# All Questions

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### Compute the degree of $\phi:(x_1,x_2,x_3,x_4) \mapsto (x_1,-x_3,-x_2,x_4)$. Does $\phi$ preserve orientation?

Consider the map $\phi:S^3 \rightarrow S^3$ given by $\phi(x_1,x_2,x_3,x_4)=(x_1,-x_3,-x_2,x_4)$. Compute the degree of $\phi$. Does $\phi$ preserve orientation? First I want to point ...
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### Inductively show that "the ordered n-tuple $(x_1,\ldots,x_n)$ of a set so that $(x_1, \ldots,x_n) = (y_1,\ldots,y_n)$ if their coords are ordered [on hold]

I believe that the $n=1$ step is more or less trivial because there is no order to a step that involves only a single element However, I have extreme difficulty determining how to prove the n=k+1 ...
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### Bound on “width” of points in a plane

Suppose we define width $w(P)$ of point set $P$ in a plane to be the ratio of the maximum distance to the minimum distance between the points in $P$. (Assume unique coordinates so that $w(p)$ is ...
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### Long exact sequence of a triple: working through the geometry

Suppose $X$ is a topological space with subspaces $X \supset U \supset A$ such that $U$ deformation retracts onto $A$. We know that $H^*(X,U) \cong H^*(X,A)$--one way to see this is to take the long ...
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### How to show that this modification of Thomae's function is Riemann integrable

I am dealing with the function $f(x)=\begin{cases} \frac{1}{n} & \text{if }\frac{1}{n+1}<x<\frac{1}{n},\:n\in\mathbb{N},\\ 0 & \text{ otherwise.} \end{cases}$ I want to show it is ...
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### Find the value of $sin(2\theta)$ when $cot(\theta) + tan(\theta) = 2.5$

I have an homework question that goes like: $cot(\theta) + tan(\theta) = 2.5$ is valid on some angles $\theta$ at section $0 < \theta < \pi/2$. Find the value of $sin(2\theta)$. (There is no ...
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### Indicate whether True or False

a) The derivative of the cubic spline interpolant at the nodes agree with those of the function. b) If $f$ is a continuous function with $f(a) f(b) > 0$, the $f$ has no roots in $(a,b)$. c) ...
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### When does a periodic but positive recurrent markov chain have a limiting distribution

So I know it's a fact that an aperiodic, finite state, irreducible (so positive recurrent) markov chain has a unique stationary distribution which is limiting. However, I am curious if there is a ...
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### Given common terms (and their position) between an arithmetic and geometric sequences, find the common ratio. [on hold]

The fourth, seventh and sixteenth terms of an arithmetic sequence also form consecutive terms of a geometric sequence. Find the common ratio of the geometric sequence
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### Converse of Fermat's Little Theorem.

If $a^n\equiv a \pmod n$ for all integers $a$, does this imply that $n$ is prime? I believe this is the converse of Fermat's little theorem.
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### Order of element in polynomial ring in Hatcher

So I've been reading Hatcher and I am unsure what they mean when they say things like $H^\ast(\mathbb{R}P^n;\mathbb{Z}_2)\cong \mathbb{Z}_2[\alpha]/(\alpha^{n+1})$ where $|\alpha|=1$. It is this last ...
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### What is the probability that a psychic correctly “predicts” the outcome of at least 5 out of 10 coin flips?

Assume the psychic is actually just randomly guessing on each flip. The attempt: let E be the event in question number of outcomes per flip = 2 chance of correctly guessing the correct outcome = ...
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### Prove that the space $\Bbb R_K$ is not regular.

Prove that the space $\Bbb R_K$ is not regular. where the basic open sets on $\Bbb R_K$ is given by $\{(a,b):a,b\in \Bbb R\}\cup \{(a,b)-K\}$ where $K=\{\dfrac{1}{n}:n\in \Bbb Z_+\}$. ...
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### Barycentric subdivisions and labeling of $(d-1)$-simplex

I am trying to prove that it is always possible to label the vertices of the $k$-th barycentric subdivision of a $(d −1)$-dimensional simplex with labels $1, 2, . . . , d$ such that each simplex ...
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### Let $A$ be a $2\times2$ matrix with real entries such that $A$ is invertible. If $Det(A)=k$,and $Det(A+kadj(A))=0$

Let $A$ be a $2\times2$ matrix with real entries such that $A$ is invertible. If $Det(A)=k$,and $Det(A+kadj(A))=0$, then find the value of $Det(A-kadj(A))$ My attempt: $Det(A+kadj(A))=0$ ...
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### Transforming a vector configuration to an affine point configuration

I am reading the first chapter of Oriented Matroids by Bjorner et. al, in which they consider the set of vectors $\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v}_6$ given as the columns of the following ...
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### PDE Proof that a linear combination of 2 solutions is also a solution [on hold]

Can someone please help? I've been trying to figure this for a few days now. Consider the first order PDE: $au_t + bu_x$ = 0, where a and b are constants. Show that if $u_1$ and $u_2$ are solutions ...
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### Troubles With The Beginning

The following is the question I'm having a bit of troubles starting: Musicnotes.com sells sheet music in the following genres: rock jazz, new age, and country. An experiment consists of recording the ...
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### If $a\equiv b \pmod{p_i}$ for $i=1,2,…,k$ then prove $a\equiv b \pmod{p_1p_2\cdots p_k}$

If $a\equiv b \pmod{p_i}$ for $i=1,2,\cdots,k$ then prove $a\equiv b \pmod{ p_1p_2\cdots p_k}$ $a\equiv b \pmod{p_1}$ implies $a-b=p_1x_1$ Similarly, $a-b=p_2x_2,\ \cdots,\ a-b=p_kx_k$ So, ...
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### Show that if $(x_1,x_2)$ is defined to be $\{\{x_1\},\{x_1,x_2\}\}$ then $(x_1,x_2)=(y_1,y_2)$ iff $x_1=y_1$ and $x_2=y_2$ [duplicate]

My Work: If you take the cartesian product of any set with two arbitrary elements $a$ and $b$, and the resulting set is $\{\{x_1\},\{x_1,x_2\}\}$, then the only possible values for $a$ and $b$ are ...
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### how would zipf's law be expressed as a logarithmic function?

I am doing a project for math class based on Zipf's law but i cannot understand how it relates to logarithms. The project handout states that "In 1949, George Zipf noticed that if you tabulate the ...
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### How to show $S_1\subset W_1$ and $S_2\subset W_2$ are independent $\implies$ $S_1\cup S_2$ is independent based on the following assumption?

Let $W_1$ and $W_2$ be subspaces of vector space $V$ satisfying $W_1\cap W_2=\{0\}$ ,how to show $S_1\subset W_1$ and $S_2\subset W_2$ are linearly independent $\implies$ $S_1\cup S_2$ is linearly ...
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### Relationships for estimating the value of a double integral with sample points

I have a double integral with function (x+y) where I am supposed to estimate the value of the double integral by using a Reimann sum with m = 2 and n = 3. For sample points, I must use lower left ...
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### Smallest $n$ such that $U(n)$ contains a subgroup isomorphic to $\mathbb Z_5 \oplus \mathbb Z_5$

I solved the following exercise: Find an integer $n$ such that $U(n)$ contains a subgroup isomorphic to $\mathbb Z_5 \oplus \mathbb Z_5$. Here $U(n)$ is the group of units modulo $n$. To solve it ...
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### Using Modulo reduction

I'm really confused on how to do modular reduction. I understand we're supposed to take the factor of the exponent? for example how would I go about doing modular reduction on: $5^{17}$ mod 16
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### Find the matrix representing T and Find the Image of T (as a span of vectors)

Let $T(a,b) = (a+b,2a-b,3a)$. a)Find the matrix representing $T$. b)Find the image of $T$ (as a span of vectors). So I found that $T$ is a linear transformation. Now would the matrix just be $A$= ...
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### How to integrate to solve a PDE with mixed partials in the integrand

Problem Statement: Determine the equlibrium temperature distribution inside a circular annulus $r_1\leq r \leq r_2$. If the outer radius is at temperature $T_2$ and inner radius at temp $T_1$. So ...
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### Basis for Vector Space iff can be Expressed Uniquely as Linear Combo of Basis

Let $V$ be a vector space and $\beta= \{ u_1,\dots ,u_n \}$ be a subset of $V$. $\Rightarrow$ $\beta$ is a basis for $V$ iff each vector $v\in V$ can be unquiley expressed as a linear combination of ...
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My studies into graphs and models following examples from Khan Academy has helped me on my goal to learn how to chart and model via the quadratic formula However while I have been successful in ...
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### How to prove that a convex polygon is cyclic

Is there an easy way to tell if a convex polygon is cyclic? I was told that if the vertices of the $n$-gon are $A_1,A_2,\ldots,A_n$, it is enough to prove that $A_1A_2A_3A_i$ is cyclic for each ...
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### Prove a conditional distribution is uniformly distributed across a given interval?

$X$ and $Y$ are independent random variables identically exponentially distributed with $\lambda$. Take $Z=X+Y$. Show that $(X|Z=z)$ is uniformly distributed over $(0<x<z)$. Then, find ...
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### How to determine the limiting distribution of a Markov Chain which can only increment up or down a state at every stage?

I have a random walk Markov chain that has states from $0$ to $N$. The conditions are that when the chain is at $0$, the chain will go to state $1$ with probability $1$. When the chain is at state ...
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### Proving a value-wise magnitude function is differentiable

Let $v: \mathbb{R^n} \rightarrow \mathbb{R^m}$ be a function such that $v(y) \neq 0,\forall y \in \mathbb{R^n}$ that is differentiable at $x \in \mathbb{R^n}$. (a) Show that the value-wise ...
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### If $x$ is an isolated point of $S \subseteq \mathbb{R}$, then $x$ is a boundary point of $S$.

Is the following proof valid? (Note: I know there is a post discussing this problem, but I am curious to see if my argument works). This problem is different from another post that is similar with ...
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### How many ways are there to choose one-half dozen donuts from $9$ varieties so that there are exactly $4$ glazed? [on hold]

How many ways are there to choose one-half dozen donuts from $9$ varieties so that there are exactly $4$ glazed? How should I approach this problem? Okay I think it's C(10, 2) because I already have ...
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### Group orderable iff all its finitely-generated subgroups are orderable

I want to proof this specifically using the Compactness Theorem from propositional logic (this is an exercise from Model Theory, Hodges). $G$ orderable means there is a total ordering s. t. $g\leq h$ ...
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### How to compute $df(e)$ explicitly?

I am reading the book. On page 244, the formula (9.2.3.4). I would like to compute the bracket on g^* induced from the Poisson bracket on C[G] explicitly in the example of $G=SL_2$. The formula is: ...
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### Is $S_1\cap S_2$ and $S_1\setminus S_2$ always linearly dependent if $S_1$ and $S_2$ are linearly dependent subsets of vector space $V$?

Let $S_1$ and $S_2$ be linearly dependent subsets of vector space $V$, are $S_1\cap S_2$ and $S_1\setminus S_2$ always linearly dependent? The counterexample for the first one I can think of is ...
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### Computing a new inverse given information from an earlier inverse:

Suppose I have linearly independent set of vectors $v_1 ... v_k \in \mathbb{R}^{k}$. I can let $B = [v_1 ... v_k]$ and by applying Gaussian elimination on the matrix $$[ B \ I_k]$$ I end up ...
Suppose that $(X, \mathcal{O}_X)$ is a ring space. If $\mathcal{F}, \mathcal{G}$ are sheaves of $\mathcal{O}_X$-modules then the assignment $$U \mapsto \text{Hom}_{\mathcal{O}_X|U}(\mathcal{F}|U, ... 1answer 20 views ### Connecting up boxes mathematically (Puzzle) How would you connect each black box once to each colored box without any lines overlapping, this is racking my brain so please help. Note that you can move the boxes where ever you want. Maybe ... 0answers 30 views ### Show \frac{y-x}{(2-x-y)^3} is not integrable on [0,1]\times[0,1], not invoking Fubini's theorem. The double integral$$I = \int_{[0,1]\times[0,1]}\frac{y-x}{(2-x-y)^3} dxdy$$does not have a finite value. The two iterated integrals have different values (Counterexample to Fubini?). Then Fubini's ... 1answer 72 views ### If f:\mathbb{Z} \to \mathbb{Z} is an isomorphism, prove that f is the identity map. [on hold] I am a little baffled by this question. Is it safe to assume that since f is an isomorphism, f (1) = 1 ? And, if it is safe to assume this, could I construct a proof by induction, by using the ... 0answers 11 views ### Lorentzian Delta Function Sifting Property Using the Lorentzian as the delta function$$\delta(x) ~=~ \lim_{\epsilon\rightarrow 0} \frac{1}{\pi}\frac{\epsilon^2}{\epsilon^2+x^2} Is there a way to rigorously prove the sifting property, ...
Let $k$ be a field, $S = k[x_0,\dots,x_r]$, $I$ a homogeneous ideal of $S$ and $R=S/I$. Let $P$ be a homogeneous prime ideal of $R$ and let $R_{(P)}$ be the homogeneous localization of $R$ at $P$. I ...