# All Questions

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### 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!
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 $m=2^l$): $\mathcal{H} = \{h_{A,b}|A\in \{0,1\}^{l\times w}, b \in ... 2answers 32 views ### Find the$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: $$\frac{\sin(A)... 1answer 150 views ### Group and Ring Homomorphisms between \mathbb{Z}_p and \mathbb{Z}_{2p} Let p be prime with p > 2. (a) Determine the number of group homomorphisms between \mathbb{Z}_p and \mathbb{Z}_{2p} (b) Determine the number of ring homomorphisms between \mathbb{Z}_p ... 1answer 43 views ### Infinite graphs satisfying a certain Ramsey property Let G be a countably infinite graph. If G has cliques of arbitrarily large finite size, then G satisfies the following property, which I will call (*): for any r\in \mathbb{N} and any r-... 3answers 460 views ### Extending a basis for a subspace of V to create a basis of V Can any set that forms a basis for a subspace of a vector space V be extended to form a basis for V? 1answer 42 views ### Convergence In probablility implies convergence in distribution? I'm currently working on the following exercise: If X_n is sequence of randon maps with values on a metric space (S,d). Show tha convergence in probablity to a randon map X implies P\circ ... 3answers 97 views ### Find \lim_{x \rightarrow 0} (\frac{\tan x}{x})^{x^{-2}} I see that this is in the 1^ \infty form, so I've taken log to get: \lim_{x \rightarrow 0} \log( \frac{\tan x}{x})^{\frac{1}{x^2}} which is equivalent to \lim_{x \rightarrow 0} \frac{\log ( \... 1answer 51 views ### Calculating the surface area of a curve$$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$$... 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 ... 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.... 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 ... 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 ... 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 ... 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? 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. 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 ... 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 ... 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 ... 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)| \... 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)}$$I think this series converge and I tried to justify using the ratio test but I ... 3answers 152 views ### An inequality related to the cosine theorem Let A,B,C be the three angles in the a triangle (with length a,b,c). Can we show that$$x^2+y^2+z^2\geq 2x y \cos A+2xz\cos B+2yz\cos C?$$for all x,y,z\in\Bbb R. I do not see whether it is ... 4answers 36 views ### Convergence of a function with e in the denominator$$\int^{\infty}_1\frac{dx}{x^3(e^{1/x}-1)}$$I'm given the hint that the function y = e^x has a tangent y=x+1 when x=0\land y=1. How do I prove its convergence and find a upper-limit for the ... 1answer 91 views ### How to organize myself around calculus? Calculus is the most advanced topic I have encountered in math. The book that I am using is clear as can be, but it has so many definitions and theorems. I would like to have all the most crucial ... 0answers 45 views ### L^2 inequality for derivatives of polynomials on triangles I'm reading a paper which states the following inequality, but the (presumably) elementary proof is cited to be in a document, which is too old to get access to. Let p: \mathbb{R}^2 \to \mathbb{R} ... 1answer 222 views ### Finding the length of a curve? Is this question impossible? Find the length of the curve$$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}$$And in turn I got the following equation: ... 2answers 93 views ### Two ways to localize a ring using a prime ideal I was reading the part about localization of the Introduction to Commutative Algebra of Atiyah-MacDonald and I have a question I was not able to solve. Let R be a commutative ring with unit 1 and ... 3answers 114 views ### Determinant of a 4x4 Matrix Find determinants of matrices A=\begin{bmatrix}a & 3 & 0 & 5\\0 & b & 0 & 2\\ 1 & 2 & c &3\\ 0&0&0&d \end{bmatrix} and B=\begin{bmatrix}x & y&... 1answer 37 views ### how am I supposed to do these problems any differently? (finding basis for row space) I'm given two problems, which look exactly the same except the second one says "consisting of only row vectors of A". here are the problems: on 5 II, I ended up row reducing and writing my basis ... 4answers 123 views ### Is \frac{5x}{3} The Same As \frac{5}{3}x? I believe they are the same but I'm not sure. Can someone please clarify this for me, and also explain why it would be the same or different. 1answer 52 views ### Solve Integral \int_{-\infty}^{+\infty} \log[F(y_k)]*F(y_k)^{\frac{1-\theta}{\theta}}*f(y_k) for MVUE proof Let \theta > 0 a parameter, Y_1 ... Y_n is a set of iid observations with marginal distribution function F_\theta(y) = [F(y)]^{1/\theta} ,-\infty < y < +\infty Show that t = -\... 1answer 90 views ### Is the existence of such a transitive model M of ZFC consistent? Questions. Q0. Does anyone know of a refutation, in ZFC, of the following statement? Q1. If not, does ZFC plus large cardinals prove its consistency? Statement. There exists a transitive model M ... 1answer 65 views ### Euler Characteristic for Convex Polyhedra It is well-known that a convex polyhedron with V vertices, E edges, and F faces has Euler characteristic 2: V-E+F=2. Conversely, given a triple (V,E,F) satisfying the above relation, can ... 1answer 77 views ### Absolute Value Trig Sum I have been trying to solve$$y(x)=\sum_{k=1}^{\infty} \frac{|\cos(kx)|}{k}$$however, this is proving to be more difficult than I had hoped, and cannot seem to figure this out. What I have figured ... 0answers 91 views ### convexity of function built from piecewise linear convex function? Let B(x):[0,1] \rightarrow [0,1] be piecewise linear increasing convex function with B(0)=0 and B(1)=1. (Think of the power of the neyman-persron test). Let E(x)=-log(B(1-e^{-x})) a logaritmic ... 1answer 31 views ### Condition for dim of the Euclidean space with orthogonal basis I would like to show that if the orthogonal basis of the \Bbb R^n Euclidean space with the standard dot product has the vectors whose elements are exclusively 1 or -1, then n \le 2 or n is ... 0answers 52 views ### Is f=t^5 + t^4 + 1 reducible or irreducible over the field of Z_2 integers modulo 2. I need to find out whether f=t^5 + t^4 + 1 is reducible or irreducible over the field of Z_2 integers modulo 2. I approached the question by substituting 0 and 1 into the function and got answers ... 2answers 178 views ### Difficult but Interesting Inequalities Problems 1.) Consider the identity$$(px + (1-p)y)^2 = Ax^2 + Bxy + Cy^2.$$Find the minimum of \max(A,B,C) over 0 \leq p \leq 1. 2.) Let n be a positive integer. Show that the smallest integer ... 3answers 201 views ### A specific kind of probabilistic proof for central binomial coefficients I'm looking for a specific kind of proof of the statement$$ \lim_{n\to\infty} \frac1{4^n}\binom{2n}{n} = 0 $$I know how to show this using Stirling's formula; I have seen the very nice elementary ... 2answers 82 views ### Prove S^{-1}\mathbb{Z_6}\simeq \mathbb{Z}_3 Prove S^{-1}\mathbb{Z_6}\simeq \mathbb{Z}_3 when S=\{\overline{1},\overline{2},\overline{4}\}. Note: S^{-1}\mathbb{Z_6}= \frac{S\times \mathbb{Z}_6}{\sim } where (x,y)\sim (u,v) \iff \exists ... 2answers 120 views ### Maximum and minimum function on an interval Let I := [a,b], where a<b. Suppose that f is continuous and 1-1 on I. Let m denote the minimum value of f on I and let M denote the maximum value of f on I. (a) Carefully ... 1answer 409 views ### Calculating tangent vectors and normal to the bezier curve. I am preparing for the graphics midterm and one of the practice problems dealing with parametric surfaces is as follows: A bilinear patch x(u,v) is given by four control points$$p_0 = (2, 0, 1) \\ ... 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\}$in which two ... 0answers 113 views ### Projections of a rational normal curve of$\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 ... 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 ... 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 ... 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 ... 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{... 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$v_1,...,v_k$,$\...
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 ...