \times n$, call it $A$. The matrix acts on a row vector, call it $x$ and returns a row vector, call it $x'$, so ...
3
votes
2answers
47 views

Partial cycles in projective resolutions of square-free algebra

Short version: Over a square-free algebra must every projective resolution of a simple module eventually terminate or contain a shift of itself as a direct summand? I suspect not, but have not ...
3
votes
0answers
116 views

Problems sampling from a $pdf$ over $SO\left(3\right)$

I have a probability density function over $SO\left(3\right)$, which I am trying to sample from. The $pdf$ is given as a generalized fourier series: $$ f\left(\omega,\theta,\phi\right)=\sum ...
6
votes
2answers
421 views

Prove that semi-simple rings are Dedekind-finite

Just to be consistent with the terminology, let me define the words I'm using. A module $M$ is simple if it has no proper non-zero submodules, and semi-simple if it can be written as a direct sum $M ...
1
vote
1answer
81 views

Prime and semiprime ideals of $A=T_3(D)$, the ring of $3\times 3$ upper triangular matrices over $D$

Let $D$ be a division ring. Could anyone tell me which are the prime and semiprime ideals of $A=T_{3}(D)$, where $A=T_{3}(D)$ the ring of $3\times 3$ upper triangular matrices with coefficients in ...
3
votes
1answer
242 views

Does the splitting principle define chern classes for vector bundles if they are known for line bundles?

Suppose we have definied chern classes $c_i$ for line bundles for all $i\geq 0$. Let $E\to B$ be a rank $r$ vector bundle. By the splitting principle there exists a map $f:Y\to B$ such that $f^*E$ is ...
1
vote
0answers
55 views

Construction binary tree

First let $\mu$ be the induced distribution of the random variable $X$ on $(\mathbb{R},\mathcal{B})$ and denote $EX=m$. We also define for all $A\in G_{n+1}$ and $\omega\in X^{-1}(A)$ ...
3
votes
1answer
617 views

Calculate the Riemann Stieltjes integral

This is not a homework question. It is a past exam question and I would appreciate some step by step help, as I never understood this concept in class. Let $\alpha(t) = n^2$ for $t\in[n,n+1).$ ...
1
vote
1answer
213 views

Anonymous graphs and graph embeddedness

What are anonymous graphs, what is graph embeddedness, and how do they relate to each other? Very confused - I could not find short answer. Thanks.
3
votes
2answers
97 views

How to solve $at + b = 0 \pmod {(a-t)}$?

Is there any way (except trying for $t=0,1,2,\ldots,a-1)$ to solve the following equation for $t$ when $a$ and $b$ are known? $$ at +b = 0 \pmod{(a-t)} \text{ with } a,b,t \in N $$
3
votes
2answers
221 views

Does this polynomial factorize further?

I just did a national exam and this question was in it; I am convinced this does not work: Given that $(x - 1)$ is a factor of $x^3 + 3x^2 + x - 5$, factorize this cubic fully. My attempt 1 | ...
9
votes
3answers
576 views

How do I know when a form represents an integral cohomology class?

Suppose $M$ is an $n$-dimensional manifold, and $\omega \in \Omega^p(M)$ is a closed $p$-form. Moreover, assume that $d\omega = 0$, so that it represents a de Rham cohomology class. I would like to ...
2
votes
4answers
346 views

Is the set of surjective functions from $\mathbb{N}$ to $\mathbb{N}$ uncountable?

I want to use Cantor's diagonalisation argument to prove that the set S of surjective functions of the form $\Bbb{N} \to \Bbb{N}$ is uncountable. The normal procedure is creating a matrix and filling ...
0
votes
1answer
123 views

Is a series (summation) of continuous functions automatically continuous?

I'm being asked to show that a given series (of rational functions) converges uniformly on a given disc, and then and asked to use this fact to show that integrating its limit function (i.e. a ...
22
votes
6answers
746 views

Why is the Derangement Probability so Close to $\frac{1}{e}$?

A derangement is a permutation of $\sigma$ of $\{1,2,3,\dots,n\}$ such that $\sigma(i) \neq i$ for every $i$. A common application of inclusion/exclusion in undergraduate combinatorics and ...
2
votes
1answer
76 views

Lipschitz constant for $F(t,y,z)=(z,f(y,z)\sin t)$

Let $f\in C^1(\mathbb R^2,\mathbb R)$. Prove that all solutions for $x''=f(x,x')\sin t$ such that $x(0)=x(2\pi)$ and $x'(0)=x'(2\pi)$ have period $2\pi$. I'm in the process of solving the above ...
0
votes
1answer
42 views

Can't establish a lower bound on a supremum

I have a sequence of functions $f_{k,j}:[0,1]\to\mathbb{R}$ defined by $$f_{k,j} = k^{\frac{1}{p}}\chi_{(\frac{j-1}{k},\frac{j}{k})},$$ for all $k\geq 1,1\leq j\leq k$. This serves as an example of ...
0
votes
1answer
51 views

About continuous functions and aritmethic progression

I've try solve this question, but I haven't sucess... The problem is the following: A continuous functions $f:[a,b]\rightarrow \mathbb{R}$ assume positive and negative values in its domain, show ...
1
vote
1answer
74 views

Bolza example like Question

I have to find $u$ minimizing $\int_0^1 F$ with $F(x,u(x),u'(x)) = (1-(u'(x))^2)^2+(u(x))^2$ with $u(0) = 0$ and $u(1) = 1$. I'm relatively new to CoV and got told i should try ...
7
votes
2answers
143 views

A simple 2 grade equations system

If we have: $$x^2 + xy + y^2 = 25 $$ $$x^2 + xz + z^2 = 49 $$ $$y^2 + yz + z^2 = 64 $$ How do we calculate $$x + y + z$$
3
votes
4answers
175 views

How to prove that $A\cap B\subseteq C$ and $A^c\cap B\subseteq C$ imply that $B\subseteq C$?

How do you solve this problem?? Suppose that $A\cap B\subseteq C$, and $A^c\cap B\subseteq C$. Prove that $B$ is a subset of $C$. I don't know where even to begin Can anyone help? Thank you
2
votes
0answers
57 views

Solve the special integral

I want to solve a integral which contains a shift version $$\int^{\infty}_{c}N [(1-e^{-1/t})]^{N-1} \frac{-1}{(t-c)^2}e^{-1/(t-c)}dt$$ This kind of integral has the form of normal integral $$ \int ...
16
votes
1answer
863 views

Can you raise $\pi$ to a real power to make it rational?

We're all familair with this beautiful proof whether or not an irrational number to an irrational power can be rational. It goes something like this: Take $(\sqrt{2})^{\sqrt{2}}$ If it's rational, ...
0
votes
0answers
98 views

Variation of 3SAT is in NP-Complete

Consider the problem of "K-3SAT", a variation of 3SAT: Given a 3CNF formula O and an integer k, the machine determines whether the formula O has a satisfying assignment in which at most k variables ...
1
vote
3answers
315 views

Please help with this probalility problem.

Vince buys a box of candy that consists of six chocolate pieces, four fruit pieces and two mint pieces. He selects three pieces of candy at random without replacement. Calculate the ...
1
vote
0answers
46 views

expanded geometric series?

I'm having some issues with the following series $$ \sum_{n \geq 0} n^p r^n $$ for a fixed positive integer $p$ and some real $r > 0$. Is there any way to avoid going through linear combinations ...
3
votes
0answers
217 views

Proving identity involving sum

I'm stuck trying to prove the following identity: $$\displaystyle\sum_{i=j}^k\frac{(-1)^{i+j}{k+1 \choose i}{i \choose j}(4S-i-k-3)!(4S-2i-1)}{(4S-i-j-1)!} $$ $$=\frac{(-1)^{j+k}(4S-2k-3)!{k+1 ...
2
votes
2answers
120 views

Rouché Theorem to calculate the number of zeros

How can I calculate the number of zeros of $\cos z+3z^3$ using the Rouché Theorem?
3
votes
0answers
74 views

Is a simplicial subcomplex of a contractible simplicial CW-complex also contractible?

Let $Y,X$ be simplical CW-complexes (By that I mean functors $\Delta^{op}\to CW$. In other words, for each natural number $n$ there exist CW-complexes $X_n$ and $Y_n$ and there are face and degeneracy ...
5
votes
2answers
59 views

number of all inconstant maps f from A to A [duplicate]

Let $ A=\{1, 2, 3,..., n\}$. Find the number of all nonconstant maps $f: A \rightarrow A$ for which $f(k) \le f(k + 1)$ and $f(k) = f(f(k + 1))$ for $k = 1, \dots, n-1$..
1
vote
2answers
73 views

Binomial coefficient series

I'm practicing for my maths term test mainly on binomial coefficients. I can't seem to find out how to prove the following identity. Any advice? $$ \sum\limits_{k=1}^n (-1)^{k+1} k{{n}\choose k} = 0 ...
2
votes
3answers
101 views

General solution of differential equation of order 3

Please ,how to find that the general solution of $u'''(t)=e(t) , t\in [0,1]$ is given by $u(t)=c_0+c_1t+c_2 t^2 +\frac12 \int_0^t (t-s)^2 e(s) ds$ $e:(0,1)\rightarrow \mathbb{R}$, and $e\in ...
2
votes
3answers
129 views

$(x+2)\cos\frac1{x+2} - x\cos\frac1x > 2$ for $x\in[1,\infty)$

Let $$f(x) = x\cos\frac{1}{x}$$ for x in $[1,\infty)$ Now I need to prove or disprove the difference $$f(x+2) - f(x) > 2$$ for all $x$ in the domain. I tried a lot but I don't seem to be getting ...
1
vote
2answers
152 views

How do I divide a set of data samples which follow a logarithmic distribution?

I'm working for the first time with Logarithmic distribution. I have a set of samples which follow logarithmic distribution. I extracted the maximum and the minimum values from the set and defined the ...
1
vote
1answer
62 views

Is restriction of scalars well-defined on subspaces?

Let $K/k$ be an extension of fields and let $v_1,\ldots,v_r,u_1,\ldots,u_r\in k^n$. If the span of the $v$'s over $K$ equals the span of the $u$'s over $K$, must the two spans also be equal over $k$? ...
18
votes
1answer
230 views

Need help with $\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx$

I need help with solving this integral: $$\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx,$$ where $\text{Li}_{s}(z)$ is the polylogarithm.
2
votes
0answers
107 views

quick integration notation question

What exactly does the base of this integral sign refer to? $$\int\limits_{[0\to 1]}$$ Is that an arbitrary contour from the point $0$ to the point $1$? (this is complex analysis) ps. Here's the ...
1
vote
1answer
64 views

Convex functions on real vector spaces

So I'm trying to solve the following problem, Suppose that $f$ is a non-zero convex real-valued function on a real vector space $V$ with $f(0) = 0$ Show that there is a linear functional $g$ on $V$ ...
4
votes
1answer
294 views

Integral basis for a number field

I need some help in solving the following problem: Suppose $K$ is a number field and $K=\mathbb{Q}(\theta)$ where $\theta\in\mathfrak{O}_K$, the ring of integers of $K$. Now among the elements in ...

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