0
votes
1answer
74 views

Compactness and cartesian product

I'm having trouble figuring out how can I show that if two sets are compact then their cartesian product is also compact. Any help is much appreciated,thank you!
1
vote
0answers
47 views

Show that the following family of hash functions is $2$-wise but not $3$-wise independent

Consider the following family of hash functions that map $w$-bit numbers to $l$-bit numbers (i.e. the range is $\{0,...,m-1\}$ where $m=2^l$): $\mathcal{H} = \{h_{A,b}|A\in \{0,1\}^{l\times w}, b \in ...
0
votes
2answers
32 views

Find the $sin$ of an angle $B$ using law of sines given side angle side

this is giving me trouble, here's what I've tried: Q.A triangle has sides $a=2$, $b=3$, and $\angle C = 60^o$. Using the law of sines, find $\sin(B)$ OK so I know the law of sines is: $$\frac{\sin(A)...
1
vote
1answer
150 views

Group and Ring Homomorphisms between $\mathbb{Z}_p$ and $\mathbb{Z}_{2p}$

Let $p$ be prime with $p > 2$. (a) Determine the number of group homomorphisms between $\mathbb{Z}_p$ and $\mathbb{Z}_{2p}$ (b) Determine the number of ring homomorphisms between $\mathbb{Z}_p$ ...
3
votes
1answer
43 views

Infinite graphs satisfying a certain Ramsey property

Let $G$ be a countably infinite graph. If $G$ has cliques of arbitrarily large finite size, then $G$ satisfies the following property, which I will call $(*)$: for any $r\in \mathbb{N}$ and any $r$-...
0
votes
3answers
460 views

Extending a basis for a subspace of V to create a basis of V

Can any set that forms a basis for a subspace of a vector space V be extended to form a basis for V?
1
vote
1answer
42 views

Convergence In probablility implies convergence in distribution?

I'm currently working on the following exercise: If $X_n$ is sequence of randon maps with values on a metric space $(S,d)$. Show tha convergence in probablity to a randon map $X$ implies $P\circ ...
1
vote
3answers
97 views

Find $\lim_{x \rightarrow 0} (\frac{\tan x}{x})^{x^{-2}}$

I see that this is in the $1^ \infty$ form, so I've taken log to get: $\lim_{x \rightarrow 0} \log( \frac{\tan x}{x})^{\frac{1}{x^2}}$ which is equivalent to $\lim_{x \rightarrow 0} \frac{\log ( \...
2
votes
1answer
51 views

Calculating the surface area of a curve

$$y=\frac{x^2}{32}$$ rotated about the y-axis, from $x \in [0,8]$ I'm using the following approach, but I keep getting the answer wrong. $$A= \int_{0}^2 2 \pi x \sqrt{1+\frac{x^2}{16^2}}\,d x$$ ...
2
votes
1answer
91 views

Showing $\lambda_1=\rho_1$ in monoidal category

For a monoidal category $\mathcal{C}$ with $\alpha_{a,b,c}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c$, $\rho_a : a \otimes 1 \rightarrow a$, and $\lambda_a: 1 \otimes a \rightarrow ...
3
votes
0answers
59 views

The order of a differential operator on a manifold is well-defined

In these notes, the author defines a differential operator of order at most $n$ on a manifold $M$ as an element of $\operatorname{Diff}^n(M) := \operatorname{span}_{0\leq j \leq n} (C^\infty(M;TM))^j.$...
3
votes
5answers
156 views

How do I succinctly note the sum of $(n-1)+(n-2)+…$?

I was playing with numbers and wanted to see how many possible connections there are in a network of $n$ nodes. I found that the answer was equal to ...
0
votes
1answer
46 views

One-To-One functions

Let A be the set with n elements and B be the set with m elements. How many one-to-one functions are there from A to P(B) (power set of B). There are n! total functions from A to B. and (2m)n from A ...
0
votes
1answer
114 views

Dollar weighted return. Formula or definition?

I was learning dollar-weighted return and I was a bit puzzled by the following and I would like to have some advice. I understand that it's basically the internal rate return, but using simple ...
0
votes
1answer
27 views

Absolute value of the sume of two complex number

I have a question about the following. $|A+B|^2$, where $A, B $ is complex number. The question is , when can $|A+B|^2$ be equal to $|A|^2 + |B|^2$?
0
votes
3answers
122 views

Finding the extrema of $f(x) = \sin x + \cos x$

Find all extrema in the interval $[0, 2\pi]$ for $y=\sin(x) + \cos(x)$ I got MAX:$(0,1), (2\pi, 1)$ / MINI:$(5\pi/4, -\sqrt{2})$ I am not sure about the max points.
1
vote
1answer
101 views

In war with exercise, any future for me?

I love theory with theorems, definitions & proofs, but i don't like exercise, I need more context around it. Is there a different way of practicing theory except given exercises, maybe some ...
2
votes
1answer
76 views

For every infinite class C of sets in V the universe is there an infinite set $x$ such that $x\subset C$?

For every infinite class C of sets in V the universe is there an infinite set $x$ such that $x\subset C$? I wasn't sure about how to phrase the question, I could have also asked, is V closed under ...
0
votes
0answers
92 views

Mean of standard deviation and confidence intervals

I have 1000 values from which I calculate the mean A and the standard deviation std A and the 95% confidence interval. I have another 1000 values from which I calculate the mean B and the standard ...
1
vote
0answers
122 views

Existence and uniqueness of a solution for the stationary transport equation

Can anyone help me with this problem? Let $\lambda > 0$, $a$ and $S \in C^1(\mathbb R^n)$. Supose that it exists $M > 0$ such that: $a(x) \geq 0$ and $|\nabla a(x)| + |S(x)| + |\nabla S(x)| \...
3
votes
0answers
93 views

Proving convergence of series [duplicate]

Prove whether the following series converge or diverge. $$\sum \limits_{n=1}^{\infty} \frac{(2n)!}{(4^n)(n!)^2(n^2)}$$ I think this series converge and I tried to justify using the ratio test but I ...
5
votes
3answers
152 views

An inequality related to the cosine theorem

Let $A,B,C$ be the three angles in the a triangle (with length $a,b,c$). Can we show that $$x^2+y^2+z^2\geq 2x y \cos A+2xz\cos B+2yz\cos C?$$ for all $x,y,z\in\Bbb R$. I do not see whether it is ...
1
vote
4answers
36 views

Convergence of a function with $e$ in the denominator

$$\int^{\infty}_1\frac{dx}{x^3(e^{1/x}-1)}$$ I'm given the hint that the function $y = e^x$ has a tangent $y=x+1$ when $x=0\land y=1$. How do I prove its convergence and find a upper-limit for the ...
2
votes
1answer
91 views

How to organize myself around calculus?

Calculus is the most advanced topic I have encountered in math. The book that I am using is clear as can be, but it has so many definitions and theorems. I would like to have all the most crucial ...
1
vote
0answers
45 views

$L^2$ inequality for derivatives of polynomials on triangles

I'm reading a paper which states the following inequality, but the (presumably) elementary proof is cited to be in a document, which is too old to get access to. Let $p: \mathbb{R}^2 \to \mathbb{R}$ ...
1
vote
1answer
222 views

Finding the length of a curve? Is this question impossible?

Find the length of the curve $$y^2=4(x+4)^3, 0 \le x \le 2, y \gt 0$$ I applied the formula for finding the length of a curve, $$L = \sqrt{1- (f'(x))^2}$$ And in turn I got the following equation: ...
2
votes
2answers
93 views

Two ways to localize a ring using a prime ideal

I was reading the part about localization of the Introduction to Commutative Algebra of Atiyah-MacDonald and I have a question I was not able to solve. Let $R$ be a commutative ring with unit $1$ and ...
0
votes
3answers
114 views

Determinant of a 4x4 Matrix

Find determinants of matrices A=$\begin{bmatrix}a & 3 & 0 & 5\\0 & b & 0 & 2\\ 1 & 2 & c &3\\ 0&0&0&d \end{bmatrix}$ and B=$\begin{bmatrix}x & y&...
0
votes
1answer
37 views

how am I supposed to do these problems any differently? (finding basis for row space)

I'm given two problems, which look exactly the same except the second one says "consisting of only row vectors of A". here are the problems: on 5 II, I ended up row reducing and writing my basis ...
1
vote
4answers
123 views

Is $\frac{5x}{3}$ The Same As $\frac{5}{3}x$?

I believe they are the same but I'm not sure. Can someone please clarify this for me, and also explain why it would be the same or different.
0
votes
1answer
52 views

Solve Integral $ \int_{-\infty}^{+\infty} \log[F(y_k)]*F(y_k)^{\frac{1-\theta}{\theta}}*f(y_k) $ for MVUE proof

Let $ \theta > 0 $ a parameter, $ Y_1 ... Y_n $ is a set of iid observations with marginal distribution function $F_\theta(y) = [F(y)]^{1/\theta} ,-\infty < y < +\infty $ Show that $ t = -\...
4
votes
1answer
90 views

Is the existence of such a transitive model $M$ of ZFC consistent?

Questions. Q0. Does anyone know of a refutation, in ZFC, of the following statement? Q1. If not, does ZFC plus large cardinals prove its consistency? Statement. There exists a transitive model $M$ ...
2
votes
1answer
65 views

Euler Characteristic for Convex Polyhedra

It is well-known that a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces has Euler characteristic $2$: $V-E+F=2$. Conversely, given a triple $(V,E,F)$ satisfying the above relation, can ...
2
votes
1answer
77 views

Absolute Value Trig Sum

I have been trying to solve $$y(x)=\sum_{k=1}^{\infty} \frac{|\cos(kx)|}{k}$$ however, this is proving to be more difficult than I had hoped, and cannot seem to figure this out. What I have figured ...
0
votes
0answers
91 views

convexity of function built from piecewise linear convex function?

Let $B(x):[0,1] \rightarrow [0,1]$ be piecewise linear increasing convex function with $B(0)=0$ and $B(1)=1$. (Think of the power of the neyman-persron test). Let $E(x)=-log(B(1-e^{-x}))$ a logaritmic ...
0
votes
1answer
31 views

Condition for dim of the Euclidean space with orthogonal basis

I would like to show that if the orthogonal basis of the $\Bbb R^n$ Euclidean space with the standard dot product has the vectors whose elements are exclusively $1$ or $-1$, then $n \le 2$ or $n$ is ...
0
votes
0answers
52 views

Is $f=t^5 + t^4 + 1$ reducible or irreducible over the field of $Z_2$ integers modulo 2.

I need to find out whether $f=t^5 + t^4 + 1$ is reducible or irreducible over the field of $Z_2$ integers modulo 2. I approached the question by substituting 0 and 1 into the function and got answers ...
1
vote
2answers
178 views

Difficult but Interesting Inequalities Problems

1.) Consider the identity $$(px + (1-p)y)^2 = Ax^2 + Bxy + Cy^2.$$ Find the minimum of $\max(A,B,C)$ over $0 \leq p \leq 1$. 2.) Let $n$ be a positive integer. Show that the smallest integer ...
3
votes
3answers
201 views

A specific kind of probabilistic proof for central binomial coefficients

I'm looking for a specific kind of proof of the statement $$ \lim_{n\to\infty} \frac1{4^n}\binom{2n}{n} = 0 $$ I know how to show this using Stirling's formula; I have seen the very nice elementary ...
0
votes
2answers
82 views

Prove $S^{-1}\mathbb{Z_6}\simeq \mathbb{Z}_3$

Prove $S^{-1}\mathbb{Z_6}\simeq \mathbb{Z}_3$ when $S=\{\overline{1},\overline{2},\overline{4}\}$. Note: $S^{-1}\mathbb{Z_6}= \frac{S\times \mathbb{Z}_6}{\sim }$ where $(x,y)\sim (u,v) \iff \exists ...
1
vote
2answers
120 views

Maximum and minimum function on an interval

Let $I := [a,b]$, where $a<b$. Suppose that $f$ is continuous and $1-1$ on $I$. Let $m$ denote the minimum value of $f$ on $I$ and let $M$ denote the maximum value of $f$ on $I$. (a) Carefully ...
0
votes
1answer
409 views

Calculating tangent vectors and normal to the bezier curve.

I am preparing for the graphics midterm and one of the practice problems dealing with parametric surfaces is as follows: A bilinear patch x(u,v) is given by four control points $$p_0 = (2, 0, 1) \\ ...
0
votes
1answer
84 views

Expressing a hypercube subset definition using set notation

The definition of a hypercube is this: The $n $-dimensional hypercube $Q_n$ is the graph with $V = \left\{{ (e_1,\dots,e_n)|e_i \in \left\{{0,1}\right\}(i=1,\dots,n)}\right\}$ in which two ...
2
votes
0answers
113 views

Projections of a rational normal curve of $\mathbb{P}^4$ (Exercise 3.9 in Harris' _Algebraic Geometry_)

In exercise 3.9 of his Algebraic Geometry book, Prof. Harris asks to show that the rational quartic curves $$C_{a,b}=[X^4-aX^3Y,X^3Y-aX^2Y^2,bX^2Y^2-XY^3,bXY^3-Y^4]$$ are projections of a rational ...
2
votes
0answers
92 views

Limit of $(\frac{1}{n^2})^{(\frac{1}{n})}$

Question: is this a valid procedure for finding the limit of $$\bigg(\frac{1}{n^2}\bigg)^{(\frac{1}{n})}$$ as $n\rightarrow \infty$. background: This is one of the examples of the failure of the ...
5
votes
1answer
329 views

Basic misunderstanding of the theorema egregium

The theorema egregium demonstrates that the Gaussian curvature, $K$, is an intrinsic property. What I think this means is that if you know the metric corresponding to the surface, then you can compute ...
1
vote
0answers
101 views

Find max distance from $(0,0)$ to line defined on ellipse.

I have got a following problem : $E = \{ \frac{x^2}{a^2} + \frac{y^2}{b^2} =1 \}$ $N$ - line (normal) perpendicular to E at $(x_0,y_0)$ Find max $dist(N,(0,0))$ So I am starting with attempt to ...
0
votes
0answers
42 views

Partial fractions in Laplace Transform

Solve: $$y''+y'+\frac{5}{4}=U_\frac{\pi}{2}(t)f(t-\frac{\pi}{2})$$ becomes: $$[s^2+s+\frac{5}{4}]Y(s)=\frac{1-e^\frac{-\pi*s}{2}}{s^2}$$ becomes: $$Y(s)=\frac{1-e^\frac{-\pi*s}{2}}{s^2[(s+\frac{...
0
votes
1answer
27 views

What does it mean when you have $\operatorname{Pr}\limits_{h\in \mathcal{H}}$

I'm asked to prove that a family of hash functions is $2$-wise independent. I'm told that: $\mathcal{H}$ is $k$-wise independent if for any $k$ inputs $x_1,...,x_k$ and hash values $v_1,...,v_k$, $\...
1
vote
0answers
58 views

semi-infinite heat equation with Dirichlet BC via Laplace transforms

I am trying to solve the heat equation for a semi infinite rod with lateral surfaces insulated and $u(x,0)$ = $u_0$ for $x>0$, $u(0,t)=u_1$ for $t>0$, and the $\lim_{t\to\infty} u(x,t)=u_0$. I ...

15 30 50 per page