Give a formula $F=M(x,y)i + N(x,y)j$ for the vector field in the plane that has the property that F points toward the origin with magnitude inversely proportional to the square of the distance from (x,y) to the origin. (The field is not defined at (0,0)). I first found the norm of the vector $|F...

I have been selected by my college to teach integration to kids in the age group of 8-12. I am an engineering major who has finished Calculus 1 and 2 but I have no idea how to teach integration from scratch to kids that small and at the same time make it fun for them. I am asked to create a lesso...

I am given ( a simplistic definition I think ) of Mandelbrot set: M- set of complex numbers $c \in \mathbb{C}$ s.t. the sequence $(z_n)$ is bounded where $z_0=0 , z_{n+1}=z_n^2+c $ Need to show that: (i)$-2\in M$ (ii) $1/4 \in M$. Am I missing something here or is it kind of trivial? (i) ...

Consider a bipartite graph $(G, U, V)$. Each $v$ in $V$ represents a soccer team, and each $u$ in $U$ represents a mini-tournament needs to be scheduled. If $u_i$ and $u_j$ share no common neighbor, these two tournament can be scheduled on the same day. Similarly, one can schedule multiple tourn...

So my question is pretty simple : Are any groups of order 6 isomorphic ? I would say no, but I know that if the groups are cyclic then yes. If the answer is indeed no, could I please have a counter-example ? Thank you.

Find a point on the surface 1/x+1/y+1/z=1 (x>0,y>0,z>0) such as the tangent plane at this point is parallel to the plane x+2y+3z=0

I see this: $<nx> = nx - ⌊xn⌋$ in one of my exercise solutions for probability theory class. I found on wikipedia that <> means average and ⌊⌋ means greatest integer with the following examples: $⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2, ⌊−2.6⌋ = −3$ Now I'm wondering why $⌊2.9⌋ = 2$ and $⌊−2.6⌋ = −3$. ...

$\int_{-\pi}^{\pi} {x^{1000}\sin x}dx$ As I understand, the definite integral equals to $0$. However, what are the ways to show that properly?

I just had a few questions concerning the epsilon delta proof of limits. To be more precise, I always get lost at the part where we take $\delta$ to be the minimum of two real numbers. Let me provide a concrete example to work with. Consider $\displaystyle\lim_{x \to 2}x^2 = 4$ Given $\epsilon >...

Here, x is the unique solution mod $\frac{n}{gcd(a,n)}$ Let $gcd(a,n)>1$ and consider the equation: $\frac{a}{gcd(a,n)}x = \frac{b}{gcd(a,n)} mod \frac{n}{gcd(a,n)}$

My question is the following : Let n ≥ 1 be an integer. Every homomorphism of groups $f$: ℤ/nℤ → ℤ is the zero homomorphism. a) Yes. b) No. c) Depends on n. I don't really know how to go with this one. Could someone re-explain briefly the zero homomorphism ? Thank you.

Could you give me some help on finding the roots (if any) of the following equation: $$ \frac{\sqrt{a+x}}{\sqrt{a}+\sqrt{a+x}}=\frac{\sqrt{a-x}}{\sqrt{a}-\sqrt{a-x}} $$ I tried to apply some classic approaches, but I had no luck... Could you lend me a hand? Thanks in advance!

Disclaimer: Probably stupid question Is there a polynomial representation of the rate of change with respect to $n$ of $$f(n)=2n-3+\sum_{m=0}^{n-3}{n-3-m}$$ and if so, how can I find it? (For those wondering, this is the maximum number of undirected connections between $n$ vertices.)

What is the purpose of linear approximation questions? For example, one question reads: (a) Use the Linear Approximation for f(x) = ln(x) at a = 1 to estimate ln(0.84)... (b) That Linear Approximation has error... How would I complete this question?

Let $L=\int^1_0 \frac{dx}{1+x^8}$. Then A. L<1 B. L>1 c. L D. L>$\frac{\pi}{4}$

I need to show that the sequence $x_n$ defined as : $x_0=0 , x_{n+1}=x_n^2+1/4 $ is bounded. Not sure how to approach this, some hints would be greatly appreciated :)

Let G be a group and H a subgroup of G. Define a relation on elements of G by saying that a ~ b if b$^-$¹ a ∈ H. This relation is : a) reflexive and symmetric, but transitive only if G is abelian. b) relfexive and transitive, but symmetric only if G is abelian. c) reflexive, symmetric and trans...

Equipped only with the knowledge that $MA = I$, $A$ is one-to-one and the output weighting matrix is $W$, how shall we prove that $(A^HWA)^{-1}A^HW = M$ is the form for all left pseudo-inverses of A? My thinking is that first we can show that $M = (A^HWA)^{-1}A^HW$ is at least one of the (possib...

I know you can't have all integers, but how do you factor this anyway? Wolfram|Alpha gives me $-\frac{1}{4} (1+\sqrt{5}-2 x) (-1+\sqrt{5}+2 x)$. Cymath gives me $(x-\frac{1+\sqrt{5}}{2})(x-\frac{1-\sqrt{5}}{2})$. The closest I can get is $(x+1)(x-1)-x$. So how do I get a nice answer like the o...

I think that when $k_n$ is big enough the limit will be 0, and when $k_n$ is small enough the limit will be going to 1. How do we make a formal analysis?

For naturals $n$, $f(n) = 1 + \dfrac{f(n) + f(n-1) + \cdots + f(1)}{n+1}$. What is $f(n)$? This is not a homework problem. Is there a general method to solve these recurrence relations? I will appreciate if someone directs me to a short tutorial/book to learn about solving recurrence relations.

Show $f(x)=\sqrt{x}$ is uniformly continuous on $[0,\infty)$ Let $\epsilon>0$, then there exists a $\delta=\epsilon^2$ such that $|x_1-x_2|<\delta$ for all $x_1,x_2\in[0,\infty)$. $\begin{align}|f(x_1)-f(x_2)|&=|\sqrt{x_1}-\sqrt{x_2}|\\&<\sqrt{|x_1-x_2|}\\&<\sqrt{\delta}\\&=\epsilon\end{a...

Does there exist a transitive action of S4 on the set {1,2,3,4,5} ? I would say no, because the cardinality of our set is bigger than 4, but I am not sure how to prove this. We can’t obtain the element {5} with the cycles of S4 , that’s the way I would go for it. Thank you.

I can not figure it out. Thanks.

I’m generally pretty good at proving limits using epsilon-delta, but on this one I’m stuck — and have been for days. This is the problem I’m talking about: \begin{equation} \lim_{x\to\infty}\frac{e^x}{e^x + x} = 1. \end{equation} I know that I need to let $\epsilon>0$ and choose an $N$ suc...

I know that because $W_t$ is a martingale, $$\int_{0}^{T} W_t dW_t = 0$$ then what should the value for this equation be: $$\int_{0}^{T} W_t^{n}dW_t$$

Are there any patterns, symbols, etc. in Pascal's triangle? For example, how do you draw a Pascal's triangle?

I need help to compute the following indefinite integral: $$\int_{-\infty}^{\infty}\frac{z^4}{1+z^8}dz$$ I need to use Cauchy's residue theorem. I can write that $z^8+1=z^8-i^2=(z^4-i)(z^4+1)$. How do I proceed? Please give a methodological answer so that I can solve other questions too.

Find all x-values where the tangent line is horizontal: y=(x+1)^4(x-2)^3

I've been working on a modification to the standard Conway's soldiers game. In Conway's soldiers, we have an endless number of soldiers in a grid of squares at and below point 0 North, and I can move soldiers left, right, up, and down by having them jump over other soldiers, which makes those ...

We would like to prove by induction that the number of the subsets of cardinality $2$ of a finite set with $n$ elements is given by $\frac{n(n-1)}{2}$. I know the reason why this is true, but how could I use it in the induction process? Something seems to be missing here...

What are the units of ℂ[ε] ? ℂ-{0} would be the units of ℂ, I get 3 other possibilities : a) {a+bε : a,b ∈ ℂ, a not equal to 0} b) {1} c) {bε : b ∈ ℂ, b not equal to 0} I would go for a) but not sure really. Thank you.

Hello I am really needing for someone to help for me to understand the following; I know that in fields, quadratics and cubics are irreducible if and only if they have roots. My question is, what is the form of these roots. For example to explain my confusion, consider the field with 121 elemen...

What does it mean that a theory admits constructive elimination of quantifiers? A theory admits elimination of quantifiers when each formula of the theory is equivalent to a quanifier-free formula, right? But what is meant when we use the term "constructive" ?

enter image description here Determine if each language is regular or not-regular. The former justied by providing a minimal DFA which accepts the said language and the latter by using the Pumping Lemma for Regular Expressions. Please help. Thanks :)

Let $f(x,y)$ be a real-valued function defined on an open set $S$ containing the origin. Prove the following by $\epsilon - \delta$ definition: If there exists: $$\lim_{(x,y)\to (0,0)} f(x,y)=L$$, and there exists: $$\lim_{x \to 0}\lim_{y \to 0} f(x,y)=L_{12}$$, then $L=L_{12}$ I'm trying to wo...

Can you help me find the likelihood function for the below? $$ X_i \sim \pi_0Gamma(1, β_1) + π_1Gamma(1, β_2) $$

Consider the following list of principal ideals (2), (3), (5), (6) in the ring ℤ/14ℤ. There are : a) Only one ideal in this list. b) Two distinct ideals in this list. c) Three distinct ideals in this list. d) Four distinct ideals in this list. I know the answer is related to divisors of 14, b...

How do I apply the squeeze theorem for finding the limit of this problem? Sorry, not sure how to format on here. lim(|x|cos^2(1/x)) x->0

I gather that mathematicians distinguish between sets of a single element and the element itself, that is $\forall x,\ x \ne \{x\}$. However, in regard to tuples, is it true that $\forall x,\ x = (x)$? $*$ Where $(x)$ is the singleton tuple with element $x$.

Can someone please help me to check over , I get a different answer then is written as the solution Suppose we are told that the weight of each gum ball ( in centigram) is given by the gamma distribution function, with $α=25$ and $β=2$ . We are wanting to know the probability that 100 gum bal...

Find the derivative of the function using the limit definition of derivative: f(x)=square root(x)

I was helping out my little brother with his math homework and I came across a question I'm pretty sure is unsolvable without the use of a graphing calculator. It's been a while since I've done math, so can anyone confirm for me? $\frac{120000}{1+48e^{-.015t} } = 24e^{.055t}$

This is a cultural question: Are there any, even moderately or historically used, units that measure solid angles which are not steradians? Basically, is there a unit x such that x:sr::grad:rad?

float cosine(float x, int j) { float val = 1; for (int k = j - 1; k >= 0; --k) val = 1 - x*x/(2*k+2)/(2*k+1)*val; return val; } int main( void ) { for( double x = 0; x <= PI/4; x += 0.9999 ) { if(cosine(x, 2) <= 0.001) { printf("cos(x) : %10g ...

In four dimensions, there can be two orthogonal axes of simultaneous rotation (of two planes), right? Does that mean that we can measure such rotations of a 3-sphere in solid angles, much as we measure the rotation of a circle (or 2-sphere- still one axis) in radians? Would there be a notion of "...

I really try hard to draw phase portrait shown below by mathematica or matlab, but I failed. Please help enter image description here

I'm working on a proof where I want any subset of $n+1$ distinct integers chosen from $\{1,2,...,2n\}$ has at least two numbers such that one divides the other. I have a feeling that this may be a problem related to modular arithmetic on the set $[2n]$, but I am having issues figuring out the equ...

I'm looking for a name for the (finite, boundless, impossible) shape created by taking a square and wrapping it so that the opposite edges are coterminous (I think that's the correct usage, it's been a while). Effectively, this is the shape formed by the Pac-Man board, where you exit the right s...

((root3/2)+(i/2))^25 I know that you have to put it in the form costheta+isintheta but i'm not sure how to go about it.

I'm wondering if I have a sufficient proof of the following: If $(a_n)$ is a sequence such that $\lim_{n \rightarrow \infty}a_n=A$, then $\lim_{n \rightarrow \infty}\frac{a_1+...+a_n}{n}=A$. My approach: For all $\epsilon > 0$, there exists $N$ such that for all $k>N$, $|a_k -A|<\epsilon$. S...

integrate and how absolute variable affect on the final result if Limit -0.5 to 0.5.

a. z^3 = (1-iroot3)8 b. z^2 - (3-2i)z + (1-3i) = 0 c. z^4 + 1 + iroot3 = 0 I know for the last two you start off using the quad form but i'm not sure what to mainly do for any of them.

How do you convolve multiple dirac-delta functions a rect function? Is the below convolution correct? Thank you! Anthonya picture of graphical convolution

Given: x^2 - 2 being irreducible in Q[x], 2x+5 and x^2 -2 relatively prime in Q[x]. Congruence class [2x+5] is a unit in the ring Q[x]/(x^2 - 2). Find the inverse u(x) given (2x+5)(u(x)) + (x^2-2)(v(x)) = 1.

I do not know how to approach this question. I have never dealt with first principle questions which involved a g(x) so I am a bit confused how I would go about it in the first place. Please help me!

$y=\frac{(2x+3)^9}{\sqrt x(x^2-x)^6}$ I switched it to $ln(y)=9ln(2x+3)-6x^{1/2}ln(x^2-x)$ and then used the log rules for derivatives I know and the product rule on the right side and wound up with $$y'=y[\frac{18}{2x+3}-\frac{3ln(x^2-x)}{x^{1/2}}+\frac{6x^{1/2}(2x-1)}{x^2-x}]$$ The 18 over 2x+...

I am looking for some latest material on convergence rate of the basic forward backward operator splitting algorithm. After googling, I found the following: http://epubs.siam.org/doi/abs/10.1137/S1052623495290179 http://link.springer.com/chapter/10.1007/978-1-4419-9569-8_10#page-1 I am newbie...

For two density functions: Suppose again that Z = X + Y . Find fZ(z) if fX(x) = fY(x) = { x/2, if 0 < x < 2, { 0, otherwise I understand this is a convolution fX *fY but I don't understand how to obtain the limits of integration since they are not -inf to +inf. X = x Y = Z - x fZ(x) = i...

I am going through my practice sheet before finals in my proofs class, and I would like critique, please. My understanding of the If-Then and the "hidden implications" is that each real number $x>0$ has at least one $y>1$ such that $xy>1$. If this is the case, then it is true, and here is my fo...

Given $(X,\mathcal{F},m)$ a measure space with $m(X)=1$ and $||f||_p<\infty$ for some $p>0$. Need to show that $\forall q \in (0,p)$ $$\int \log |f| dm\leq \log(||f||_q),$$ $$ \log ||f||_q \leq \frac{\int |f|^qdm -1}{q},$$ $$\lim_{q\to 0} \frac{\int |f|^q dm -1}{q}=\int \log |f| dm$$ and finall...

enter image description here This is from Abbott's understanding analysis, but I can't really figure it out

Microtubules play a role in the migration of chromosomes to opposite ends of a mitosing cell during anaphase. Microtubules are hollow tubes 24 to 25 nm in diameter composed of 13 parallel rows. The parallel rows are called protofilaments and are made of heterodimers called tubulin subunits. A tub...

So when I graph 2x - y = 6 and x + 2y = -2, I see them intersecting at points (2,-2). HOWEVER, when I set them equal to each other (2x - 6 = -1/2x -1) I don't get 2 for x. Can someone please clarify how to do this?

Prove that $\displaystyle \lim_{x \to a} x^2 = a^2$ Let $\epsilon > 0$, and let $\delta = \min(\frac{\epsilon}{2|a|+1}, 1)$. Suppose $x \in\mathbb{R} - \left\{a\right\} $ and $|x-a| < \delta.$ Then $|x-a| < 1$ which implies $ -1 < x-a <1 $ which implies $ a-1 < x < a+1$ which implies $2a-1< ...

Suppose that $a < b$ and $f: [a,b] \to\mathbb{R}$ is a continuous function such that the range of $f$ contains $[a,b]$. Prove that $f$ has a fixed point.

What is the correct Levenshtein distance between the following strings? hahaha ahahah These sites report different values: http://andrew.hedges.name/experiments/levenshtein/ (6) http://planetcalc.com/1721/ (2) The first site seems to be saying that you must compare each letter in place and ...

Why do some people used closed intervals when referring to those intervals? Concavity uses open intervals, so why does increase/decrease use closed intervals?

Let A be a set (of real numbers); define $\mathcal{C}^\omega (A)$ as the set of all real-valued functions that are defined, bounded, and analytic on A. My question is simply this: how did $\mathcal{C}^\omega (A)$ get its name? Who named it? What does $\omega$ mean in this context? How does it...

How can it be useful if I have a job which involve no mathematics. Some times it is the ugly truth we face. We learn mathematics , and we get a job which does not require anything what we have learnt. Any ideas ?

Please help me find this limit. I have already tried to use L'Hopitals rule and logarithmic differentiation but it turns into a quadratic.

$h(y)= \ln(y^2 \cos y)$ Treating it like a normal variable like an x isn't working for me, the way we used y's in earlier problems where you get a y' in there doesn't seem right, so I'm not quite sure here.

So the question was, Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be three times differentiable and $f'''$ is bounded, find constants $a,b,c$ such that $$f''(x) = \lim_{h\rightarrow 0} \frac{af(x-h)+bf(x)+cf(x+2h)}{h^2}$$ My Solution: Expand $f (x-h)$ and $f(x+2h)$ by Taylors theorem and solve...

$\int\limits_{0}^{2\pi}e^{cos\varphi}(cos\varphi-sin\varphi)d\varphi$ I think $e^{i\varphi}=z$ $\to d\varphi=\frac{dz}{iz}$ $cos\varphi=\frac{z^2+1}{2z}$ $sin\varphi=\frac{z^2-1}{2iz}$ $\oint\limits_{|z|=1}^{}e^{\frac{z^2+1}{2z}}(\frac{z^2+1}{2z}-\frac{z^2-1}{2iz})dz$ $z=0$ - essential ...

T: M33 -> M33 defined by T(A) = 1/2(A+A^T) What is the basis for the kernel of this linear transformation? So how do I solve this?

How many ways can we assign four different jobs to five different employees, assuming it is possible to assign more than one job to any employee?

Can some one please help me on the attached image. Is it possible to user Fourier series to find the solution or is there any other way? Assuming g(x,t) = Summation n = 1 to infinity Sin(nx/L) , what could be the possible solution.

Prove that $\lim \frac{n^2}{n^2+n+1} = 1$ Let $\varepsilon > 0$ and let $N = \frac{1}{\varepsilon}.$ Then $n > N$ implies $n > \frac{1}{\varepsilon} \implies \frac{1}{n} < \varepsilon.$ But $\displaystyle \frac{1}{n} = \frac{n+1}{n(n+1)} = \frac{n+1}{n^2+n} > \frac{n+1}{n^2+n+1} = \bigg|\fr...

The examples they use on my book are basically from basic arithmetic like 1, 2 and 3 to calculus and calculating derivatives, which is really annoying because I can't build the fundamental skill required to do harder questions. Anyhow, I have to solve an non-factorable inequality. Normally, I wou...

Given the equation: r=4cos3(theta)...how can I find the domain of each petal? Help!

Here is a picture of the textbook question cause I'm not completely sure how to format it. I don't know how the answer for d) is 10. https://gyazo.com/1fff6acd3b0076a313aacfd119743d21

How to prove in the set of $A$ = {1, 2, 3, ..., 2n-1}, when starting with any $i$ element $A$, if $i + n$ mod $(2n-1)$ as many as $2n-1$ times where the new result is added also by $n$ then all of the elements of A become the results of the operation or in other words all of the results are diffe...

I am taking a course about differentiable manifold and I need to prepare a representation about Chern class. Although now I am familiar with the properties of Chern class, but I can not find good examples of the applications of Chern class. As a topological invariant of vector bundle, I really w...

We need to find residue $\frac{1}{cos^2z}$ $cosz=0$ $z=\frac{\pi}{2}+\pi k$ - order 2 poles as the next?

enter image description here in the exercise 1.3Y, what does 1.3.10.2 commutes with the maps 1.3.10.1 mean? I can't see any relation between 1.3.10.1 and 1.3.10.2.

In some papers I've seen the following set: $$\mathbb{N}_0$$ What's the set $\mathbb{N}_0$?

In Boyd's Convex Optimization Textbook, page 157, it is stated: $ \mathrm{sup}\{a_i^\top x\; |\; a_i\in\mathcal{E}_i \} = \bar a^\top x + \mathrm{sup}\{u^\top P_i^\top x\; |\; \lVert u \rVert_2 \leq 1 \} = \bar a^\top x + \lVert P_i^\top x \rVert_2 $ But I do not see how $\mathrm{sup}\{u^\top...

Let $f:[-1,\infty]\to R$,$f(x)=(x+1)^2-1$.Then the solution set of the equation $f^{-1}(x)=x$ is $\left\{-1,0\right\}$.Is this statement true of false? $f(x)=(x+1)^2-1\Rightarrow f^{-1}(x)=\sqrt{x+1}-1$ Now solving $f^{-1}(x)=x$,we get $\sqrt{x+1}-1=x$ $\sqrt{x+1}=x+1$ $x=0,-1$ It appears that...

I'm currently taking a Transition to Advanced Mathematics course, which is entirely proof-based, so it's pretty new territory. Up until now, all the classes I've taken were fairly computational, so studying was just doing practice problems. I've already made flash cards of all the definitions (wh...

Here is the problem. I have 3 marbles in a bag, a red one, a blue one and a green one. Let's say you win a prize if you pick the red marble and let's say that you can pick two times in the bag and each time you have to put the marble back in the bag. Here are then the possible results if you say ...

Suppose we have two power series, with coefficients $a_n, b_n$, both with radius of convergence, $R$. Let the power series with the coefficients $a_n+b_n$, with radius $R'$ I need an example for scenario where $R'<R$. The other cases are easy NY the way. Written from mobile. Thanks in advanc...

Let {X(t)~Pois(λt)} be a Poisson process with parameter λ, and 0 How do can find the pdf of S1|X(t)=n and use it to find E[S1|X(t)=n]. Also, what is the pdf of Si|X(t)=n for i=1....n? what would be E[S1+S2+...+Sn|X(t)=n]

Let ABC be a triangle. Let B' and C' denotes respectively the reflection of B and C in the internal angle bisectors of angle A. How do I prove that the triangles ABC and AB'C' have same incenter.

I am having difficulty -normalizing sequence $1,x,x^3...$ in $L^2([-1,0]\cup[1,2],dx)$ So I begin by choosing basis $1,x,x^2...$ and i need to use Gram-Schmidt? or how should i approach this? I have attempted to integrate x from -1 to 0 and add integral from 1 to 2, but I don't think I understand...

Assume you designed a PID controller to let a given system track a unit step. Will the controlled system exhibit the same behaviour with regard to step inputs with different amplitudes?

Hello I am trying to solve this but i am confuse how can do that. Are these sequences are graphic? Explain your answer. a. 4, 3, 3, 2, 1. b. 6, 3, 2, 2, 1, 0.

Assume that $E \subset [0, 2\pi]$ has positive measure. For any sequence $t_n$ of real numbers, do we have$$\lim_{n \to \infty} \int_E \cos(n x + t_n)\,dx = 0?$$

I have conflicting answers with different methods. Here is the question: The number of defects per yard in a certain fabric, Y , was known to have a Poisson distribution with parameter $\lambda$. The parameter $\lambda$ was assumed to be a random variable with a density function given by: $f (...

The recurrence is given as follows :- f(i) = f(i-4) + floor (((i + 2)(i + 6)/12)^2) ( ^ symbol stands for raised to power ) f(0)=1 , f(1)=3, f(2)=7 , f(3)=14 I tried finding its general term but failed miserably . Can someone help in finding its general term ?

This is an old exam question from linear algebra that I am working on, The problem statement is: Part 1 Can you find $2×2$ matrices A,B, and C, so that $A^2≠0$, $B^2≠0$ and $C^2≠0$ but AB=0,BC=0 and CA=0? Part 2 Can you find $2×2$ matrices A,B, and C, so that $A^2≠0$, $B^2≠0$ and $C^2≠0$ ...

Anyone help me to solve this: $K_{i,j}$ denotes a complete bipartite graph of (i, j) vertices. A Ki is a complete undirected graph of n vertices. a. How many vertices are there in a $K_{i,j}$? b. How many edges are there in the graph ? i*j c. What is the total degree of the graph...

How do I find all real numbers a such that 4 < a < 5 and a(a-3{a}) is an integer. here {a} is an fractional part of a.

Let H and K be a finite subgroups of a group G whose orders are relatively prime, that is, gcd(|H|,|K|) = 1. Show that H ∩K={e}, where e is the identity in G. I really have a very small knowledge regarding abstract algebra, since I've only explored this topic today, and I have to answer this bef...

If corr(X,Y)=0.9, will there exist a third variable Z such that Corr(X,Z)=corr(Y,Z)=-0.9? I tried reasoning out there is one such variable using the regression equation Y=a+bX but reached a dead end.

How can I tune a PID to control a system with a parameter? Can I know beforehand how to change the PID parameters in function of the value of the system parameter?

Is a subsemigroup of a Clifford semigroup also a Clifford semigroup? I can prove that the maximal group image of a Clifford semigroup is a Clifford semigroup, but I am not sure whether any subsemigroup is also a Clifford semigroup!!

For this following graph Hamiltonian path and Eulerian path exist or not. Basic definition A cycle that uses every vertex in a graph exactly once is called a Hamilton cycle, and a path that uses every vertex in a graph exactly once is called a Hamilton path. So I think it follows Hamilton path,...

This is part of a homework assignment. Any hints will be useful, I haven't made any progress. I need to evaluate: $\int_{|z-1|=1} \bar{z}^n dz, n \in \mathbb{Z}$

Right now there is an event occurring in Heroes of the Storm where a special hero (Cho'gall) is unlocked if you play with another player currently playing that hero. I ran into a bit of an intuition problem when I started thinking about the optimal strategy for attempting to unlock this hero. ...

State true or false $(1)$If $f$ is a one-one mapping from set $A$ to set $A$,then $f$ is onto. $(2)$If $f$ is an onto mapping from set $A$ to set $A$,then $f$ is one-one. I do not understand whether it is true or false.If true,why true.If false,why false.No concrete reasoning,example and count...

Sorry if my question is really basic, I'm quite new to the field of quaternions. I face a problem where I would like to find a vector perpendicular to a quaternion. I know how to find an arbitrary vector perpendicular to a vector, but I would like to select the one pointing in the direction of th...

Any one help me to show the prove for this? Let the undirected graph G = (V, E) be a tree. Prove that by adding one edge to G you create a cycle in G.

Let $S^kV$ be the $k$-th symmetric power of tautological representation of $U(n)$ how to see that it's irreducible? I'm trying to do it using weight, but with no benefits..

$$L = \{w|w \space does \space not \space contain \space 000\}$$ $$L2 = \{w | xwy \in L \space for \space some \space x,y \in \space (0+1)^*\}$$ Is L2 regular? I am thinking regular language is closed under concatenation, but it seems that this language should be irregular. And I have no idea how...

I consider level sets $E_t = \{u <t\}$ and $N_t = \partial E_t$ in a manifold $M$ on dimension 3. Is this equality true : $\int_{E_t}|\nabla u| = |\partial E_t|$ ? Thanks

I can't solve this, no matter how I try

I am in a need of proving the following identity. Hope someone can help me. Let {x_n, n=>0} be a homogenous Markov Chain. Show that P(X_{n+1}=k_1,...,X_{n+m}=k_m (conditiong bar) X_0=i_0,...,X_n=i)=P(X_1=k_1,...,X_m=k_m (conditiong bar) X_0=i) I know I should use multiple conditiong and the Marko...

$$ D = \begin{vmatrix} 1 & 1 & 1 & \dots & 1 & 1 \\ 2 & 1 & 1 & \dots & 1 & 0 \\ 3 & 1 & 1 & \dots & 0 & 0 \\ \vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\ n-1 & 1 & 0 & \dots & 0 & 0 \\ n & 0 & 0 & \dots & 0 & 0 \\ \end{vmatrix} =n*1*(-1)^\frac{n(n-1)}{2} $$ I don't quite understand th...

Let $\mathbb{Z}/2$ act on the $m$-sphere $S^m$. If the action is not trivial, can we conclude that the action is homotopy equivalent to the antipodal action? That is, the following diagram commutes up to homotopy? Can we generalize this to $S^1$-actions on $S^{2m+1}$ and $S^3$-actions on $S^{...

Let $A$ be a real $n\times n$ matrix and $w,x$ real $n\times 1$ vectors. For fixed $A$ and $w$ solve the following for $x$: $(x^\top A x)w - (x^\top w) (A+A^\top) x = 0$ Any hints? I do not really feel confident on what kind of rearrangements are legit since multiplication is generally not com...

If $X\sim N(\mu_x,\sigma_x^2)$ and $Y\sim N(\mu_y,\sigma_y^2)$ what is $$\text{Var}{(X/Y)}$$

Let $U$ and $V$ be open subsets of $\mathbb{R}$. Let $f:U\rightarrow V$ be differentiable at $x_0$ and $g:V\rightarrow\mathbb{R}$ be differentiable at $f(x_0)$. Then $g\circ f$ is differentiable at $x_0$. What is wrong with the following "proof" of the above fact: $$(g\circ f)'(x)=\lim_...

Let $Z_1,Z_2,Z_3,Z_4 \in\mathbb{C}$ such that $Z_1+Z_2+Z_3+Z_4=0$;$|Z_1|^2+|Z_2|^2+|Z_3|^2+|Z_4|^2=1$ then the minimum value of $|Z_1-Z_2|^2+|Z_2-Z_3|^2+|Z_3-Z_4|^2+|Z_4-Z_1|^2$ is ?

Page 20 http://www.cs.umb.edu/~offner/files/pi.pdf he says that "We see immediately that we have $\Delta^{k}a_{m} = \sum (-1)^{j} (\frac{k}{j})x_{j}$." Why exactly is that? There must be a more formal way to prove this in mathematical terms.

Circles are drawn passing through the origin O to intersect the coordinate axes at points P and Q such that $(m)(OP)+(n)(OQ)=k$, then the fixed point satisfying all such circles is? (A) $\left(m,n\right)$ (B) $\left(\dfrac{m^2}{k},~\dfrac{n^2}{k}\right)$ (C) $\left(\dfrac{mk}{m^2+n^2...

If I wanted to know whether a line integral $\int_C f \cdot d\textbf{r}$, where $d\textbf{r}$ is $(dx,dy),$ is independent of path, what is the best way of doing this? I know it is path independent $\iff$ it's conservative. So I could show the $\nabla \times f=0$, I could find two different paths...

How can i prove that every topological space can be realized az the quotient of some hausdorff space?! I tried to show this by using the intersection of two open sets in x(for f:z-->x)

When I read Evans' PDE, I want to use the Theorem 2 in picture below to get $||u_m(0)||_{H_0^1(U)}\le||g||_{H_0^1(U)}$. So ,I let the $T=0$, then I get $$ ||u_m(0)||_{L^2(U)}\le C||g||_{L^2(U)} $$ There is a little difference , I don't how to deal it .The question is from the 362 of Evans' PDE, ...

is there a way to add a user defined convergence criteria to an ode solver so that the solution is stopped? I am using the solvers in to solve fluid dynamics problems and I am not sure whether the already used tolerances are sufficient. I wold like to have some control over the convergence crite...

I'm trying to solve the following task: Let $f:\mathbb{R}\rightarrow \mathbb{R}$ is measurable and $h\in\mathbb{R}$, we define the measurable function $\tau_hf:x\rightarrow f(x+h)$. Let $1\leq p<q\leq \infty$ such that $\frac{1}{p}+\frac{1}{q}=1,f\in L^P(\mathbb{R}),\ g\in L^q(\mathbb{R})$. Reca...

Is there a sharper bound for the following sum $$S:=\sum_{d \in (Z/qZ)^{*}} \overline{d},$$ where $a$ is any integer and $\overline{d}$ is the inverse of $d$ modulo $q>0$? Thanks in advance.

In a game of drawing cards, $4$ cards will be drawn from the deck randomly. In the deck of cards, $0.4$ of them are black, $0.6$ of them are white. Among the black cards, $20\%$ of them are worth $1$ point, $50\%$ of them are worth $0$, and $30\%$ of them are worth $-1$ point. Among the white car...

For $x \notin \pi\mathbb Q$, that is, a real $x$ that is not a rational multiple of $\pi$, consider the set $$\{(\cos nx,\sin nx):n = 0,1,2,...\}.$$ It is known that this set is dense in the unit ball $B(0,1)$ of $\mathbb R^2$. Could someone please give me a proof or reference for a proof?

Please, anyone, help me to tell is it correct? Draw the picture of the specified graph (including any isolated vertices): Graph H has vertex set {v1,v2,v3,v4,v5} and edge set {e1,e2,e3,e4} with edge-endpoint function as follows: Edge Endpoints e1 {v1, v4} e2 {v1, v2} e3 {v1, v...

What is lim(1/x)^sinx when x goes to zero?

How do I write perfect mathematical symbols on this site. I have problem to write perfectly with all symbols placed correctly.

I'm trying to solve a question where I need to write $\frac {1}{1+z^2} $ as a power series centered at $z_0=1$ I can't use taylor expansion. So my first thought was to rewrite the function in a form where I can apply the basic identity: $$ \frac{1}{1-x} = \sum_{n=0}^\infty x^n $$ So let's rewr...

For $f \in L^2(\mathbb{R})$, let $$Tf(x) := \int_0^1 f(x+y)\,dy.$$Do we necessarily have that$$S: L^2(\mathbb{R}) \to L^2(\mathbb{R}),\text{ }Sf = f - Tf$$is onto?

Is there a Sylvester's criterion for negative semidefinite matrices? I suspect such a criterion to be: All principle minors with odd dimension are non-positive. All principle minors with even dimension are non-negative. Please include a reference if it exists.

I have the integral, $$I(R) = \int_{C_R}\frac{1}{z(z-1)^2} dz$$ with the property that $$\left|\frac{1}{z(z-1)^2}\right| \leq \frac{1}{R(R-1)^2} \quad |z|=R>1$$ Where $C_r$ is the contour defined by the circle of radius $R>1$ so $C_r $ can be parametrized as $z(\theta)=Re^{i\theta}$ The goal...

Show that ${\mathbb R[x]}/(x^4−1)$ is a finitely generated $\mathbb R[x]$-module. I know that $ \mathbb R[x]$ is a PID, but I'm not sure where to go from there.

Is this formula P(A|C)=P(A|B)*P(B|C) correct according to Bayes' theorem?

$$ \begin{align} \left(v_1^{\mathrm T}\cdot A\cdot v_2\right)\cdot\left(B\cdot v_3\right)\stackrel{?}{=}&v_3\cdot\left(\begin{pmatrix}v_1^{\mathrm T} & v_1^{\mathrm T} & v_1^{\mathrm T}\\v_1^{\mathrm T} & v_1^{\mathrm T} & v_1^{\mathrm T}\\v_1^{\mathrm T} & v_1^{\mathrm T} & v_1^{\mathrm T}\end{p...

I'm looking at the following optimization problem: $$ max_{t\in(0, T)} Y(t) = \sum_{s=1}^\infty P(s, a t) d(T-t)\int g(x) h(x; s) dx$$ where $h()$ is given by $$H(x; s) = Prob(\min \leq x) = 1 - \prod_{i=1}^s Prob(x_i \geq x) = 1 - \prod_{i=1}^s(1-F(x))\\ h(x; s) \equiv \frac{\partial H(x; s)}...

Let P(x) = x^{2} ax + b be a quadratic polynomial with real coefficients. Suppose there are real number s ≠ t such that P(s) = t and P(t) = s. Prove that b - st is a root of the equation x^{2} =ax = b - st = 0.

I didn't understand why we may suppose $a=0$ in this proof: I'm reading Conway's Complex Analysis book, page 31. Any help is welcome

The question is "For two consecutive multiples of 5, the smaller number is greater than half of the larger one. Find the least values of these two number." I am stuck!

For which numbers $n$ exists a group of order $n$ with center {e} ? And how many groups are there for given order ? The first such numbers are $6,10,12,14,18,20,21,...$ Groups with order $32,40,64$, for example have not center {e}. For $n=18$, we have $2$ groups with center {e} Is there a...

IF a set A has K elements,formulate a conjecture about the number of elements in p(A). I can't even understand the term conjecture.

I have a problem asking me to find $\int_C \textbf{f} \cdot d\textbf{r}$ where $\textbf{f}$ = $(siny,xcosy)$, and the curve $C$ is any closed circle. I'm struggling with this, so far I have found that $d\textbf{r}$ is $(-rsin\theta,rcos\theta)d\theta$, which is then $(-y,x)d\theta$ in cartesian ...

To show that S is closed under addition, is it enough to say the following? Let u and v be in R^n. Then u + v is in R^n. Therefore A(u) + A(v) = A(u+v) and S is closed under addition. Thank you in advance!

If $ n $ be a positive integer $>1$, prove that $$2^{n(n+1)}\gt(n+1)^{n+1}\biggl(\frac{n}{1}\biggr)^{n}\biggl(\frac{n-1}{2}\biggr)^{n-1}...\biggl(\frac{2}{n-1}\biggr)^{2}\biggl(\frac{1}{n}\biggr)$$

Let A and B be two real $nxn$ matrices. Part 1) Show that if A and B are symmetric, then for any $\vec x \in R^n$, $$((A^2 + B^2)\vec x, \vec x) \ge((AB+BA)\vec x, \vec x)$$ Hint: look at $((A−B)^2 \vec x,\vec x)$ Part 2) Find a counterexample to Part (1) if the assumption of symmetry is d...

math.stackexchange community. I have joined to inquire on a hypothesis a friend of mine has recently proposed. Please note: before posting this, I have repetitively told him that his logic is flawed in a plethora of ways. But I'll humor him, just this once. This dear friend of mine claims that w...

Lets say we have a series random variables $h_{i,j}$ Now if $\gamma(i) = \sum^{J}_{j=1} {h_{i,j}}^2$ has a an MGF $M_{\gamma(i)}(s)$ and $\beta(i,\tilde{i}) = \sum^{J}_{j = 1} |{h_{\tilde{i},j}-h_{i,j}}|^2$ has an MGF $M_{\beta(i,\tilde{i})}(s)$ if the random variables ($h_{i,j}$) are identical...

I know that the result is true for closed interval $[0,1]$ by using intermediate value property. But in the case where we consider open interval $(0,1)$ does the solution change?

I have a vector A with direction $\vec D$. I also have another vector B with direction $\vec R$. (think of $\vec R$ to be the reflection of $\vec D$ at point B.) From the reflection formula:$$ \vec R = \vec D-2*(\vec D \cdot \vec N)*\vec N $$ (we know both $\vec R$ and $\vec D$) How can I fin...

I have a quick question about the proof of rejection sampling. Suppose we know how to sample from a distribution with $Y$ pdf $q$, and want to sample from a distribution $X$ with (known) pdf $\pi$. Suppose we also know that $\pi(x) \leq M q(x)$ for some constant $M>0$ and for all $x$. Then rejec...

I'm looking for an equation that looks like the image below. I am comfortable constructing functions that pass the vertical line test, but as this is a relation, I am not really sure how to start. Would this need to be defined implicitly? An explanation on how you constructed it would be great...

Is there a countably compact space which is neither sequentially compact nor compact? Many thanks!

What is a Kernel and how can it be describe in the real world and how can it be defined well and precisely. Tried asking my professor and he just tell us it is just abstract idea.

The cylinder $x^2$ + $y^2$ = $x$ divdies the unit sphere S into two regions $S_1$ and $S_2$, where $S_1$ is inside the cylinder and $S_2$ outside. Find the ratio of areas $A(S_2)$/$A(S_1)$ I think the area is not surface of a revolution, right? So, I think I should parametrize it, but I can't. ...

Find a standard basis vector for r3 that can be added to the set [v1, v2] to produce a basis for r3 v1=(1,1,1), v2=(2,-1,3) So, from what i know, a standard basis vector in r3 is either v1=(1,0,0), v2=(0,1,0) or v3=(0,0,1). I have no clue but I would like to assume that I just add the standard ...

May you help on how to start, or where to look for the following question? By using the $n$-th roots of the unity, show that: $\cos\left(\frac{2\pi}{n}\right)+\cos\left(\frac{4\pi}{n}\right)+\ldots+\cos\left(\frac{2(n-1)\pi}{n}\right)=-1$ $\sin\left(\frac{2\pi}{n}\right)+\sin\left(\frac{4\pi}{...

Suppose $Z$ is a complex (Wishart) matrix. Let $a=\frac{1}{[Z^{-1}]_{kk}}$, where $Z^{-1}$ is the inverse of $Z$ and $[Z^{-1}]_{kk}$ represents the $(k,k)$-th entry of $Z^{-1}$. When I was reading an article, I saw that they define $Z_{kk}^{\text{SC}}$ as the Schur complement of $[Z]_{kk}$, a...

Context: there are three candies in a basket (5 red, 4 blue, 5 green). How many ways can you choose 2 candies with different color? Should I use permutation or combination w/ this? I tried using: 14!/(14-2)! = 182 ways; but it's marked as wrong.

$\oint\limits_{|z-3|=4}^{}\frac{z}{cosz-1}dz$ I think $cosz=1$ $z=2\pi k$ to include only $z=0$ and $z=2\pi$ But how next?

Need a relation between a and b that is only reflexive. Example: a < b is only transitive. Thanks

Consider the quantity, $$\frac{\sum\limits_{i=0}^{n} A_i}{\sum\limits_{i=0}^{n} B_i},$$ where $A_i \leq B_i$ for any $i$ and $A_i$ is increasing with $i$. Let $S_i$ be exponentially decreasing with $i$. Let all these numbers be positive strictly less than $1$. Which additional conditions do I ne...