0
votes
1answer
9 views

How can I prove that $g\cdot H \cdot g^{-1}$ is also finite and has the same number of elements that $H$?

Suppose $G$ a group and $H$ is a finite subgroup of $G$ also $\forall g \in G$ the set $g\cdot H \cdot g^{-1}=\{ g\cdot h \cdot g^{-1} : h\in H\}$, is a subgroup of $G$. Prove that $g\cdot H ...
0
votes
0answers
20 views

Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...
0
votes
1answer
4 views

Assigning a specific value to components of a vector

So far, I've run into this twice and I'm not exactly sure how to make this connection myself, but in this case, I've been asked to find the dot product of (i+j+k) dot (3i+2j-5k) I understand ...
3
votes
1answer
18 views

How to draw the 5 dimensional hypercube graph with 56 edge crossings?

I'm probably doing something stupid but I can't seem to think of a way to draw $Q_5$ with $cr(Q_5) = 56 $. In this paper the author says drawing a hypercube graph with $\leq56$ edge crossings is easy ...
2
votes
2answers
48 views
+100

solving 2nd order pde with dirac delta

I want to find the functional form of the Green function G(x,t) for a parabolic differential equation: $$ ...
5
votes
0answers
54 views
+50

Siegel's Theorem for Genus 0 Curves over Function Fields

Let $V$ be a quasi-projective algebraic curve over a field $k$. Are there finitely many morphisms from $S$ to the triply-punctured sphere $\mathbb{P}^1 -\{0,1,\infty\}$? This is true if $k = ...
1
vote
3answers
59 views

Is there a name for this type of expression?

Forgive me if this seems like a silly question. I know that the following expression is an example of a polynomial: $a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$ but I am wondering if there is a ...
6
votes
1answer
440 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
2
votes
1answer
34 views

Proving a Set is a Vector Space

for each $\left(x_1,y_1\right)\,,\, \left(x_2,y_2\right)$ that is an element of $\mathbb R$x$\mathbb R$ define $$(x_1, y_1) + (x_2, y_2) = (x_1 +x_2 + 2 , y_1 +y_2 -3)$$ And, for each $(x,y)$ that is ...
0
votes
0answers
7 views

Question about usage of $\leq$ in definition of Nash equilibrium

Quick definition: Given $g$, a strategy N-tuple $u^* = (u_1^*,...,u^*_N)$ is said to be a Nash equilibrium if: $$J_i(u_i^*, u^*_{-i}) \leq J_i(u_i, u^*_{-i}), i \in N$$ where $J$ is ...
0
votes
1answer
101 views
+100

Product of 2 dirac delta functions

Find the functional form of the Green function G(x,t) for a parabolic differential equation (i.e. heat diffusion etc): $$ ...
0
votes
1answer
41 views
+50

What rotation rules can be applied to stacked cubes to make a 3D spirograph?

If you arrange building blocks for example toy cubes so that every next cube is tilted over its base by 30 degrees and rotated to it's right by 12 degrees, it would wind through space in a helix ...
0
votes
0answers
17 views

Understanding $\Theta$-notation rigorously

Let $f$ and $g$ be functions on $\mathbb{N}$. If $f(n) \in \Theta(g(n))$, we say that (for sufficiently large input) "the function $f(n)$ is equal to $g(n)$ to within a constant factor". What does ...
0
votes
0answers
8 views

Combinatorial Convex Optimization: Russian paper

I'm looking for an electronic version of the paper: David Yudin and Arkadi Nemirovski. Informational complexity and effective methods of solution of convex extremal problems. Economics and ...
0
votes
0answers
14 views

How to prove this decomposition

There are two vectors l=$(l_1,l_2)^T$, m=$(m_1,m_2)^T$, and a symmetric matrix S=$\begin{bmatrix}s_{11}&s_{12}\\s_{12}&s_{22}\end{bmatrix}$. Then, ...
0
votes
0answers
7 views

Abel inversion where axisymmetric function is multiplied by $cos(\phi)$

I have a problem seems similar to Abel inversion, but the axisymmetric function is multiplied by $\cos{\phi}$, making the integrand non-axisymmetric. Here is a picture of the problem: Each chord is ...
1
vote
5answers
50 views

Dartboard puzzle.

Given a dartboard of radius r and infinite darts.How many minimum darts you need to throw so that you can be sure that the next dart you throw is strictly less than r distance from some previous dart? ...
0
votes
2answers
25 views

Solving for a Limit Given a Limit

$$ \text{Given}\; \lim_{x \to 1} \frac{f(x)-4}{x-1} = 10, \;\text{evaluate}\; \lim_{x \to 1} f(x) $$ I'm wondering if anyone can give me some tips on how to approach this problem. I ...
4
votes
1answer
16 views

Why do polynomial regressions have larger variance at the end?

In reading the book "An Introduction to Statistical Learning with Applications in R", I came across this graph: It shows that the point-wise variance is larger at the ends of the regression curve. ...
0
votes
0answers
8 views

Restriction over pdf such that an integral inequality holds $\int_{-\infty}^{+\infty}\left(F(x)-\frac{2}{3}\right)xf(x)dx\geq 0$

Let $f(x)$ be a pdf in $(-\infty,+\infty)$ and $F(x)$ it's cdf. Assume both are smooth. I need to find restrictions over the pdf such that the following inequality holds: ...
0
votes
0answers
8 views

Question related to Square Integrable Martingale

Let $X_n$ and $Y_n$ be martingales with $EX_n^2<\infty$ and $EY_n^2<\infty$ for all $n$. Show that $EX_nY_n-EX_0Y_0=\sum_{m=1}^nE(X_m-X_{m-1})(Y_m-Y_{m-1})$. I tried to expand the right ...
1
vote
0answers
25 views

does the equivalence class of an element in a set is the set itself?

what does equivalence class mean? I am trying to understand I think that I am a little confused so let's take this example: Suppose that we have the relation $2x+3y$ is a number less than or equal ...
0
votes
0answers
43 views

Elliptic curve is self dual.

How to prove $E[p^\infty] \cong Hom ( T_E, \mathbb{Q}_p/\mathbb{Z}_p(1)) $ where $T_E$ denotes the Tate module of $E$ ?
2
votes
0answers
19 views

Is it true that $A\cong B$ implies $A = B$ when $A$ and $B$ are ordered structures

In Immerman's book "Descriptive complexity" he says that $A \cong B$ implies $A = B$ when $A$ and $B$ are totally ordered structures. See: http://i.stack.imgur.com/BjXKE.png (Descriptive ...
3
votes
5answers
50 views

Intuitive explanation for p ∨ q → r ≡ ( p → r) ∧ (q → r)

Although, it is possible to prove the above equivalence using truth tables, I don't know how to prove it without using truth tables.Can someone explain it in plain english?
1
vote
0answers
22 views

If a linear operator between two normed linear spaces is continuous at one point, then it is continuous at all points.

Let $f : \langle V_1, \|\cdot\|_1\rangle \to \langle V_2, \|\cdot\|_2\rangle$ be linear. Then if $f$ is continuous at some $v \in V_1$, then it is continuous on all of $V_1$. Without appealing to ...
3
votes
0answers
13 views

Solvable non homogeneous Sturm-Liouville with non homogeneous boundary conditions

I was just presented with this problem in my PDE Methods course which involves a non homogeneous Sturm-Liouville problem, which states as follows: Find the conditions under which the following SL ...
1
vote
1answer
41 views

If $f$ is differentiable and $f'$ is bounded then relation between upper sum , lower sum and the integral

Let , $f:\mathbb R\to \mathbb R$ be a differentiable function such that $f'$ is bounded. Given a closed and bounded interval $[a,b]$ and partition $P=\{a=a_0<a_1<\cdots <a_n=b\}$ of $[a,b]$ . ...
0
votes
2answers
15 views

An 11-gon with complex numbers

Let $A_1 A_2 \dotsb A_{11}$ be a regular $11$-gon inscribed in a circle of radius $2$. Let $P$ be a point, such that the distance from $P$ to the center of the circle is $3$. Find $[PA_1^2 + PA_2^2 ...
0
votes
0answers
37 views

The sum of $\frac{k^{2}}{k-1}$ from 6 to 12

Can you evaluate this sum by using the properties of the sigma notation ? Or I must develop this and evaluate them one by one ?
4
votes
1answer
173 views

A theorem from the theory of groups

Let $K$ be a (not necessarily normal) subgroup of the group $G$ : $K < G$ A fixed element $g\in G$ can act, from the left, on all elements of $G$, thus generating a bijection of $\,G\,$ onto ...
0
votes
0answers
16 views

Convergence of sum of nets

In an arbitrary topological vector space $S$, if I have two nets ${x_\alpha}$ and ${y_\alpha}$ such that $x_{\alpha} \to x$ and $y_{\alpha} \to y$, can we say that $x_{\alpha} + y_{\alpha} \to x + ...
1
vote
1answer
38 views

Generating Series and Recurrence Relation and Closed Form

We have the following recurrence relation: $b_n=2b_{n-1}+b_{n-2}$ and initial conditions $b_0=0, b_1=2$ I use the generating series method to solve as following: Let ...
0
votes
1answer
33 views

How many distinct roots $ax^5+bx^3+cx+d$ has

$a,b,c>0$ How many distinct roots $ax^5+bx^3+cx+d=0$ has? question doesnt clarify which kind of root it has. and I dont understand why the question didnt say 'may has' . because by ...
5
votes
3answers
36 views

Intersections of Planes, Points…

I'm in sixth grade and learning geometry. Can someone tell me if I'm correct? The intersection of a point and a point is a point. The intersection of a point and a line is a point. The intersection ...
2
votes
2answers
22 views

Probability that two sets do not intersect

I'm trying to understand this simpler problem so I can apply the process to a more difficult homework problem. Let $U$ be a set with $n$ elements. Select $2$ independent random subsets $A_1, A_2 ...
5
votes
1answer
46 views

From Primitive Recursive to Recursive by Iterating over more than one Argument?

Is the only way a function can be recursive and not primitive recursive by growing faster than primitive recursion allows (as with Ackerman's function)? If so, then consider the following. Primitive ...
-5
votes
2answers
76 views

Prove $R$ is a finite ring [on hold]

Suppose that $R$ is a ring and $D(R)$ is the set of all elements $x\in R$ such that there exist $b,c\neq 0$ with $xb=cx=0$. Prove that if $1<|D(R)|<\infty$ then $|R|<\infty$. ($|S|<\infty$ ...
9
votes
2answers
219 views

Is transfinite induction needed to remove all the elements from an uncountable set?

Given any uncountable set S, would I need to use transfinite induction to prove if I remove single elements recursively, I will be left with the empty set? It seems like this can be thought of as an ...
0
votes
1answer
9 views

Growth of Fourier coefficients of piecewise linear function

Suppose $f$ is a periodic continuous piecewise linear function. What can be said about the growth (or decay, rather) of the Fourier coefficients $\hat f(n)$ as $n\to\infty$, other than the ...
1
vote
0answers
18 views

non-countable subset of $\mathbb 2^{\mathbb Z}$ with finite pairwise intersection. [duplicate]

Does a non countable subset of the power-set of $\mathbb Z$ exist so that the intersection of any two elements is a finite set? If we ask for the sets to be pairwise disjoint then the answer is a ...
14
votes
7answers
384 views

Intuition for a physical real line vs. a physical “hyperreal line”

As a mathematical structure, I have no problem with the hyperreals. But I came across the following from Keisler's book "Elementary Calculus: An Infinitesimal Approach". "We have no way of knowing ...
6
votes
4answers
114 views

Can we find uncountably many disjoint dense measurable uncountable subsets of $[0,1]$?

Can we find uncountably many disjoint dense measurable uncountable subsets of $[0,1]$? Obviously we may as well assume all the subsets have measure $0$. If I didn't specify the subsets were ...
1
vote
1answer
355 views

Binary constraint integer programming problem

Hi I have a question to the folowing question: Explain how to use integer variables and linear inequality constraints to ensure: A) let x and y be integer variables bounded at 1000. How can you ...
11
votes
2answers
189 views
+200

Smallest integer $k$ so that no Sudoku grid has exactly $k$ solutions

Inspired by this question, consider hints on a Sudoku board. A regular puzzle has a unique solution. It is clear that there are puzzles with 2 or 3 solutions, and therefore, I guess, puzzles with say ...
1
vote
0answers
15 views

Proof of conformal mappings onto polygons (Stein)

Recently I am reading Stein and Shakarchi's Complex Analysis and I find a great difficulty in understanding the proof of theorem 4.6 in Chapter 8 (p.242-244), which talks about conformal mappings onto ...
0
votes
3answers
55 views

recurrence relations and generating functions - I need a hint

I need to find "closed form" to the recurrence relation given by: $a_{n+1} = \sum_{k=0} ^ {n} k a_{n-k}$ and $a_0 = 1$. I have tried using generating functions but it is no good. Any help would be ...
0
votes
0answers
20 views

Lower bound for $\Pi(n)$ - viability of probabilistic theory

Can somebody check the validity of my arguments below, and tell me why its wrong or right? Consider the sequence of non-negative integers. Let $a_0=0, a_1=1, ..., a_i=i,...$ Divisiblilty of $a_i$ ...
2
votes
1answer
52 views

How to prove that G is a cyclic Group?

Suppose $G$ is a finite Abelian group and, $\forall\ n\in \mathbb{N} $, there exist at most $n$ elements in $G$ which satisfy $x^n=1$. Prove $G$ is cyclic. Thanks for your help.
1
vote
1answer
21 views

Using a linear function as a routine to determine a matrix

Let $F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ be a linear function, i.e., $$F(\alpha x + \beta y) = \alpha F(x) + \beta F(y)$$ Suppose you are given a routine that returns $F(x)$ given any ...

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