1
vote
1answer
22 views

Solve L'Hopitals problem

$$\lim_{x\rightarrow \frac{\pi}{2}} \frac{\sec x}{{\sec^2 3x}} $$ I used LH: $$\lim_{x\rightarrow \frac{\pi}{2}} \frac{\sec x \ tan x}{6\sec 3x \sec 3x \tan 3x}$$ then: $$\lim_{x\rightarrow ...
1
vote
2answers
129 views

how to solve $\displaystyle \frac{a^3}{a^2+2b+c}+\frac{b^3}{b^2+2c+a}+\frac{c^3}{c^2+2a+b}\geq\frac{3}{4}$

$a,b,c>0,a+b+c=3,$ prove that: $$\frac{a^3}{a^2+2b+c}+\frac{b^3}{b^2+2c+a}+\frac{c^3}{c^2+2a+b}\geq\frac{3}{4}$$
0
votes
0answers
46 views

Difference between the “Hazard Rate” and the “Killing Function” of a diffusion model?

I posted this question on Cross Validated - but I think it applies here too. Also, it increases the chances of the question being answered. Link here If this is not acceptable - administrators ...
0
votes
0answers
7 views

Solve DOE system with polar coordinates?

I am studying for a exam and one of model questions is solve a DOE system using polar coordinates. I've research and didn't find any reference about this subject. System in question is $$ ...
1
vote
2answers
55 views

What's the importance of proving that $0,1$ are unique?

I had a course in the construction of numbers last semester. I understand the potencial of most of the proofs, for example: I guess I can answer decently why commutativity is important. But when it ...
0
votes
0answers
14 views

Fiding the most general antiderivative of a function bounded by two x's.

At first I thought this problem would simply become a definite integral since it appears two be bounded by two x's. However, I feel as though I may be wrong and I'm curious as to how I would approach ...
1
vote
1answer
28 views

Use L'Hopital's with this problem?

The problem is: $$\lim_{x\rightarrow 0^+} \left(\frac{1}{x}\right)^{\sin x}$$ I know the answer is $1$ because I checked with my graphing calculator, but how exactly do I get there? I got this far: ...
18
votes
1answer
327 views

Does there exist regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ (treated as a surface) includes 5 points which form a regular planar pentagon?
2
votes
0answers
64 views

What methods are known to visualize patterns in the set of real roots of quadratic equations?

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations ...
0
votes
1answer
14 views

How to perform the following change of variables?

Suppose I have a function $f(t,x,y)$ such that $$f_t = \nabla^26f(t,x,y).$$ I want to perform a change of variables so that for $f(t',x',y')$ we have $$f_{t'} = \nabla^2f(t',x',y').$$ Expanding the ...
4
votes
0answers
33 views

Can anyone improve on this work and find a closed form of $\zeta(3)$?

This was something I and another user came across independently, although he decided to post it on reddit. So while its already online, let me reproduce it here with the hope that someone will be able ...
0
votes
0answers
15 views

UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$

suppose $X_1,X_2,\ldots,X_n$ be random sample of $N(0,\sigma^2)$. how can I calculate UMVUE of parameter $(1-\sigma^2)^{-\frac{n}{2}}$
1
vote
1answer
58 views

$(1)$ $a\boxplus b=a+b-ab$ for all $a,b \in F$. $(2)$ $a\boxdot b=1-t^{\log_t{(1-a)}\log_t{(1-b)}}$ for all $a,b\in F$

Let $1<t\in \mathbb{R}$ and let $F=\{a\in \mathbb{R}: a<1\}$. Define $\boxplus$ and $\boxdot$ on $F$ as follows: $a \boxplus b=a+b-ab$ for all $a,b \in F$. $a \boxdot b=1-t^{\log_t ...
5
votes
3answers
17 views

Significant Figures in quantity?

I am still having trouble grasping significant figures. How many significant figures would be in this quantity? $01010.0$ I feel like it is 3, but not quite sure.
1
vote
0answers
19 views

$\operatorname{Im} A = (\operatorname{ker} A^*)^\perp$

Let $A:\mathbb{R}^m \to \mathbb{R}^n$ be a linear transformation. We know that there is a unique transformation $A^*:\mathbb{R}^n \to \mathbb{R}^m$ such that $$\langle Ax,y\rangle = \langle x,A^*y ...
2
votes
3answers
47 views

Minimum value of $\cos x+\cos y+\cos(x-y)$

What is the minimum value of $$ \cos x+\cos y+\cos(x-y). $$ Here $x,y$ are arbitrary real numbers. Mathematica gives (with NMinimize) $-3/2$. But I don't know if this is correct and if so, how to ...
-2
votes
1answer
18 views

Probability and Statistics - to understand expectation and variance better

A strange clause in a version of Dungeons and Dragons says: roll a d6 (a six-sided die with faces from 1 to 6). If the value rolled is 3 or less, roll a d8 else roll a d10. Add the two values ...
0
votes
1answer
17 views

Discrete Math logically equivalent?

Show that p ↔ q and (p ∧ q) ∨ (¬p ∧ ¬q) are logically equivalent. How would I go about doing this? Do I use a truth table or a more "algebraic" process?
-1
votes
2answers
30 views

Please help me with this sigma notation

Use the properties of summation to evaluate the following: $$\displaystyle \sum^{4}_{i=1}(i^2-3i)$$
0
votes
1answer
15 views

If I have $m \times n$ matrix A and a vector $x \in \mathbb{R}^m$, Can I make Ax working? Will it be possible for Ax to do row operations?

If I have $m \times n$ matrix A and a vector $x \in \mathbb{R}^m$,where m>n, Can I make Ax working? Will it be possible for Ax to do row operations? If yes, then how do they operate?
8
votes
4answers
624 views

What are some easily-stated recently proven theorems?

What are some easily-stated relatively recently proven theorems? I don't mean they were necessarily easy to prove, just easy to state. Here are a few examples: The proof of Fermat's Last Theorem ...
11
votes
1answer
74 views

Distribution of $\sum\limits_{i=1}^{N}X_{i}$ conditionally on $\sum\limits_{i=1}^{N}X_{i}^{2}$ for i.i.d. standard normal $X_i$s

Assume that the random variables $X_{i}$ are i.i.d $\mathcal{N}\left(0,1\right)$, then: $$S_N=\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)\qquad\qquad ...
2
votes
1answer
25 views

transformation of uniformly distributed random variable f(x)=1/2pi into Y=cosx

Let X be a uniformly distributed function over $[-\pi􀀀;\pi]$. That is $ f(x)=\left\{\begin{matrix} \frac{1}{2 \pi} & -\pi\leq x\leq \pi \\ 0 & otherwise \end{matrix}\right.\\ $ Find the ...
-2
votes
0answers
20 views

An compute of Riemannian geometry [on hold]

According to Einstein summation convention , $g_{ij}$ is metric tensor,and $f$ is a real function. Show that : $$ g_{jl}g^{ij}\frac{\partial f}{\partial x^i}g^{kl}\frac{\partial f}{\partial ...
1
vote
1answer
21 views

Solve for $x$ in the following inequality

$2x+x^3+7\gt0$ I have no idea what to do here, because my level of knowledge goes up to binomials and such. Thank you
-1
votes
2answers
40 views

proof of isosceles triangle?

How do you prove this isosceles triangle? Given line AC is congruent to line BC Prove: Angle A=Angle B I've gotten to the angle bisector and SAS(side- angle- side), and I believe there is one more ...
0
votes
1answer
14 views

Given the end-vertices of two line segments, how do you calculate the point at which they intersect?

Given only the vertices of each line segment, and it's assumed they intersect, how do I calculate the point at which they intersect (in two and additionally three dimensions)?
-2
votes
0answers
23 views

Find a metric space $(X,d)$, such that $\partial B_r(x)\neq S_r(x)$, where $S_r(x)$ ={ y $\in$X: $d(x,y)=r$} and $B_r(x)$ ={ y $\in$X: $d(x,y)<r$}

Find a metric space $(X,d)$, such that $\partial B_r(x)\neq S_r(x)$, where $S_r(x)$ ={ y $\in$X: $d(x,y)=r$} and $B_r(x)$ ={ y $\in$X: $d(x,y)<r$}
5
votes
0answers
68 views
+100

Help provide a proof of the Helly–Bray theorem

Given a probability space $(\Omega,\cal F, \Bbb P)$, the distribution function of a random variable $X$ is defined as $F(x)=\Bbb P\{X \le x\}$. Now if $F_1,F_2,...,F_{\infty}$ are distribution ...
3
votes
2answers
36 views

Borel sets: alternative characterization for metric space

For any topological space $(X,\tau)$, the Borel $\sigma$-algebra $\mathcal{B}$ is the $\sigma$-algebra generated by the open sets. In other words, it is the intersection of all $\sigma$-algebras on ...
-2
votes
1answer
21 views

Notation methods for the following things?

I go to a high school that rushes concept and does not ever talk about notation. I want to be prepared for college, and not be swamped by all this notation I don't know. From SE, I would like to know ...
2
votes
1answer
27 views

If I had a matrix A, what is the meaning of $A^T Ax$, given $A^T$ is transpose of A and x are vectors of variable?

If I had a matrix $A$, what is the meaning of $A^TAx$, given $A^T$ is the transpose of $A$ and $x$ is a vector? Is it operation on $x$ by the result of the multiplication of two matrices, or is it ...
8
votes
1answer
1k views

Vectors, Basis, Dual Vectors, Dual Basis and Tensors

I'm trying to understand tensors and I know they have something to do with the basis and the dual basis of a vector space and a dual space. First I will give a concrete example to make clear what I ...
3
votes
6answers
414 views

Show that if $x>0$, then $\ln(x)\geq 1-\frac{1}{x} $

Show that if $x>0$, then $$ \ln(x)\geq 1-\dfrac{1}{x}. $$ I tried a few things but so far nothing has worked, I could use a hint.
-2
votes
2answers
15 views

How far is the bottom of the ladder from the house?

A 13 foot ladder is leaning against a house. The distance from the bottom of the ladder to the house is 7 feet less than the distance from the top of the ladder to the ground. How far is the bottom ...
0
votes
1answer
16 views

Poisson sampling

Suppose I have a pdf $f(S)$. $f(S)$ describes the size of firms in the economy. Also define the Bernoulli variable $X_{f} \in \{0,1\}$ where $P(X_{f}=1)=g(S_{f})$ and $P(X_{f}=0)=1-g(S_{f})$. $S_{f}$ ...
2
votes
3answers
81 views

$f:\mathbb R^{2} \rightarrow \mathbb R$ s.t ${f(x,y)}={{xy}\over {x^{2}+y}}$ is not continuous at the origin

$f:\mathbb R^{2} \rightarrow \mathbb R$ is defined as $${f(x,y)}={{xy}\over {x^{2}+y}}$$; when $x^{2}+y\neq 0$ and $$f(x,y)=0$$ otherwise. To show this is not continuous at the origin . ...
11
votes
3answers
167 views

Show that $(1+\frac{1}{n})^n+\frac{1}{n}$ is eventually increasing

I would like to find a way to show that the sequence $a_n=\big(1+\frac{1}{n}\big)^n+\frac{1}{n}$ is eventually increasing. $\hspace{.3 in}$(Numerical evidence suggests that $a_n<a_{n+1}$ for ...
1
vote
2answers
21 views

Problem with injective functions on an explanation of the Birthday problem

The Wikipedia article on the Birthday problem gives an "abstract proof" of the problem, in which the birthday function $$ b:\mathcal{S} \mapsto \mathcal{B} $$ where $\mathcal{S}$ is the set of ...
1
vote
2answers
26 views

Question about product topology notation

Instead of using the general form, I will use a simpler one such as $\mathbb{R} \times \mathbb{R}$ (which is $\mathbb{R}^2$ of course). Now the notation says that the open sets are the union of the ...
-3
votes
1answer
8 views

Calculus :Work of an inverted right circular cone

A tank in the shape of an inverted right circular cone has height 6 meters and radius 4 meters. It is filled with 5 meters of hot chocolate. Find the work required to empty the tank by pumping the hot ...
2
votes
1answer
45 views

Calculate the pushforward of smooth map between manifolds

Let $\Phi : GL(n)\rightarrow Sym(n)$ be defiened by $\Phi (A)=AA^T$. I can not see how to get the "right" pushforward, I.e I want help in understanding the pushforward $\Phi _*:M_I(n)\rightarrow ...
0
votes
1answer
390 views

general formula for an orthogonal projection of a point onto a line

Could someone confirm this or correct the mistakes because this seems somehow wrong although I double checked it. $(m_x,m_y)$ are coordinates of a point , $(p_x,p_y),(k_x,k_y)$ are coordinates of a ...
4
votes
1answer
16 views

What is the probability that a customer will not use a credit card? Pays in cash or with a credit card?

So I'm doing some basic probability problems for homework, and we just recently went over the Inclusion-Exclusion prinicple, which I'm assuming this problem deals with, which is as follows. ...
-1
votes
1answer
17 views

discrete finite summation of non-linear functions

Can anyone have idea for dealing with the two following series summations ∑_(i=1)^n▒1/(a+bx_i )=c ∑_(i=1)^n▒x_i/(a+bx_i )=d I need to find the values of 'a' and 'b'; 'c' and 'd' are known. x_i is ...
2
votes
3answers
52 views

Notation: $f(A)$ when $f$ is a function $f:A\to B$.

I've seen the following notation with no previous clarification: $f(A)$, when $f$ is a function $f:A\to B$. Am I correct to assume $f(A)$ should be the image of $f$? E: I'd appreciate downvoters ...
2
votes
1answer
23 views

Euclidean Geometry (Potential Menelaus Theorem)

I have a strong suspicion that this problem applies Menelaus's theorem, but I can't see it. I also tried algebraic manipulation (such as trying to re-write BD/DC in terms of AB or CP), but to no ...
2
votes
0answers
53 views

solving definite integral problems without complex line integral

It is well known that some definite integrals such as $$\int_{0}^{\infty} \frac{dx}{a+\cos{x}}$$ $$\int_{0}^{\infty} \frac{\sin{x}}{x}dx$$ are solved by using complex analysis techniques. (It uses ...
0
votes
0answers
37 views

Distribution of the test statistic?

Let $\mathbf{x}_i \sim \mathcal{N}(\boldsymbol\mu, \boldsymbol\Sigma)$. I am trying to find a distribution of the following test statistic $ T(\mathbf{x}) = \frac{\bar{\mathbf{x}}^T ...
2
votes
1answer
18 views

Optimizing number of 6-digit strings differing in at least two places

A certain province issues license plates consisting of six digits (from 0 to 9). The province requires that any two license plates differ in at least two places. (For instance, the numbers ...

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