2
votes
1answer
172 views

Wronskian is independent to the choice of a basis

How to show that the Wronskian is independent to the choice of a basis of $V$? Let $f_1(x), \ldots, f_n(x)$ be a basis of a vector space $V$. Then the Wronskian is the determinant of a matrix whose ...
2
votes
1answer
814 views

Vertex Clique Cover and Chromatic Number

I have heard that the Vertex Clique Cover Number is equal to the Chromatic Number of the complement of a graph. But, I can't find a reference. Is this true? And, is it true for all graphs or just ...
1
vote
3answers
99 views

Is this a consequence of independence?

Suppose the real random variables $X$ and $Y$ are independent; i.e. $$\mathbb{P}[\{X \leq x\} \cap \{Y \leq y\}] = \mathbb{P}[\{X \leq x\}] \cdot \mathbb{P}[\{Y \leq y\}].$$ Does it follow that ...
0
votes
1answer
97 views

If $A\vec{x}=\vec{b}$ and $B\vec{x}=\vec{b}$ inconsistent, then $(A+B)\vec{x}=\vec{b}$ inconsistent?

The matrices $A$ and $B$ are of $n \times m$. Suppose there is a $\vec{b}$ such that both $A\vec{x}=B\vec{x}=\vec{b}$ have infinitely many solutions. Is it true that $(A+B)\vec{x}=\vec{b}$ also has ...
0
votes
2answers
169 views

Question on equivalence of inner product

How to show that if $A, B \in M_{n}(\mathbb{C})$, and $x,y\in \mathbb{C}^{n}$ then the following two conditions are equivalent, 1.) $\langle Ax,y\rangle =\langle Bx,y\rangle$ 2.) $\langle Ax,x ...
3
votes
1answer
468 views

Proving that a process is a Brownian motion

Let $B$ be a Brownian motion with natural filtration $(\mathcal{F}_t)_{t\geq 0}$ and let $\mathcal{H}_t$ be the $\sigma$-algebra generated by $\mathcal{F}_t$ and $B_1$. Define $$A_t = ...
3
votes
0answers
174 views

What's the origin of the terminology “Normalization” in commutative algebra?

Since the terminology "normal", "normalized", etc has different meanings in mathematics (some geometric in flavor, like when referring to perpendicularity) and I just read in Eisenbud's book on ...
8
votes
2answers
554 views

Interesting but elementary properties of the Mandelbrot Set

I suppose everyone is familiar with the Mandelbrot set. I'm teaching a course right now in which I am trying to convey the beauty of some mathematical ideas to first year students. They basically know ...
2
votes
1answer
427 views

poker hand probability question

8 pokers hands are dealt from a shuffled deck without replacement. a. Find the probability that at least one of the 8 hands is a heart flush(all five cards are hearts). Pr(at least one of 8 hands is ...
1
vote
1answer
532 views

Solving an equation with trig functions and two different angles

I am trying to solve this equation derived from matrix multiplication (where $a,b,c,d$ are constants): $$-a \cos(\theta) \sin(\alpha)-b\sin(\theta) ...
6
votes
1answer
459 views

Solving $n$-queens with determinants

I keep reading about a proposed method of finding solutions to the $n$-queens problem using determinants, but I can't find any specific details anywhere. Can somebody explain to me how to find ...
0
votes
2answers
194 views

Die probability question

A standard balanced 6 sides die is rolled is rolled 10 times. a. Find the probability that five of the rolls are odd numbers and five of the rolls are even numbers. P(odd) = (10 choose ...
3
votes
1answer
118 views

Variant of the Lévy hierarchy on formulas

Consider the following variant of the Lévy hierarchy on formulas : let $\Phi$ be the set of all meaningful formulas on the alphabet $\in,=,\vee,\wedge,(,),\neg,\exists,\forall$ and a countable set of ...
3
votes
2answers
1k views

Singular value decomposition of product of matrices

Given SVD(A) and SVD(B) and B is a diagonal matrix, is there a way or method to construct SVD(AB) ?
4
votes
1answer
195 views

For any irreducible variety $X$ and any point $x$ in $X$, dim$\mathscr{T}(X)_x \geq$ dim$X$, with equality holds in a dense open subset of $X$

For any irreducible variety $X$ and any point $x$ in $X$, $\mathrm{dim}\mathscr{T}(X)_x \geq \mathrm{dim}X$, with equality holds in a dense open subset of $X$. Here, $\mathscr{T}(X)_x$ denotes ...
1
vote
1answer
104 views

Another $\operatorname{arctan2}$ question

In a previous question I asked about how to apply $\operatorname{arctan2}$ to: If $\sin(\theta) = \frac{-1}{2}$ and $\cos(\theta) = \frac{\sqrt{3}}{2}$ which is found as: $\operatorname{arctan2}( ...
2
votes
0answers
86 views

Is my use of Lagrange multipliers correct?

Given $n$ positive values. $x_i \ge 1 (1\le x_i \le n)$. Their sum is $k$. $$ \sum_{i=1}^{n}x_i = k $$ Define the following value: $$ \sum_{i=1}^{n}x_i(x_i-1) $$ Now use Lagrange multipliers to ...
1
vote
0answers
190 views

Is it possible to calculate integral of $|\sin t|$ using Cauchy integral theorem (or residue theory)

Is it possible to calculate integral $$ \int\limits_0^{2\pi} |\sin t| dt \qquad (1) $$ by the complex function theory instruments, i.e. $$ \frac{1}{2\imath}\int\limits_\mathcal{C} ...
4
votes
4answers
2k views

What base is Roman Numerals?

What is the base for Roman Numerals? It starts off with unary then goes back and forth between multiples of 5 and 10.
3
votes
1answer
85 views

Reducing $Ax = u \vee Ax = v$ to $By = w$

Let $\mathbb{Z}_2 = \{0,1\}$ be the integers mod $2$. Let $A \in \mathbb{Z}_2^{m \times n}$ and $u,v \in \mathbb{Z}_2^m$. Consider the problem of determining whether there is a vector $x \in ...
1
vote
2answers
290 views

How do I generate the set of binary strings with elements that are unique under reversal?

What is the most efficient way to generate the set $S$ of unique binary strings of a certain length, $L$, s.t. all strings are unique under the reversal operation? For example, if $L = 2$, the ...
1
vote
2answers
708 views

Prove $n$ is composite when it divides $(n-1)!$

I am having trouble solving this. Any tips of how to get this proof started would be greatly appreciated. Let $n$ be a number in $\mathbb{N}$. Prove that if $n$ divides $(n-1)!$ then n is composite. ...
3
votes
2answers
111 views

Proving $S+T$ is a subspace of $\mathbb{R}^{n}$

$S$ and $T$ are subspaces of $\mathbb{R}^{n}$ and is defined as $S+T = \{v+w \mid v \in S \; and \; w \in T\}$. I need to show that $S+T$ is a subspace of $\mathbb{R}^{n}$. Instinctively, $S+T$ is ...
0
votes
1answer
205 views

In the existence of a short exact sequence, the projective dimension of $B$ is less than the larger of projective dimensions of $A$ and $C$

If there is an exact sequence of $R$-modules $0 \rightarrow A \stackrel{\alpha}{\longrightarrow} B \stackrel{\beta}{\longrightarrow} C \rightarrow 0$, then $\mathrm{pd}(B) \leq \mathrm{max}\{ ...
23
votes
6answers
5k views

Does $ \int_0^{\infty}\frac{\sin x}{x}dx $ have an improper Riemann integral or a Lebesgue integral?

In this wikipedia article for improper integral, $$ \int_0^{\infty}\frac{\sin x}{x}dx $$ is given as an example for the integrals that have an improper Riemann integral but do not have a (proper) ...
12
votes
7answers
2k views

How to prove the sum of squares is minimum?

Given $n$ positive values. Their sum is $k$. $$ x_1 + x_2 + \cdots + x_n = k $$ The sum of their squares is defined as: $$ x_1^2 + x_2^2 + \cdots + x_n^2 $$ I think that the sum of squares is ...
1
vote
1answer
154 views

Periodic solution of an ODE

I'm suck with the following question I'm asked to find all possible real values of $a$ for which the solution to the following ODE is periodic. The ODE is: $$x''=-\nabla U(x),\quad ...
2
votes
2answers
2k views

Order of product of two elements in a group

Let $G$ be an abelian group. Let $a, b \in G $ and let their order be $m$ and $n$ be respectively. Is it always true that order of $ab$ is $lcm(m,n)$? What if $m$ and $n$ are coprime to each other?
1
vote
2answers
106 views

Showing countability of $V:=\{f\in F^{\mathbb{N}}:\operatorname{supp}(f)$ is finite $\}$ and uncountability of $V'$

Let $F:={\mathbb{Z_2}}$ be the field with two elements and let $F^{\mathbb{N}}:=\{f:\mathbb{N}\to\mathbb{Z_2}:$ with usual pointwise addition and multiplication by scalers $\}$ Then define ...
6
votes
0answers
105 views

What is Birkhoff's ergodic theorem for $GL_n$?

I was wondering about the general setup of a "noncommutative" ergodic theory. Instead of measurable maps into $\mathbb R$, we should consider measurable maps into $GL_n(\mathbb R)$, and instead of ...
4
votes
4answers
337 views

Evaluating $ \int {\frac{1}{2+\sqrt{1-x}+\sqrt{1+x}}}\mathrm dx $

I would like to evaluate: $$ \int {\frac{1}{2+\sqrt{1-x}+\sqrt{1+x}}}\mathrm dx $$ $$ \frac{1}{2+\sqrt{1-x}+\sqrt{1+x}}=\frac{\sqrt{1-x}+\sqrt{1+x}-2}{2(\sqrt{1-x^2}-1)} $$ The substitution $ x ...
6
votes
4answers
14k views

Calculating the Modular Multiplicative Inverse without all those strange looking symbols

I am sure all those symbols are really easy for you guys to understand, but I would appreciate it if someone could bring it down to earth for me. How could I do this on a basic calculator? or with a ...
1
vote
3answers
281 views

Finding Speed at a Single Point

I am given this information: A question asks me to estimate the speed at point P to the nearest integer. I understand that this has something to do with the tangent line at P, but I'm not sure how ...
4
votes
1answer
175 views

which topology is used on $\mathbb{C}^ 2$ to prove that the complex addition continuous?

The title says it all. The motivation behind this question is that I'm stuck with a theorem in topology, which states that if $f,g:X \rightarrow \mathbb{K}$ (where $\mathbb{K}$ is either ...
1
vote
1answer
386 views

Number of solutions of Frobenius equation

I have one problem which needs to count the number of solution of the equation $$2x+7y+11z=42$$ where $x,y,z \in \{0,1,2,3,4,5,\dots\}$. My attempt: I noticed that that maximum value of $z$ could ...
2
votes
2answers
1k views

How is diagonal matrices a subspace of upper triangular matrices?

I'm kind of confused that the diagonal matrices is a subspace of the upper triangular matrices. Suppose I have the following matrices: $$U=\begin{bmatrix} a & b\\ 0 & d \end{bmatrix} ; \; ...
3
votes
2answers
248 views

Does $\gcd(40,80)$ = $40$ or $20$

Which of the following correct? $\gcd(x,x\times2) = x$ or $\gcd(x,x\times2) = x/2$ I am a programmer. I am new to mathematics.
3
votes
3answers
162 views

The sum of a polynomial over a boolean affine subcube

Let $P:\mathbb{Z}_2^n\to\mathbb{Z}_2$ be a polynomial of degree $k$ over the boolean cube. An affine subcube inside $\mathbb{Z}_2^n$ is defined by a basis of $k+1$ linearly independent vectors and an ...
1
vote
1answer
67 views

The fraction of k-juntas with low influences in all of the coordinates

Let $f:\{-1,1\}^n\to\{-1,1\}$ be a boolean function. Define the influence of the $i$'th coordinate of $f$ as follows: $$\operatorname{Inf}_i(f)=\Pr_{x}[f(x)\neq f(\hat x_i)]$$ where $x$ is uniformly ...
15
votes
5answers
2k views

If $a^3 =a$ for all $a$ in a ring $R$, then $R$ is commutative.

Let $R$ be a ring, where $a^{3} = a$ for all $a\in R$. Prove that $R$ must be a commutative ring. Please guide me with a proof. Thank you for your kindness.
0
votes
1answer
77 views

Rotate two vectors until one of them is “upright”

I've have two vectors A and B. I want to rotate both of them by the same angle until A is "upright" (has only a positive y component). Is it possible to calculate B without using functions like sin or ...
1
vote
1answer
133 views

What are the anti-involutions of complex projective space?

An anti-involution of $\mathbf{P}^n (\mathbf{C})$ is an anti-homography $\phi$ such that $\phi^2 = \mathrm{Id}$. Is it true that all anti-involutions are induced by anti-involutions on ...
9
votes
3answers
598 views

Square free finite abelian group is cyclic

How do I show every abelian group whose order is square free is cyclic without using the fundamental theorem of finite abelian groups? I tried something like this Let $|G| = p_1p_2...p_n$ By ...
0
votes
1answer
83 views

How does existence of an invertible linear form give info about the behavior of injective endomorphisms

The following question is related to this post: Example of a an endomorphism which is not a right divisor of zero and not onto I was told there is a simple motivation for this problem from a linear ...
2
votes
1answer
113 views

Sipser's definition of a space constructable function

I have a problem with definition of space constructable function. As I understood we use this definition just for simplification of further proofs and idea behind this definition is very clear, but ...
3
votes
3answers
175 views

Prove that $(I-T)(I+T)^{-1}$ is an involution

I have to prove that if $V$ is a finite-dimensional vector space over a field of characteristic not 2, and $T$ is an endomorphism such that $\det(I+T) \neq 0$ then $T \mapsto (I-T)(I+T)^{-1}$ is an ...
4
votes
0answers
328 views

Sum of odd prime and odd semiprime as sum of two odd primes?

How to prove that each sum of odd prime and odd semiprime can be written as sum of two odd primes $(p_1+p_2p_3=p_4+p_5)$ ? Since we know that each prime number greater than $3$ is of the form $6k\pm ...
1
vote
1answer
51 views

Two submodules $H$ and $H^*$ that are both direct factors but $H \cap H^*$ is not a direct factor

Fix a prime number $p$. I think the module I am working has a standard notation but I am using some notes that are not following any textbooks I know of. Let $J = \{ \frac{l}{p^m} + \mathbb{Z} : l ...
12
votes
4answers
2k views

Groups of order $pq$ without using Sylow theorems

If $|G| = pq$, $p,q$ primes, $p \gt q, q \nmid p-1 $, then how do I prove $G$ is cyclic without using Sylow's theorems?
1
vote
1answer
257 views

Euclidean isometries of $\mathbb{R}^3$ vs isometries of surfaces

Euclidean isometries in $\mathbb{R}^3$ are compositions of a translation and an orthogonal transformation. Each Euclidean isometry is a surface isometry that preserves length of rectifiable curves, ...

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