2
votes
1answer
120 views

Integral Power Rule Step-by-step

Real quick, two things: I'm sorry if my notation or terminology is incorrect, and I know what I'm asking isn't strictly necessary for my studies, but writing something out step by step helps me to ...
1
vote
2answers
84 views

How to show that for $a_n := \left( \frac{(2n)!}{n!n^n} \right)$, $a_n \rightarrow 4/e$

I'm asked [in a homework problem set] to show that for $a_n := \left( \frac{(2n)!}{n!n^n} \right)$, $a_n \rightarrow 4/e$. I'm told to show that this function is Riemann integrable for $ln(a_n)$. I ...
2
votes
0answers
56 views

Why continuous growth (based on e) is being simply scaled to match non-limit cases (limit of the (1+1/n)**n formula)?

The constant $e$ is the maximum exponential growth that is possible when it is done continuously, i.e. when what can be called "continuous breeding" occurs. That is the meaning of the ...
0
votes
1answer
29 views

How to find the largest n for which one can solve a problem with a given algorithm?

This is from Discrete Mathematics and its Applications I am working on 15b and 15f. First I converted the rate to say that 10^9 operations are carried out in one second. so for 15b. In one ...
1
vote
1answer
41 views

Vector Fields on Real Numbers

I'm looking at vector fields on the manifold $\mathbb{R}$, in the sense that a vector field $v$ is a smooth function from $\mathbb{R}$ to $\mathbb{R}\times T_p\mathbb{R}$. These seem so simple that ...
-2
votes
1answer
38 views

T/F question on free groups

Is this statement True or False- If F is a free group with basis {$x,y$} and H is the subgroup generated by {$x^2,y^2,xy,yx$} then H is a free group of rank $3$. What should be my approach to solve ...
2
votes
1answer
136 views

Surface of revolution and curvatures

Let $f(x)$ be a smooth function. Consider a surface of revolution, \begin{equation} M(u, v) = (f(v) \cos(u), f(v) \sin(u), v). \end{equation} (a) Calculate coefficients of the first and second ...
0
votes
1answer
173 views

what is the area of a symmetric lens with the circles having a radius of 35' being 60' apart

I need this answer for work. Ive tried for hours to figure this out. If anyone could help it would be much appreciated. I need to find the area of a symmetrical lens of two circles who's radius is ...
2
votes
1answer
223 views

Cesaro summable series

A series $\sum_{k=0}^{∞}a_k$ is said to be Cesaro summable to an $L\in R$ if and only if $\sigma_n = \sum_{k=0}^{n-1}(1 - \frac{k}{n})a_k$ converges to $L$ as $n$ → $∞$. Let $s_n = ...
1
vote
1answer
52 views

Relation between the connectivity and the minimum degree of a graph

Prove that for every graph $G$, (a) $κ(G) ≤ δ(G)$; (b) if $δ(G)≥n−2,$ then $κ(G)=δ(G).$ Here $κ(G)$ is the connectivity, which is the minimum size of a cut set of $G$. And $δ(G)$ is the ...
0
votes
0answers
59 views

What is so enlightening in the Ferrers diagrams?

I've studied the Ferrers' diagram. And It's not clear why it's useful, the only property I noticed until now is that once one partition is drawn with a Ferrers' diagram, it's conjugate could show ...
1
vote
2answers
144 views

Stochastic Calculus - Ito decomposition

I have got one question about Ito decomposition. Suppose $W_t$ is a Brownian Motion: $X_t = W_t^2 + \int_0^t(W_t^3-1)du$ How to get $dX_t$? I am quited comfused by the integral. Should we calculate ...
2
votes
3answers
112 views

Solve for $y''+2y'+y = \frac{e^{-t}}{t^{2}}$

I need some help in finding the solution to this second-order non-homogeneous DE. I know how to find the solution of the reduced equation $$y''+2y'+y=0.$$ The characteristic equation ...
1
vote
1answer
36 views

If $f$ is a non-constant entire function, can $f(z + T_n) = f(z)$?

If $f$ is a non-constant entire function, can $f(z + T_i) = f(z)$ for all $i \geq 1$ if $T_i \to 0$ as $i \to \infty$ and $\{ T_i \}_{i \geq 1} \neq 0$ for each $i \geq 1$? My idea is to write ...
1
vote
2answers
142 views

How to obtain the chromatic polynomial of $C_5$?

I've been reading some books on chromatic polynomials, I am a little confused at the procedure that is needed to obtain it. I've read in a book that the chromatic polynomial is obtained by ...
0
votes
2answers
432 views

$A$ and $B$ are finite sets. How many Partial Functions exist between them?

so I have following question $A$ and $B$ are finite sets How many Partial Functions exist between them ? $f:A\to B$ Can someone give me a solution/hint/website where they may explain me a ...
0
votes
1answer
46 views

Probability of getting disease and Markov chain

I am studying marcov chain. The question is . There are 5 people ( 4 diseased / 1 healthy) Two people are selected randomly and assumed to interact. If one is diseased and the other is healthy, ...
1
vote
1answer
37 views

Express $A^{2014}$ as a polynomial of least possible degree

Given $A=\left( \begin{array}{ccc} 8 & 18 \\ -3 & -7 \end{array} \right)$ find the polynomial $p$ of least possible degree such that $p(A)=A^{2014}$. The eigenvalues of $A$ are $-1$ and ...
1
vote
0answers
47 views

Groups and Symmetries Question on permutations and cycles

Let a=(123)(456) be in S10. If H is a cyclic subgroup of S10 and a is in H, what are all the possible orders of H? My attempt: The order of a is 3 and the order of S10 is 10!. So we must find the ...
2
votes
1answer
787 views

Solve for $y'' - 4y=0$

I wanted to check my answer. So I have the characteristic equation: $$r^{2}-4=0\Rightarrow r^{2} = 4 \Rightarrow r = \pm 2.$$ So the general solution is: $$y = c_{1}e^{-2t}+c_{2}e^{2t}$$ Is this ...
0
votes
1answer
149 views

How to give a big O estimate/visualize for these while loop?

This is from Discrete Mathematics and its applications I am currently working on problem 4. I was able to see that for problem 2, that one operation one will run n times for every n(meaning in ...
0
votes
1answer
50 views

Hom$_{Set}(G_1 \times G_2,R) \cong $ Hom$_{Set}(G_1,R) \otimes_R $Hom$_{Set}(G_2,R)$?

I find this isomorphism in one example in my class notes, but I can't see when or why it holds. If someone could help me with it I would appreciate it. Let $G_1,G_2$ be groups and $R$ a ring, then: ...
0
votes
1answer
13 views

Bounding $f$ on a larger disk given a bound on a smaller disk

Suppose I have a function $f$ holomorphic in $\{z: \operatorname{Re}(z) > 0, \operatorname{Im}(z) > 0\}$. Let $C_{1}$ denote the circle centered at $2 + 10i$ of radius 1 and let $C_{2}$ denote ...
0
votes
2answers
60 views

Degree of Splitting Field to Prove Irreducibility

Let $f(x) \in F[x]$ have degree $n>0$ and let $L$ be the splitting field of $f$ over $F$. Show that if $[L:F]=n!$ then $f(x)$ is irreducible over $F$. My approach: I attempted to prove the ...
4
votes
2answers
48 views

What is the module structure here?

Let $A$ be a local ring with maximal ideal $\mathfrak{m}$, and let $M$ be an $A$-module. I want to turn the following object into an $A/\mathfrak{m}$-module: $$A/\mathfrak{m} \otimes_A M$$ I ...
2
votes
1answer
189 views

Finding the minimal polynomial from characteristic equation

I am attempting to find the minimal polynomial of a matrix. My characteristic equation turns out to be $x^3 - x$ which factors out to $x(x-1)(x+1)$. Now, I am reading that the minimal polynomial is ...
0
votes
1answer
101 views

Find Point on Polyhedron Nearest Given Point

Given the 8 vertices (Y, R, M, B, C, G, K, W) of a polyhedron with 12 faces, 18 edges, and 8 vertices and another point somewhere in 3D space (P) find whether the point is within, on, or outside the ...
1
vote
1answer
51 views

Verifying a proof of martingales.

I am trying to prove the following: Let $T$ be a stopping time bounded by $c$, and let $(X_n)$ be a martingale, then $E(X_T)=E(X_0)$. Here is what I did: $\int ...
1
vote
1answer
103 views

Growth faster than polynomial, slower than exponential.

Assume $F(n)$ is a positive function. If $F$ is growing faster than a polynomial then is it growing exponentially fast? Is this statement true? Can we find a function $F(n)$ such that ...
0
votes
3answers
23 views

Determine the boundedness of a complex function

How can I determine whether {$\frac{z}{1+z^2}$; z $\in$ $\mathbb{C}$ \ {-i, i}} is bounded? My textbook is very poor at describing boundedness for complex functions. Thanks for the help!
1
vote
0answers
30 views

Maximum value of a given function.

The question is to find the maximum value of the function: $f(x)= \frac{|x|-2 -x^2}{|x| +1}$. What I tried to do was maximise the numerator and minimise the denominator to obtain the maximum value ...
3
votes
1answer
95 views

First examples for topology of non-Hausdorff spaces

I have absolutely no intuition about non-Hausdorff spaces. I would like to understand the topology of non-Hausdorff spaces (in particular spaces obtained by "bad" group actions). As a first example, ...
0
votes
1answer
380 views

Proving Decimal Representation

Prove that if the decimal representation of a nonnegative integer n ends in 5 or 0 then 5 | n. (Hint: As a first step show that if the decimal representation of a nonnegative n integer ends in d0 then ...
0
votes
1answer
32 views

Decreasing family of path connected open sets containing two points has a common path.

I was trying to use van Kampen theorem on fundamental groupoids to prove the van Kampen theorem on fundamental groups. I got stuck at the following technicality. Let $\{U_i\}$ be a decreasing ...
1
vote
0answers
39 views

Homework : Anti log expression

I have this expression $x(r) = y(a)r^a$ where $r$ is a random variable and I want to express the expression in terms of $r$. The objective is to substitute the variable $r$ into the pdf of $r$, ...
2
votes
0answers
33 views

Efficient algorithm for calculating hypervolume

Given a $d$-dimensional hyperrectangle that spans from the origin to the integer coordinates $l_1,l_2,l_3,\cdots,l_d$. If $V$ is the hypervolume of the solid formed by all points in the ...
1
vote
1answer
48 views

Question about permutations and cycles (Groups and Symmetries)

Let $a = (123)(456)$ be in $S_{10}$. Find a permutation $b$ in $A_{10}$ such that the disjoint cycle form of does not contain any $3$ cycles and $a$ is in $\langle b \rangle$. Attempt: I'm assuming ...
2
votes
0answers
76 views

Examples of forcings which add no “definable” Aronjain tree

Maybe a bit board question but: Fixing a regular cardinal $\kappa$ in the ground model, I am looking for examples of set forcing notions which preserve regularity of $\kappa$ and add no new $\kappa$ ...
0
votes
1answer
87 views

Find $\int_C ydx+zdy+xdz~$ where $C$ is the curve of the intersection of the two surfaces $z=xy$ and $x^2+y^2=1$

Find $\int_c ydx+zdy+xdz~$ where $C$ is the curve of the intersection of the two surfaces $z=xy$ and $x^2+y^2=1$ , traversed once in a direction that appears counterclockwise when viewed from high ...
0
votes
0answers
38 views

Eigenvector of canonical form matrix

So if you have the canonical matrix: $$ \begin{pmatrix} \lambda &1\\ 0 & \lambda \end{pmatrix} $$ the eigenvector is $(1,0)$. I've found this from multiple sources. But where does that come ...
2
votes
1answer
42 views

Equivalence proof

Let $(f_n)$ and $(g_n)$ be sequences with e.g.f $F(t)$ and $G(t)$ respectively. Show the equivalence of the following: $$g_n=\sum^n_{k=0}{n\choose k}\ f_k$$ $$G(t)=F(t)e^{t}.$$ ...
3
votes
2answers
27 views

Consider this primality test: Fix an initial segment of primes (e.g. 2,3,5,7), and combine a $b$-pseudoprime test for each b in that list…

Consider this primality test: Fix an initial segment of primes (e.g. 2,3,5,7), and combine a $b$-pseudoprime test for each b in that list. For several such initial segments, find the first $n$ for ...
5
votes
3answers
181 views

a characterization of $L^p$ space

The following question should be part of the questions I recently asked here Prove or disprove a claim related to $L^p$ space If $g \in L^p(\Omega, \lambda)$ where $\Omega$ is a bounded subset of ...
0
votes
1answer
45 views

For the following functions, the domain and codomain is {a,b,c,d} which ones are one to one and which are onto? Give reasons for each

For the following functions, the domain and codomain is {a,b,c,d} which ones are one to one and which are onto? Give reasons for each. a) f(a) = b, f(b) = a , f(c) = c, f(d) = d b) f(a) = b, f(b) = ...
4
votes
0answers
45 views

Is it possible to construct a 1-D linear differential operator with given spectrum $0\leq\lambda_0\leq \lambda_1\leq\dots\leq\lambda_n\le\dots$?

Suppose one is given with a sequence $S$ of non-negative real numbers $0\leq\lambda_0\leq \lambda_1\leq\dots\leq \lambda_n\leq\dots$. Under what conditions on $S$, is it possible to construct a Linear ...
0
votes
2answers
70 views

Determine if $h(x)=\sin(x^2)$ is uniformly continuous on $(-\infty,\infty)$ [duplicate]

Normally I know how to do these kind of questions using different kind of methods, but on this specific one I have no idea what to do: Determine if $f(x)=\sin(x^2)$ is uniformly continuous on ...
0
votes
1answer
27 views

The two planes 2x-y+3z=2 and -4x+2y-6z=3 are parallel. Let V= the set of P1P2: P1 is an element of equations one and P2 is an element of equation 2.

The two planes 2x-y+3z=2 and -4x+2y-6z=3 are parallel. Let V= the set of P1P2: P1 is an element of equations one and P2 is an element of equation 2. Is this a vector space? P1P2 should have a line ...
0
votes
1answer
43 views

Trapped volume of (n-1)-sphere inside an n-simplex

Assume that we take an $n$-simplex (with side length of 2 units) and place a unit $(n-1)$-sphere at each vertex. For $n=2$, half of a circle is enclosed inside the $2$-simplex. For $n=3$, solid angle ...
1
vote
1answer
32 views

Proving eigenvector of addition operator

If $v$ is an eigenvector of operators $S$ and $T$, then $v$ is also an eigenvector of $aS + bT$ , $a, b \in F$ I know that if I let $ v_1,\space v_2 \in V$. By definition of an eigenvector, $T(v_1) ...
0
votes
2answers
132 views

The multiplication of two even numbers gives an even number

I am given the following proposition: If $m$ and $n$ are even integers, then $mn$ is also an even integer. This is my strategy: An integer $m$ is said to be even if it is divisible by 2 (integer). ...

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