3
votes
2answers
142 views

What is a Theta Function?

What exactly is a theta function $$\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)= 1 + 2 \sum_{n=1}^\infty \left(e^{\pi i\tau}\right)^{n^2} \cos(2\pi n z) = ...
2
votes
1answer
21 views

How do people find the number of ways you can put together a rubiks cube?

Just curious. How do people actually find the number of ways you can put together a rubiks cube? How do you find the number of choices? Do you use the same permutation formula? Insight would be ...
0
votes
1answer
13 views

Calculate $\mathbb{E}(T^2)$ and $\mathbb{E}(\int_0^T X_s \,d s)$ for exit time $T$ of Brownian motion $(X_t)_{t \geq 0}$

Let $T$ be the exit time of from the interval $[-b,a]$ of a standard Brownian Motion $X_t$, then how would we go about calculating the following two expectations: $E[T^2]$ (and) $E[\int_0^T X_tds]$? ...
1
vote
4answers
43 views

How do I write this proof formally?

How can I formally prove that $$\max\{\lvert x+y\rvert _i \} \leq \max\{ \lvert x_j \rvert \} + \max\{\lvert y_k \rvert \}$$ Where $x,y$ are the components of a $n$-vector with $1 \leq i,j,k \leq ...
2
votes
2answers
19 views

Infinite subspaces for a vector space that cannot be spanned by a single element

If a vector space (over an infinite field) cannot be spanned solely by a single element, does it mean it has infinite subspaces? I couldn't find an example that contradicts this
1
vote
2answers
30 views

A theoretic question about cosine general solution.

I have to find the extremas of: $f(x)=x-\tan({x\over 2})$ .$(\pi\le x\le\pi)$ Last result is $\cos({x\over 2})=\pm{1\over \sqrt{2}}$. I get that: ${x\over 2}=\pm{\pi\over 4}+2\pi k$ which derives: ...
1
vote
1answer
69 views

Differential operators on the polynomial ring

Let $A$ be a commutative algebra over complex numbers. If $a\in A$ we define $m_a$ to be a linear map which sends each $x$ to $ax$. The zero map $A\to A$ is said to be a differential operator of an ...
1
vote
1answer
10 views

Help me understand where the factor 2 comes in, in this conformal mapping.

I want to "Show that the mapping $f(z) = z + \frac{R^2}{z}$ takes the two concentric circles, $|z|=R$ and $|z| = R'> R$, onto a line segment and an ellipse." (These are depicted in a figure. The ...
1
vote
1answer
10 views

distribution of distance between two points whose coordinates are normal random variables

let there be two random variables $(X_1,Y_1)$ and $(X_2,Y_2)$, where $X_1\sim N(m_1,s)$, $X_2\sim N(m2,s)$, $Y_1\sim N(n,t)$, $Y_2\sim N(n,t)$. What is the distribution of $\|(X_1,Y_1)-(X_2,Y_2)\|$?
3
votes
2answers
52 views

Proving $\left(a+\frac{2}{a}\right)^2+\left(b+\frac{2}{b}\right)^2\ge \frac{81}{2}$ for all positive real $a,b$ such that $a+b=1$

I approached this problem in two different ways, but only one was successful. I'll post the latter as an answer, while here follows the first approach: I expanded the squares: ...
2
votes
2answers
21 views

Simple equation re-arrangement

I have a simple re-arrangement of an equation which I can't seem to solve, help would be much appreciated. I'm trying to re-arrange the equation: $e^{-3t}\frac{dy}{dt} - 3e^{-3t}y = C$ where $C$ ...
1
vote
1answer
17 views

Finding coordinates of the third point of a triangle from given?

In ABC triangle we know the coordinates of A and B vertices. We also know lengths of 2 edges shown in the picture and the third edge is calculatable. What is the most efficient functon to find x3 and ...
1
vote
1answer
22 views

An integral related to the derivative of Legendre polynomials

I want to calculate the integral $$ I=\int_{-1}^{1} \Big(\frac{\mathrm{d}P_{n+1}(t)}{\mathrm{d}t}\Big) \Big(\frac{\mathrm{d}P_{m+1}(t)}{\mathrm{d}t}\Big) \mathrm{d}t $$ where $P_n(t)$ is Legendre ...
0
votes
0answers
15 views

Clocks in moving inertial frames

A clock $C$ is at rest at the spatial origin of an inertial frame $S$. A second clock $C'$ is at rest at the spatial origin of an inertial frame $S'$ moving with constant speed $u$ relative to $S$. ...
2
votes
3answers
116 views

Is it possible to solve the Zebra Puzzle/Einstein's Riddle using pure math?

A coworker of mine posted a problem in our local communication software that seems to be a simpler variation of the Zebra Puzzle/Einstein's Riddle. I know how to solve it intuitively, by using ...
-4
votes
0answers
31 views

Could someone please explain these two theorems from Topology? [on hold]

I'm trying to understand the concept of box and product topology..and these two theorems confuse me. I'd really appreciate it if someone could explain these two theorems in a simple way. Thanks!
2
votes
0answers
19 views

In how many ways can different ways can 3 children share 8 identical sweets so that each child gets at least one?

In how many ways can different ways can 3 children share 8 identical sweets so that each child gets at least one? I have tried this problem by listing all the possibilities and I got an answer of 21. ...
0
votes
2answers
26 views

Help Showing a Relation is/isn't a Partial Order

Define the relation $\le$, as $(a,b)\le(c,d)$ if and only if $a+b\le c+d$ and $a\le c$. Is this a partial order? I know it's definitely not if we remove the $a\le c$ (because then it's not ...
1
vote
2answers
12 views

What is an example of transcendental extension such that a monomorphism cannot be extended?

Let $E/F$ be a field extension and $\alpha\in E$ be transcendental over $F$. Let $\bar F$ be the algebraic closure of $F$ and $\sigma:F\rightarrow \bar{F}$ be a field monomorphism. What is an ...
1
vote
1answer
670 views

Proof that Gauss-Jordan elimination works

Gauss-Jordan elimination is a technique that can be used to calculate the inverse of matrices (if they are invertible). It can also be used to solve simultaneous linear equations. However, after a ...
2
votes
2answers
74 views

Using exclusively the definition of limit proof that $\lim_{x \to 0} \frac{x^3-2x+x}{\sin(x)} = -1$

Using exclusively the definition of limit proof that $$ \lim_{x \to 0} \frac{x^3-2x+x}{\sin(x)} = -1 $$ I have to learn how to prove limits by the delta-epsilon definition, I know how to do basic ...
1
vote
0answers
14 views

Norms and traces example

Example: Let $L=\mathbb{Q}(\sqrt{d})$ be a quadratic extention of $F=\mathbb{Q}$ with square-free integer $d$.Then, $g_{a+b\sqrt{d}}(X)=(X-a-b\sqrt{d})(X-a+b\sqrt{d})=X^2 -2aX+(a^2 -db^2),$ so, ...
0
votes
1answer
81 views

How to prove that for Brownian motion in $(a, b)$ $\mathbb{E}^x[\min(H_a, H_b)] = (x-a)(b-x)$?

i'm wondering if anyone can help me with proving the fact that for BM in the interval $(a,b)$ and with $$H_y = \inf\{t>0: X_t = y\},$$ the following is true: $$\mathbb{E}^x[\min(H_a, H_b)] = ...
2
votes
3answers
34 views

Is this permutations or combinations?

I am a bit confused. When we use the multiplicative principle are we finding the number of permutations or combinations. An example of using this principle is where I have $5$ shirts $3$ pairs of ...
1
vote
1answer
26 views

What is the meaning of this statement about complex structure?

I get confused when in papers it is said that: "Something is holomorphic (Complex, symplectic, etc ...) in some Complex structure" What is the meaning of this in general? For example in a paper ...
1
vote
0answers
10 views

create zero-order hold in matlab [on hold]

I'm trying to create a zoh of a given signal, but the plot is weird. here's what I've done: given that: $|X(w)|=0 for |w|\ge7.5\pi$ ...
1
vote
1answer
8 views

Extermal curve for specific problems?

I ran into a quiz question last month. how we can find the Extermal curve for following problem. $$ \int_1^2 \frac {\dot {x}^2}{t^3} dt $$ where $x(1)=2, \ x(2)=17$
7
votes
2answers
249 views

How to show that differential operator can be defined in terms of certain commutator operators

Let $U$ be any open subset of $\mathbb{R}^n$ (or, more general, of some smooth manifold). Define $\mathcal{D}_{-1}(U):=\{0\}$. For any two linear operators $A$ and $B$, the commutator operator $[A,B]$ ...
1
vote
2answers
25 views

Circular measure

Hi everyone, This is a question from a June 1984 cambridge past paper. I'm getting stuck with the part (c) and the 'hence show...' Please someone can help, I'd be very grateful.
2
votes
1answer
57 views

Minimize Product of Sums of Squared Distances

The Question Given two sets of vectors $S_1$ and $S_2$,we want to find a unit vector $s$ such that $$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
0
votes
1answer
367 views

Find the general solution of the PDE

Find the general solution of the PDE $ xu_x-xyu_y-y=0 $ for all $u(x,y)$ and find the parametric form of the solution of the PDE which follows the side condition $ **u(s^2,s)=s^3** $ I got part ...
0
votes
2answers
26 views

Number Theory - Multiple of $36$ problem

Let $N$ be the greatest integer multiple of $36$ all of whose digits are even and no two of whose digits are the same. Find the remainder when $N$ is divided by $1000$. $$N = \overline{abcd....} ...
2
votes
1answer
24 views

Determine if n is a prime.

Let $n$ be a positive natural number. You know the following facts about $n$ . Firstly, $n<10^{6}$ . Moreover, not a single integer $k$ between $1$ and $10^{4}$ divides $n$ . Does it ...
0
votes
3answers
45 views

If $A \cap B \cap C = \varnothing$, is one $A \cap B$, $B \cap C$ or $C \cap A$ empty too?

How do I give counter example to this? Prove or find a counter example to the following claim: For all sets $A$, $B$, $C$ if $A\cap B\cap C=\varnothing$, then either $A\cap B=\varnothing$ or ...
3
votes
0answers
24 views

Borel $\sigma$-algebra

Since the Borel $\sigma$-algebra is generated by the family of open sets, does that mean that every Borel set is essentially some countable union/intersection of open sets or a complement of open ...
0
votes
1answer
54 views

General differentials operators (Grothendieck definition) and polynomial rings

Let $A$ be an algebra over some field $\mathbb{k}$. A linear map $f:A\to A$ is said to be a differential operator of an order $\le n$ if for all $a_0,a_1,\ldots a_n\in A$ we have ...
1
vote
1answer
21 views

A problem on finding the nearest points to the origin on the intersection of two surfaces

Suppose we are to find the points nearest to the origin on the curve of intersection of the two surfaces $g^{-1}_{1}\{ 0 \}$ and $g_{2}^{-1}\{ 0 \}$, where $g_{1}: (x, y, z) \mapsto x^{2} - xy + y^{2} ...
0
votes
5answers
55 views

Mathematical induction

Prove that $9$ divides $n^3 + (n+1)^3 + (n+2)^3$ where $n$ is a nonnegative integer. I have seen many questions on this site that contain the answer to this problem and I already know the solution, ...
0
votes
0answers
10 views

How to make parity-check matrix

I'm working on exercise I found in book. There is a coding matrix, which elements are given as logarithm's values. Here is a matrix. $$G_c =\left( \begin{matrix} 2 & 1 & 1 & 0 & 1 ...
2
votes
0answers
16 views

A generalization of “any countable limit ordinal is the union of a sequence of increasing ordinal”

Using the fact that every countable ordinal is isomorphic to a closed subset of $Q$, I find out that any countable limit ordinal is a union of s sequence of increasing ordinal. Now I'm trying to ...
17
votes
3answers
1k views

Mathematically, why was the Enigma machine so hard to crack?

Mathematically, why was the Enigma machine so hard to crack? In laymen terms, what was it exactly that made cracking the Enigma machine such a formidable task? Everything I have seen about the ...
0
votes
1answer
14 views

What is the physical meaning of 2 nodes being same while fitting an interpolating polynomial?

When we are trying to find out constants for Newton's interpolating polynomial, we use divided difference method to find the constants. Then we have Hermite-Genocchi formula to find those constants ...
13
votes
2answers
2k views

Proving the “freshman's dream”

$(x+y)^p = x^p + y^p$ holds in any field of characteristic $p$. However all the proofs I have seen use induction and some relatively nasty algebra despite how fundamental this fact seems. What is the ...
0
votes
0answers
13 views

Isn't $\gamma_{n,g}=0$ if $g>n-3$?

Let $\gamma_{n,g}$ the number of ways of putting down x $y's$ on the intervall $[0,n-1]$ with the $y's$ separated by at least two $z's$ and let $\gamma_n=\sum_{g=0}^{n}\gamma_{n,g}$. Maybe a stupid ...
13
votes
1answer
357 views

Why does Group Theory not come in here?

Here is a list of questions that I find quite similar, for the one and only reason that they all show much "symmetry". Which is at the same time my problem, because I don't have a very precise notion ...
0
votes
1answer
12 views

modular problem in arithmetic

hello can someone please help me to solve this problem: 2008 mod 71, 9 square mod 41, 34 suare mod 71 b)determine all a and b that verify a square mod 41=40 b square mod 71=20 ...
0
votes
0answers
10 views

Scaling in the parameter of the rotation matrix

For the distance function $(\Delta s)^2 = (\Delta r)^2 + (r \Delta \theta)^2$, the rotation matrix is $R(\theta) = \begin{pmatrix} cos\ \theta & - sin\ \theta \\ sin\ \theta & cos\ \theta ...
2
votes
1answer
19 views

Prove that if $R$ is a principal ideal domain, then either $R \cong S$, or $S$ is a field.

Let $R$ and $S$ be integral domain and suppose that $\phi: R \rightarrow S$ is a surjective ring homomorphism. Prove that if $R$ is a principal ideal domain, then either $R \cong S$, or $S$ is a ...
1
vote
2answers
36 views

approximate $\int_{0}^{0.5}{\frac{\sin(x)}{x}}dx$

By using Maclaurin series, approximate the value of $$\int_{0}^{0.5}{\frac{\sin(x)}{x}}dx$$ to within an error $0.0001$, where $x$ is in radians. My attempt: Since we know the Maclaurin series of ...
-1
votes
1answer
21 views

Approximation for the Summation of Sequence of Powers of Sines Functions.

Let $z_1,z_2,...,z_m$ be real numbers such that $0<z_1,z_2,\ldots,z_m<\pi/2$, $z_1>z_2>...>z_m$ and $n$ an integer such that $n>0$. Prove that: ...

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