1
vote
1answer
58 views

Direct sum of $3$ subspaces

$V_1$,$V_2$,$V_3$ are subspaces of vector space $V$. How to prove that if $V_1 \cap \left(V_2+V_3\right) = V_2 \cap \left(V_1+V_3\right) = V_3 \cap \left(V_2+V_3\right)=\{0\}$ so $V_1\oplus V_2 ...
0
votes
1answer
58 views

Which math discipline should i learn to become familiar with rewriting equations?

In my self study of calculus, I've found that there are examples in the books i read where the author rewrites an equation or expression either as part of a logical step in a proof, or to simplify it ...
3
votes
2answers
123 views

Give an example of a non-abelian group G containing a proper normal subgroup N such that $G/N$ is abelian.

Give an example of a non-abelian group G containing a proper normal subgroup N such that $G/N$ is abelian. I KNOW THERE IS A QUESTION OF THE SAME NAME. However, I need more involved assistance. My ...
1
vote
4answers
229 views

Determining number of lattice paths

Question: Determine the number of lattice paths from $(0,0)$ to $(6,6)$ that take steps in $ \{\ (1,0) , (0,1) \}\ $ that do not go through the point $(3,3)$. I'm not sure what my professor means by ...
1
vote
1answer
124 views

Orthogonal matrices, their determinant and eigenvalues

Once again! Let What to do? Find eigenvalues for $A, B, AB, BA$. How I want to do this: $A = \begin{pmatrix} cos(a) & -sin(a) & 0 \\ sin(a) & cos(a) & 0 \\ 0 & 0 & 1 ...
1
vote
2answers
54 views

Conditional Expectation with transformation

Let $Y_1, Y_2,\ldots, Y_n$ be independent $N(0,1)$ random variables. Define $ X_i =\sum_{j=1}^n c_{i,j}Y_j$, $i=1,2,\ldots,n$, where $c_{i,j}$ are real constants. Show that $E(X_i\mid ...
1
vote
1answer
89 views

Improper integral and the ordinary differential equation

How does one calculate a derivative from improper integral like this? $$\int_t^{\infty}\frac{\sin(x-t)}{x}dx\qquad\ $$ It's been said that this particular integral (as a function) satisfies the ...
14
votes
3answers
6k views

The Best Strategy and Highest Possible Score for the “Threes!” Game.

[There's still the strategy to go. A suitably robust argument that establishes what is statistically the best strategy will be accepted.] Here's my description of the game: There's a $4\times 4$ ...
6
votes
1answer
144 views

Cohomological Whitehead theorem

Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...
1
vote
1answer
54 views

Solving differential equation, given initial condition

I'm stuck on this problem: $$\frac{dy}{dx}=\frac{\ln(x^3(y^2+1))-2\ln(\sqrt{(y^2+1)}(x))}{yx}$$ given y(1)= 2. I tried separating the variables using log rules to get this: $$ yx\frac{dy}{dx} = \ln ...
0
votes
1answer
29 views

Compute $S_3$ acting by conjugation on the set $X$ of $6$ subgroups of $S_3$

I know that the subgroups of $S_3$ are $\{e\}$, $\langle(12)\rangle$, $\langle(13)\rangle$, $\langle(23)\rangle$, $A_3$, and $S_3$. What I also know is that conjugation is $C_g(H) = gHg^{-1}$. Thus in ...
2
votes
1answer
92 views

$\exists f:\mathbb{R}\rightarrow \mathbb{R},$ continuous, non-constant, with uncountably many extrema?

I couldnt think of any; by intuition I don't think any can exist, but I can't figure out how to prove it. If it existed then the set of extrema would have to be uncountable but I think this might ...
4
votes
2answers
111 views

The Affine Tangent Cone

I'm failing to see how exactly is the tangent cone at a singular point on a curve picking out all the different tangent lines through this singular point (say the origin in $\mathbb{A}^2$)? Could ...
3
votes
2answers
77 views

Why do we want *unitary* representations of locally compact groups into $B(H)$?

This is related to a previous question of mine but I have a more philosophical issue with the material. Everywhere I have looked for representations of locally compact groups into $B(H)$, everyone ...
0
votes
2answers
70 views

Seeing quotient groups

Can someone explain simply what a quotient group is? I've read a lot of convoluted and unnecessarily tough descriptions on it but it seems like a really simple idea. Question Show that $$8 ...
2
votes
2answers
645 views

Eigenvalues of Matrices and Eigenvalue of product of Matrices

If $n\times n$ matrix $A$ has eigenvalues $1,-1$ and $n\times n$ matrix $B$ also has eigenvalues $1,-1$, can I then say something about eigenvalues of $AB$ and $BA$?
0
votes
1answer
118 views

Counterexample to disprove that $P(A-B) = P(A) - P(B)$?

Assuming $P(A)$ is the power set of $A$, would this be a correct counterexample for the statement that $P(A-B) = P(A) - P(B)$? Let $A = \{2, 3\}$ and $B = \{2\}$, therefore $C = \{3\}$. $P(A-B) = ...
2
votes
0answers
109 views

Equivalence of two norms in complete space

Let $X$ be a vector space with two norms $\| \cdot \|_1$ and $\| \cdot \|_2$ such that $\| x \|_1$ $\leq$ $\| x \|_2$ for all $x \in X$. If $X$ is complete in both norms, prove they are equivalent. ...
2
votes
1answer
133 views

Show that $L^1\subsetneq (L^\infty)^*$ [duplicate]

How does one show that $L^1\subsetneq (L^\infty)^*$? I am having trouble in this. Any help would be appreciated.
2
votes
1answer
72 views

Show that a set is a ring

Let $R\not=\{0\}$ be a commutative ring with unity. Let $I$ be a prime ideal in $R$. Let $S=R-I=\{x\in R|x\not\in I\}$. Let $F$ be a field that contains $R$ as a subring with the same unity. Show that ...
2
votes
1answer
295 views

String transformation

There are $n$ light bulbs place in circle and colored with Red, Green, Blue. After 1 second, from left to right, 2 consecutive bulbs which have different color will both change to extant color. ...
1
vote
0answers
85 views

Topological equivalence of any norm on $\mathbb C^n$

In University I have been told that every norm on $\mathbb C^n$, for any $n\in\mathbb{N}$, is equivalent to every other such norm. I have a proof for this on any vector space on $\mathbb R$. Trouble ...
0
votes
5answers
995 views

Probability [Sum of digits is even for a random number] [closed]

A number of 6 digit numbers is written down at random. Probability that the sum of the digits is an even number is? Please answer with explanation.
0
votes
2answers
340 views

Nonempty interior feature of a proper cone

one of feature of proper cone is solid which means a proper cone has nonempty interior what dose nonempty interior mean ? I was reading Boyd convex optimization and I saw this term "Nonempty ...
1
vote
2answers
44 views

Evaluating an integral over inifinty with polars leads to an integral of cosine over inifinity, how can this be resolved?

So I have the integral $$\int_0^\infty\int_0^\infty\frac{yx^2}{x^2 +y^2}e^{-(x^2 +y^2)} \,dx\,dy$$ And converting this into polars gives: $$\int_0^\infty r^2 e^{-r^2}\,dr ...
2
votes
0answers
57 views

Integral on the circle

It's a standard fact that to calculate integrals of the form $$\int_{0}^{2\pi}\mathcal{R}(\cos(\theta),\sin(\theta)) \ d\theta$$ with $\mathcal{R(x,y)}$ a rational function in two variables without ...
5
votes
1answer
262 views

Prove that $\frac{a^2}{a + b} + \frac{b^2}{b + c} + \frac{c^2}{c + a} \ge \frac{3}{2}$

Let $a,b,c$ are positive real numbers such that, $a^2+b^2+c^2=3$. Prove that $$\frac{a^2}{a + b} + \frac{b^2}{b + c} + \frac{c^2}{c + a} \ge \frac{3}{2}$$
1
vote
1answer
548 views

How to calculate this particle's position, velocity and acceleration, each as functions of time.

I'm given a particle of mass $m$, at position $x$, moving through 1-space dimenion with velocity $v=\gamma(d-x)$ for constant $\gamma, d.$ I'm also given that the particle starts from $x=0$ at $t=0$. ...
0
votes
3answers
67 views

Vanishing of non top-order Chern classes

Let $E \to B$ be a rank-$r$ complex vector bundle and denote by $c_1(E)$, $\ldots$, $c_r(E)$ its Chern classes. Then $c_r(E)$ is just the Euler class of the realization of $E$ as a real vector bundle ...
7
votes
3answers
3k views

How to find continued fraction of pi

I have always been amazed by the continued fractions for $\pi$. For example some continued fractions for pi are: $\pi=[3:7,15,1,292,.....]$ and many others given here. Similarly some nice continued ...
1
vote
0answers
66 views

Heineken and Mohamed Groups

I was trying to understand the costruction of the Heineken and Mohamed groups. (II) Every proper subgroup of $G$ is subnormal and nilpotent. Lemma 1. If $G$ is a non-nilpotent soluble group ...
1
vote
1answer
43 views

When the points of coproduct are the coproduct of points

Let $C$ be a category with coproduct. The Yoneda functor $C^{op}\to Psh(C)$ preserves all limits but not colimits. Suppose that $C$ has a terminal object, say $*$. My question: can we say anything ...
1
vote
2answers
121 views

non-sequential sequence function

if i remember correctly (i had one workshop on numerics years ago, sorry for my lack of knowledge) there is a way to create some sort of hash function that gives you a non sequential sequence. This ...
2
votes
0answers
46 views

Infinitude of composite numbers of the form $n\# \pm 1$

Are there infinitely many composite numbers of the form $$n\# + 1$$ where $n$ is a prime number? What about $n\# - 1$? Here $n\#$ denotes the primorial function of $n$, i.e. the product of all ...
1
vote
2answers
537 views

Probability [Distinct balls in distinct boxes]

6 different balls are put in 3 different boxes, no box being empty. The probability of putting balls in the boxes in equal numbers is ? Could anyone pleases give the answer and explain it.
0
votes
1answer
239 views

Finding closed form of a series where the first term is $n=0$

"The figure below shows the quantity of the drug atenolol in the blood as a function of time, with the first dose at time $t = 0$. Suppose atenolol is taken in $75$ mg doses once a day to lower blood ...
0
votes
1answer
57 views

hamiltonian mechanics

In $\mathbb{R}^{2n}$, $\omega=\sum dx_i \wedge dy_i$ is a canonical symplectic form, and H is an hamiltonian function, i.e. $\dot{x}= \frac{\partial H}{\partial y}$, $\dot{y}= -\frac{\partial ...
1
vote
6answers
8k views

Application of Composition of Functions: Real world examples?

Do you know of a real world example where you'd combine two functions into a composite function? I see this topic in Algebra 2 textbooks, but rarely see actual applications of it. It's usually plug ...
1
vote
0answers
169 views

Physics of Skiing

I am conducting a research paper on the physics of skiing, specifically how ski parameters affect the ski's ideal carve. I have come across this paper, which is incredibly relevant but am having ...
1
vote
0answers
63 views

Is there a name for this problem?

I was just wondering if this is (or is related to) a well-known problem. say we have some counting number N. what is the smallest set S of counting numbers such that for each n=2,...N, there exists ...
1
vote
2answers
60 views

Looking for good literature on Markov Chains with explicit calculations

I am currently starting my thesis on Markov Chains and am looking for good books and papers that include explicit calculations. I have taken a small course on Markov Chains so the subject is not ...
0
votes
1answer
43 views

Categoricity of integers under “less-than”?

I came across the following remark in Stewart Shapiro's The Governance of Identity: Consider, for example, a structure S with a single relation, <. The axioms of S are that < is a linear ...
1
vote
2answers
82 views

Proving the following sequence is Cauchy

The sequence $ \{\pi_n\}$ can be defined by $\pi_1 = 3.1, \pi_2 = 3.14, \pi_3=3.141, ...$ for each $n$, $\pi_n$ shows the first $n$ decimal digits of $\pi$. How can I show that $\{\pi_n\}$ is a ...
0
votes
3answers
70 views

Showing divergence of a series

Suppose for each $n$, $a_{n+1} - a_{n} = \alpha > 0$ with $\alpha$ independent of $n$ and $a_1 > 0$. Show that the series $\sum_{n=1}^{\infty}\frac{1}{a_n}$ diverges. My solution is as such: ...
1
vote
1answer
38 views

Approximation of continuous functions by a functions with vanishing second derivative

Denote by $C^n[-\infty,+\infty]$ the class of functions which: have finite limits at $\pm \infty$; and are differentiable $n$ times on the line, with all these derivatives bounded. Denote by $C^3_0$ ...
1
vote
3answers
452 views

Number of surjections from $\{1,…,m\}$ into $\{1,…,n\}$

Let $m,n$ be two integers such that, $m\ge n$. Compute the number of surjections from $\{1,...,m\}$ into $\{1,...,n\}$ There are $n^m$ functions (total). we subtract from $n^m$ the number of ...
0
votes
1answer
46 views

Sin x tends to x as x tends to zero

If ABC is a sector of a circle with centre A, why does the area of triangle ABC approach the area of sector ABC as angle BAC approaches zero?
14
votes
4answers
636 views

Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...
1
vote
3answers
39 views

An approximation: 2*number of events = twice the probability

My statistical mechanics textbook uses an approximation to derive a well-known result. The approximation is: Suppose for an event, the probability of an outcome is P. For n events, the probability of ...
1
vote
1answer
64 views

Bifurcation in coupled differential equations

I have the coupled differential equations $$x'(t) = -33x(t) + 3x(t)y(t),$$ $$y'(t) = 5y(t) - \frac{5y^{2}(t)}{11} - 2x(t)y(t).$$ This system has equilibrium points at $(0, 0)$ - a saddle point and (0, ...

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