let A be contained in M such that A is nonempty but clu(A) is the empty set. Assume A does not equal M. Let y be an element of M-A and let B=A union {y}. show that clu(B) is the empty set

I know the theorem states that if you have $n^2 + 1$ real numbers in a sequence then you are guaranteed to have an increasing or decreasing monotonic subsequence of length $n+1$. I'm trying to understand the proof of this theorem using the pigeon hole principle. I was following this answer here,...

Please help, i'm learning about introduction of abstrac algebra and now i'm facing a question related to Ecludian domain. Sadly up until now i coudnt find any clue how i should start proofing this question, i do hope anyone can help. Known that R Euclidean Domain, A is a matrix nxn with it's ent...

What I need to do is find values of t where matrix equals 0. Apparently, those two numbers are -1 and 0, but how do I calculate them? Do I search them with determinant? enter image description here

Prove that every set of $2k$ integers has elements of opposite parity differing by at least $2k-1$ or elements of equal parity differing by at least $4k-2$.

Let $f$ be defined on $[0,1]$ by $f(x):=1$ if $x \not= 1$ and $f(1):=0$, Show that the Darboux Integral exist and find its value. I know I want my partition to be $P_\epsilon := (0, 1-\epsilon/2, 1+\epsilon/2,2)$ I'm having trouble defining my Upper sum and Lower sum

If $a≡_nb$ for some $n≥2n$, then $n|a−b|$, thus $n|(a−b)(a+b)=a^2−b^2$. Thus $a^2≡_nb^2$. I am confused where to add the induction to prove that for any $a^k≡_nb^k$

Let $f:\mathbb{R}\rightarrow\mathbb{C}$ be continuous with a dense image. How does one show that the preimage under $f$ of an open ball in $\mathbb{C}$ is unbounded?

Let C be defined by y=f(x).Suppose there's a tangent at (4, 3). The tangent equation is y=2x+5. Let C be the curve y=xf(x)/1+x^2 Find the y-int of tangent to C at the point with x-condition that =4 For this question what I did was just use y=mx+b, make it y=(2)(4)+5 which equals 13. So woul...

Could you please check me? cosh (ln(sqrt5)) = ? cosh (x) = e^x - e^-x/2 So it should be like this e^ln(sqrt5)) - e^-(ln(sqrt5))/2 sqrt5-1/sqrt5/2 sqrt5/2+1/(2*sqrt5) 3/sqrt5 Thanks.

Consider $L_i$ the $i$-th component of angular momentum defined by $$L_i = \sum_{r,s}x^rp_s\epsilon_{rsk},$$ being $x^r$ the $r$-th component of position and $p_s$ the $s$-th component of momentum, though of as coordinate functions on the phase space $T^\ast\mathbb{R}^3$. I'm trying to show tha...

I'm given a subset \begin{equation} S = \{f \in C[-1,1]: f^{-1}[\{0\}]=\{0\}\} \end{equation} and am supposed to determine if it is a subspace of $C[-1,1]$. Now, I don't think there exists a function that is continuous on $[-1,1]$ whose inverse determined at $0$ is equal to $0$, and therefore th...

What is a method that I can decompose a full rank matrix, say H, into a series of toeplitz (T) and diagonal (D) matrices? That is how to do: $H=T_1P_1T_2P_2...T_NP_N$

If $x$ is in the interval $[0, {\pi \over 2}]$ and $y$ is in the interval $[{\pi \over 2}, \pi]$ and $tan x={4 \over 3}$ and $csc y={13 \over 5}$, evaluate $$sin(x+y)$$ The final answer is supposed to be ${-33 \over 65}$ I started by graphing $cscy={13 \over 5}$ on a cartesian plane. Because o...

I'm tryng to do a exercise about normal distribution. But I think the statement is wrong,because de the value of Z is very big. μ=2000 hours standard deviation=40 Exercise: Probability between 1450 hours and 1580 hours Recipe: Z=(X-μ)/standard deviation

function {(2,4)(-8,9)} is a set of number for f(x) f(x) -> px + q 2p + q = 4 -8 + q = 9 10p = -5 p = -2 I only get p ,but I wasnt sure whether correct or not . Help me

Is it possible to evaluate $$\int_{-\infty}^\infty \frac{x^2}{(x^2+1)^2} \, dx $$ using the residue theorem, as opposed to Calc 1 methods?

I need help with this proof. I am stuck on this question and don't know how to do it: Prove that: $$ \forall n \in \mathbb{N}, \sum\limits_{i=2}^{2^n} \frac{1}{i} \geq \frac{n}{2}$$

I have a doubt to prove by natural deduction of this: $\vdash\forall x \forall y \varphi (x,y) \rightarrow \forall x \varphi(x,x)$ First of all, I took $\vdash\forall x \forall y \varphi (x,y)$ as a hipothesis And eliminate $\forall y$, to deduce: $\forall x \varphi(x,x)$. My question is: May I c...

does somebody know about the Borromean rings or knots in general and can summarize the key steps of the proof? i don't get it.

So using the fact that root 3 + x/4 is less than or equal to root (x+3) which is less than or equal to 2 on the interval [0,1], I have to prove that root 3 plus 1/20 is less than or equal to the integral of root (x^4 + 3) dx from 0 to 1, which is less than or equal to 2. How do I go about doing...

suppose G is a nonabelian group of order pq, where p and q are both prime number. prove that if the nontriaval element a belongs to G ,then =C(a).

Here is the expression in question: $$\lim_{n\to\infty} \sum_{k=1}^{n} \Big[2 + \frac{3}{n}k\Big]^2 \Big(\frac{3}{n}\Big)$$ The first thing I noticed was this: $$\frac{3}{n}$$ I figured that as n approaches infinity, this value would approach zero, so the value for the entire expression would...

This is my first encounter with homomorphisms and I'd like to have my proof verified. Question: Let $G = (\mathbb{Z}, +)$ and $H = \{6^{n} \mid n \in \mathbb{Z} \}$. Define $f: G \to H$ by $f(x) = 6^{x}$. Show this is a homomorphism. Attempt: We want to show $f(a + b) = f(a) \cdot f(b)$ for all...

Suppose $F=\mathbb{F}_{p^n}$ is a degree-$n$ extension of $\mathbb{F}_p$. My questions concerns the action of the multiplicative group $(\mathbb{F}_p)^{\times}$ on $F$ by left multiplication. then we can write $F=\mathbb{F}_p[x]/\langle \pi\rangle$ for some monic irreducible polynomial $\pi\in \m...

I do not understand this question AT ALL. f(x)=lnx/x for xe(0, infinity) a) Show that e is the only continuous number of f(x) b) Show that f'(x) > 0 for all xe(0, e) and f'(x) <0 for all xe(e, infinity) c) Explain why part b implies that f(e) is an absolute maximum value of f on (0, infinity) d...

I have the Gram-Schmidt algorithm memorized, so that I can always compute an orthonormal basis, when I need it (on pen and paper, I don't studying mathematical / scientific computing ... yet). Could I think of the Gram-Schmidt algorithm geometrically in the two / three-dimensional cases? What d...

You are catering the awards banquet. One hundred will be attending. 8 do not eat wheat,sugar or meat. 68 eat wheat, 48 eat wheat but are vegetarians, 18 eat wheat but not sugar,50 eat wheat and sugar, 20 people eat wheat and are not vegetarians, 20 eat sugar and not wheat. How many eat wheat and ...

I understand that free variables in Lambda calculus are those that are not bound to a specific metavariable inside of an abstraction, while bound variables are the direct opposite. The idea that confuses me is that of parenthetical placement inside of an expression/series of expressions. express...

I have a field = $F_2$ and a polynomial $x^2 + x + 1$ over that field. I understand that since the polynomial is degree 2, it has no root in $F_2$, so it is irreducible. But why is $F_2[X]/(x^2+x+1)$ a field with 4 elements? (0, 1, x, x+1) Also when I do the multiplication table of the elements...

I am looking for the solution of the following problem, Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a twice differentiable function with $f(0) =2, f^{\prime}(0)=2$ and $f(1)=1.$ Then there is $x \in (0,1)$ such that $f(x)f^{\prime}(x)+f^{\prime \prime}(x)=0.$ I think we have to use mean value t...

There are $n$ marbles and $r$ boxes. One at a time, each marble is selected and randomly (uniformly) placed in one of the $r$ boxes. Let $S$ be the number of empty boxes. Compute $E(S)$ and $Var(S )$. Here is my work: Let $X$ = the box is empty This gives me: \begin{equation} X_i = \begin{cas...

I am not certain what to do here. Please assist by setting up the first two or three steps of this very simple system of first order ordinary differential equations. My differential equations textbook states to use the "elimination method" to crack this. We may also solve by the matrix method whe...

I am having trouble simplifying this equation to fit the truth table. A'B'C'D'+ A'B'C'D + A'B'CD' + A'B'CD + A'BC'D + A'BCD' + AB'C'D + AB'CD' + AB'CD + ABCD Any Help would be awesome, thanks

How do you evaluate: $$\displaystyle \int_{0}^{\infty} \frac{1}{\sqrt{2 \pi s}} e^{-z^{2}/2s} \cdot \frac{1}{2}e^{-s/2} \, ds=?$$

Can you please teach me how to solve this? There are other similar questions that I have, but I don't understand how to solve them. Another question is: How many 10-bit binary sequences contain an odd number of 1s?

Hey people of Stack exchange, I missed one class this week and missed the lecture on Anti derivatives, I am just wondering how to solve this question Solve the Initial Value problem. $\frac {dy} {dx} = x^3$, $y(0) = 7$ $y= ?$

Sorry, don't have many ideas on how to start this one. Is it because of some relation between y and z?

I'm looking for a way to solve the above equation only using U-substitution without any trig. Again the problem is: integral 0 to 1 of (ln(x+1))/(x^2+1) dx

I have an interesting dynamical systems problem that has had me stumped for a few hours now, so I'm hoping I can get some help. The problem is concerned with flows on the torus. The model is given by $\dot {\theta}_1=\omega_1$ and $\dot {\theta}_2=\omega_2$, where $\theta_{1,2}$ are the phases of...

We have f as a function of $\Bbb R \to \Bbb R$ and is of Class $C^2$. $f(0)=1/2$, $f'(0)=1$, $f''(x) \le 1 $ $\forall x \in[-1,1]$. Show that over [-1,1] we have: $$|f(c+h)-f(c)-f'(c)h|\le \frac{h^2}{2}$$, for $c, c+h \in [-1,1]$. I know that f is of Class $C^2$ if all the partial derivatives up...

I am just wondering how I'd 'solve the following inequalities' 1) |x+1| < 4 2) |3x+2| ≥ 2-x Thanks

Question: Evaluate $$\sum_{k=0}^n {2k\choose k}{2n-2k \choose n-k}$$ Hint: use the fact that $$(1-4x)^{-1/2} = \sum_{n\ge0} {2n \choose n}x^n$$ For some reason the hint has me more lost than the problem but I'm sure it is included for a reason. My gut feeling is to approach this with a binomial s...

cot B/(csc B - 1) + cot B/(csc B + 1) = 2 sec B What I've done: (cos B/sin B)/(1/sin B) - 1 + (cos B/sin B)/(1/sin B) + 1 (cos B/sin B)/(1 - sin B)/sin B + (cos B/sin B)/(1 + sin B)/sin B (cos B(1 - sin B))/sin^2 B + (cos B(1 + sin B))/sin^2 B (cos B - cosBsinB + cosB + cosBsinB) / sin^2 B ...

How do I solve ? $$\frac{d}{dx}e^{sinx}$$ A very detailed explanation is expected as I am a beginner.Thanks for any help.

Could someone help me with this question please? Question

I have a discrete signal: $x=[i_1, i_2, ..., i_n]$ I would like to do the cross correlation between sub-groups of (k number of) consecutive number of $x$, for example: $xcorr([i_1, i_2, i_3], [i_2, i_3, i_4])$ What is the most efficient way to do this, computationally? I have been thinking of ...

Original question: Consider carrying out the state minimization algorithm. We'll let Partition 0 be the name of the initial partition of the states into sets N = {Q0, Q1, Q2, Q3, Q4} and F = {Q5}. The next is Partition 1, the next Partition 2, and so forth. Based on this graph this question w...

Let $X$ and $Y$ be two random elements in a Hilbert space. Then I have seen in a paper to use the following $E[<X,Y>] = E[<E[X|Y], Y>]$ Here $E[X|Y]$ is also an element of the Hilbert space

Second part of the question: If C is similar to A and also to B then __ (Not sure if this is correct but I answer: A is similar to B?) Please help me with the first part of the question, I'm very confused.

Can Any one give me some hints? Please dont give me the answer i still want to left some room for myself to think .May anyone give me some advice so that i can solve the questions Questions: Le K be a nonempty compact subset of R. Prove that there exists a bounded continuous function f :R->R s....

Use Mathematical Induction to prove that for [; n>=1 ;], that $ b_n=(1/2)((3^n)+1) $ Solution: Basic case: For [; n = 1 ;] $$ b_1 = 2 = 1/2((3^1)+1) $$ Assume that for some k $$ b_k = 1/2(3^k + 1) $$ Then $$ b_{k+1} = 3b_k - 1 $$ $$ = 3(1/2)(3^k + 1) - 1 $$ ...

A box with 4 closed sides has $n$ straight lines going all the way across it. The base case is of course only 1 line which separates the box into 2 colors, black and white. How can I show using induction that for any $n$ lines, the box can still be colored with only black and white. I am famil...

I was hoping for some confirmation of a proof to the following preliminary exam question: Fix an $n \times n$ matrix $A$ with entries in an algebraically closed field $k$. Let $C$ be the space of $n \times n$ matrices over $k$ that commute with $A$. Observe that $C$ is a vector space over ...

At a party, ten men throw their hats into the center of a room. The hats are mixed up and each man randomly selects one. (Where we assume that if a hat is chosen, it can't be chosen again). A) Define $X_i$ , such that for i = 1,2,..,10, appropriately such that $X = \sum_i^{10} X_i$ = X So if ...

Prove that if $g(x):=0$ for $0\le x\le1/2$and $g(x):=1$ for $1/2\lt x\le1$ then the Darboux Integral of $g$ on $[0,1]$ is equal to $1/2$. My answer: Lets define a partition $P_\epsilon =(0,1/2,1/2+\epsilon,1)$ thne the upper sum $U(g,P_\epsilon)=0(1/2-0)+1(1/2+\epsilon-1/2)+1(1-(1/2+\epsilon)=0+...

Please show me steps how you would solve it. I'm really lost and detailed explanation would help me learn and understand it better.

I'm having a difficult time determining if the following function is convex: $$f(X) = \log {\rm det}(X^T A X), $$ where $A \in \mathbb{R}^{r \times r}$ is a symmetric positive definite matrix and $X \in \mathbb{R}^{u \times r}$ with $r < u$ and $X'X = I$. I've worked on find the second derivati...

So my question is is the fourth root of 1 = i? Where i^4 = 1? or if it is still 1 nonetheless?

Suppose I have two cumulative probability distributions: $P_a(x_a)$ and $P_b(x_b)$. How do I combine these two distributions to find the combined probability that one random variable is lower than the other. As in, how do I find $P(x_a < x_b)$? I'm familiar enough with statistics to be able to c...

I have found the inequality $$ x\leq cy+\log(1/c)(x-y)\log(x/y), $$ for all $x,y>0$ and $c>1$. Why is this inequality true?

I am working on a computer science assignment. I am suppose to create a function which calculates this equation !http://imgur.com/a/9rIDY Now the 2nd image shows the calculation for the table1. How would the calculation change for the table in pic 1.

Consider F is a closed set in $\mathbb R$, is it true that F $\cap$ $\mathbb Q$ is dense in F? I failed to show it...$\sqrt{2}$

Let $Q:=[0,2] \times [0,2]$ ; then how to evaluate ${\int\int}_Q\lfloor x+y\rfloor dxdy$ ? where $\lfloor . \rfloor$ denotes the greatest integer function . Please help . Thanks in advance

Let $W$ be the following subset of $\mathbb{R}^{3}$: $ W = $ { $(x,y,z) : x + 2y + z = 2$ }. Define a new vector addition rule that makes $W$ closed under vector addition.

$P(X>k)$ =$\displaystyle \sum_{i\ge k+1}p(1-p)^{i-1}$ =$(1-p)^k$ I don't understand why it can be equal to $(1-p)^k$?

I´m trying to simplify the following fractions 1/(z-d/2)^2-1/(z+d/2)^2. And I got the the answer (z+d/2)^2/(z-d/2)^2-(z-d/2)^2/(z+d/2)^2. But I not sure what I can do with the denominators.

Let $R = (r_{ij})$ be an $n\times k$ real matrix with only positive entries, and consider the convex optimization problem $\max f(x) = \sum_{i=1}^n \log \sum_{j=1}^k r_{ij} x_j$ such that $\sum_{j=1}^k x_j = 1$ and $x_j\ge 0$ for all $j=1...k$. Assume that $R$ is such that this problem has a un...

The perceptron bears a certain relationship to a classical pattern classifier known as the Bayes classifier. When the environment is Gaussian, the Bayes classifier reduces to a linear classifier.This is the same form taken by the perceptron. What is a Gaussian Environment referred to in ...

Square building blocks of side length $a$ are used to build a tower. Each layer has to contain a certain amount of blocks to hold the tower standing. Let $F_n$ be the number of blocks required in the $n^{th}$ layer from the top. Then $F_n = F_{n-1} + F_{n-2}$ where $F_1 = F_2 = 1$ A tower of $...

This is a question that I had on one of my midterms. I completely bombed this one, I still have no idea how to solve it. Any help would be appreciated :) -- Find the volume of the solid region bounded above by the plane z = x, on the sides by the cylinder x^2+y^2=y, and below by the xy-plane. ...

The question asks to fully solve for $$(sin{\pi \over 8}+cos{\pi \over 8})^2$$ My question is, is this a double angle formula? And if so, how would I go about to solve it? I interpreted it this way; $$(sin{\pi \over 8}+cos{\pi \over 8})^2$$ $$=2sin{\pi \over 4}+(1-2sin{\pi \over 4})$$ Have...

The question is: Suppose that $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is linear, and that $f^{2015} = I$. Show that $f$ is diagonalizable. [hint: If f is a multiple of the identity, it’s easy. Otherwise, it suffices to prove that f has two distinct eigenvalues.] My proof so far: Let $\vec{...

Prove that floor(2x)=floor(x)+floor(x+(1/2)). enter image description here

I'm trying to find a bounded function $f$ and an open set $A$ such that $f$ is integrable over $\overline{A}$ but not over $A$. Is there such a function and set?

I was wondering whether a homogeneous system of linear equations with more variables than equations always has a solution other than the trivial solution. If so do you mind explaining why.

The left-hand side becomes A*I_n - B*0_n,n = A, correct? How can A = det(A) just from the information given?

Suppose I have the following pdf $f(y;\theta)=\frac{1}{\theta^2}ye^{-y/\theta}$ and I am told that $E(Y)=2\theta$ and $Var(Y)=2\theta^2$. Is $\frac{1}{2}\bar{Y}$ an unbiased estimator of $\theta$? Is $\frac{1}{2}\bar{Y}$ an efficient estimator of $\theta$? First, is it unbiased. Conside...

(a) Prove that if f:A1 -> A2 and g:B1 -> B2 are bijections then h:A1xB1 -> A2xB2, h(a,b)=(f(a),g(b)) is a bijection. Please refer to image of problem for clearer formatting.

How many solutions are there to the equation x1 + x2 + x3 + x4 + x5 + x6 = 25 where each xi is a non-negative integer, 3 ≤ x1 ≤ 10, 2 ≤ x2 ≤ 7 and x3 ≥ 5 I have been able to do all my counting problems but this one. I can not find an equation for max number per variety. I know that I will subtr...

The problem is states that there are two bins, Bin #1 which has 3 pens and 7 pencils; and Bin #2 with 8 pens and 4 pencils. (a) A bin is chosen at random and an object is drawn. What is the probability that it is a pencil? I am thinking that this is just P(pencil | bin #1) * P(bin #1) + P(penci...

Let $C$ be the curve of intersection of $z=xy ; x^2+y^2=1$ traversed once in a direction that appears counterclockwise when viewed from High above the $xy$-plane ; then how to evaluate $\int_Cydx+zdy+xdz$ ? I have found a parametrization of the curve which is $\Big(\cos t ,\sin t , \dfrac {\sin ...

Considering a parabola of the form $y=ax^2+bx$ I was curious what relation $a$ and $b$ would need to have in order for the arclength of the parabola between its zeroes to be twice the distance between its zeroes. To simplify the calculus, I set the integral from the first zero to the vertex equ...

Consider a class with 4 students having min goals as{1, 3, 4, 5} and max goals as{2, 5, 8, 6} fine the best way to divide the class in such a way that the match is competitive i.e. the maximum difference between goals scored by the team is minimized My solution: after taking the average of the m...

if I want to prove that " there does not exist any homomorphism f : → Can I just prove there only exist f is trivial homomorphism? Thanks

Consider the two-dimensional system x' = -x^4 + 5µx^(2)-4µ^(2) y'=-y I found that there are four critical points for this system. but i do not know how the how draw the phase portraits for various values of µ and draw the bifurcation diagram.

While reading an article about perfect numbers I noticed 1 thing. All of them are even. I tried to find some logic behind this unusual property, because I thought that the logic might be too naive to be stated, so they haven't stated. But unfortunately , I couldn't get any success. Can someone he...

Let R-Q be the subspace of R with the usual metric.Is there a function, f:R-Q-->R-Q such that f is continuous and f does not have a fixed point? I get that if we consider the metric R every continuous function from R to R has a fixed point.But we consider R-Q ,firstly i cant think of a continuo...

Let the curve C be the mirror image of the parabola $y^2= 4 x$ with respect to the line $x+y+4=0$. If A and B are the points of intersection of C with the line $y=-5$, then the distance between A and B is. Answer =4. I have solved it as, first translating the coordinate to (-2,-2), then re...

If $H <G$, then $H$ is called normal if $gHg^{-1} \subset H$ for all $g\in G$. Why is the definition not $gHg^{-1} = H$? We are not assuming commutivity for $G$. Take $gH = Hg$, multiply both sides by $g^{-1}$ and we get $gHg^{-1} = Hgg^{-1}$. Why is $Hgg^{-1} \neq H$ always?

Give an example of an injective function f : N → (0, 1), and prove that your function is injective.

Determine whether the function $f\colon\mathbb{Z}_{3}\oplus\mathbb{Z}_4\to\mathbb{Z}_{12}$ given by $f([a],[b])=[4a+9b]$ is a ring isomorphism. How does I even solve this? please help me

If $\sin{x \over 2}={2 \over 3}$, $0 \le x \le {\pi \over 2}$, evaluate $$\cos x$$ I'm not sure how to start. I have graphed $\sin{x \over 2}={2 \over 3}$ on a cartesian plane but I am stuck. Am I supposed to set $\cos x$ to a half identity?

show this differential equation x''+sin(x)= 0 is not structurally stable ? I do not know how to covert it to a system of ode.

enter image description here Should I use the definition of matrix: det(A)=Σ sgn( σ )a[1σ(1)]a[2σ(2)]...a[nσ(n)] ? I don't understand what is a[iσ(i)] ? Where i=1,2....n. Or is there another way to solve it?

Suppose that f : A → B is a function. If S ⊆ A, then we define f(S) to be the set f(S) = {f(x) : x ∈ S}. (So for example, if f : R → R is given by f(x) = x^2 , then we have f({1, 2, 3}) = {1, 4, 9}, we have f({−2, 2, 3}) = {4, 9}, and f([−2, 2]) = [0, 4].) a) Prove that if S ⊆ A and T ⊆ A, then f...

If every degree's vertex of graph $G(V,E)$ bigger than $\delta$. Prove $G$ has two cycle of length $\delta+1$.

In the wikipedia article on currents https://en.wikipedia.org/wiki/Current_%28mathematics%29 it is written that If $\omega$ is an m-form, then define its comass by $||\omega|| = \sup\{|\langle \omega,\xi\rangle| : \xi$ is a unit, simple m-vector$} I don't understand this definition. Can some...

Integrate $f(x)$ from $0$ to $1$, such that $f(x) = \frac{x^2lnx}{sqrt(1-x^2)}$ I started off with the substitution $x=siny$, which resulted in the integrand reducing to $sin^2y.lnsiny$ Then I used the property of definite integrals that integral of $f(x)$ is the same as integral of $f(a+b-x)$...

Okay, I have a collumn stochastic matrix of order $280\times 280$, the entries are given in an url in some webpage in row format. I need to find all the eigen values and eigen vector corresponding to the eigen value $1$ How can I proceed for that? What are the quick online tools are there?could ...

If A, B, C and D are four variables, the correlation matrix will be 4x4 matrix with elements, [1 Corr(A,B) Corr(A,C) Corr(A,D); Corr(B,A) 1 Corr(B,C) Corr(B,D); Corr(C,A) Corr(C,B) 1 Corr(C,D); Corr(D,A) Corr(D,B) Corr(D,C) 1] I have several questions: 1) What is the maximum possible valu...

So I have a cut $(P,P')$ on some network and its capacity is 13. Now I'm told to assume that the current flow on the network is the max flow, is the cut of minimum capacity? So far all we've learned is that the max flow is at most equal to the capacity of the cut but I we haven't dealt with mini...

In Euclidean Geometry, parallel lines are equidistant. In hyperbolic geometry, it appears that parallel lines are $not$ equidistant. Is there a proof that supports this, or is it supposed to be trivial, or obvious because lines in hyperbolic geometry are/ can be curves. !https://graphics.stan...

I understand why the empty set is a lower bound and A is an upper bound. The only problem I am having is putting my thoughts into a mathematical solution. Can anyone help out? Thanks. Let A be a set and F ⊆ P(A) be a family of subsets of A. Prove that ∅ is a lower bound for (F;⊆) and that A is ...

Let $\mathcal{C}$ be a monoidal category with the tensor unit $I$. Then there is a "forgetful" functor from it to $Sets$: \begin{equation} \mathrm{Hom}(I,-). \end{equation} But in general, this functor is actually not a forgetful functor as it may not be faithful. My question is Is there a...

Show that the exact solution of $$\epsilon \frac{d^2y}{dx^2}+\frac{dy}{dx}-2x=0$$ is $$y=(x-\epsilon)^2+\frac{1+2\epsilon}{\exp(-1/\epsilon)-1}\{\exp(-x/\epsilon)-1\}-\epsilon^2$$

I have a dataset, which is the hourly mean wind speed of each day(24 points each day), for 20 years. And I'm planning for using this dataset to estimate the probability distribution of hourly mean wind speed The problem is that, clearly, the wind speed between consecutive hours are correlated, t...

Let $f:[0,1]\times[0,1]\to\mathbb{R}\cup\{\pm\infty\}$ be a function such that, for some $\hat{x}\in(0,1)$, $\lim_{x\to\hat{x}}f(x,y)=\infty$ for each $y\in[0,1]$ and $f$ is continuous in $[0,\hat{x})$. Also let $x_{\varepsilon}\in[0,\hat{x})$ be a sequence such that $x_{\varepsilon}\to\hat{x}$ a...

I was studying a definition in an article where author mentioned number of interactions between two actors in social networks. I am attaching the snapshot please help me out how to interpret it. Thanks enter image description here

Here is the problem "Assume that the functions I :lR3 -> lR (F,g) : lR2 -+ lR are differentiable and they satisfies F(x1,x2) = I( x1, x2, g(x1,x2) ). Find the partial derivatives of F in terms of those of I and g" My attempts at the problem ; Since its the first time i e...

$a_n=1+n\sin\left(\frac{n\pi}{2}\right)$ So the above sequence is obviously not convergent because of the sine, but how could one prove this? Thanks for any pointers. Tom

I am given the series $$\sum_{n=2}^\infty \frac{1+2^n}{3^n}$$ So I used ratio test for this one to see if this converges and I found out that it converges, however the answer key tells me there is a geometric sum for this and I don't know how to transform this geometrically because of the numera...

I need to create a spreadsheet outlining the percentage of the tonnage of a material vs. the number of sacks of drying agent it takes to solidify the material?

If given a GF F(x) = 1/(1-rx)^2, how do I find the coefficient for the term x^n? I can tell F(x) = A(x)^2, where A(x) is the GF for the sequence 1, r, r^2, r^3, r^4, .... but I don't know how to find the coefficient at x^n. Thanks.

I need help to solve the following integral: \int_{0}^{x}\int_{0}^{y}\frac{s^{m}t^{n}}{‎\sqrt{‎‎(x-s)^2+(y-t)^2}}dtds‎ thank you

Suppose $\Omega$ is bounded and that $1 \leq p \leq q \leq \infty $ . Prove the following statements: $L_q(\Omega) \subset L_p(\Omega)$ For $ k \in N^+ : W^k_q (\Omega) \subset W^k_p (\Omega)$ I have tried to used Holders ineq, for the first one, but doesnt seem to be working. If anybody has...

What is the easiest method to notice the symmetry of the following function without using any graphical tool: $$g(x)=\frac{1}{\pi \sqrt{4-x^2}}$$

Let {fn} n ∈ N, be a sequence of infinitely differentiable functions (smooth function) on [a,b] such that for all integer k >= 0, there exist a real M_k such that |fn^(k) (x)| <= M_k for all x ∈ [a,b]. Show that there exist a subsequence that converges uniformly with all it's derivatives to a inf...

I am particularly interested in how Ron Gordon came up with the parametrization in his anser to this question: Inverse Laplace transform $\mathcal{L}^{-1}\left \{ \ln \left ( 1+\frac{w^{2}}{s^{2}}\right ) \right \}$

Evaluate: $$\int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}\frac{\sin^4(nx)}{\sin^2(x)}$$ Anyone able to provide a detailed prove of this? I can't seem to figure it out, I've tried using Euler's formula, but no luck.

I am trying to set up the 1D wave equation BVP for an elastic rod. Namely, an elastic rod of length L is fixed at x = 0, and is initially stretched by a length c, and so the end of the rod is at x = L + c. At time 0, it released. Also at time 0, velocity is approximately 0. The governing equation...

The question reads as follows: A high-tech company purchases a new computing system whose initial value is $V$. The system will depreciate at the rate $f = f(t)$ and will accumulate maintenance costs at the rate $g = g(t)$, where $t$ is the time measure in months. The company wants to determine ...

A subset of R2 is radially open if it contains an open line segment in each direction about each of its points. If A is a subset of the plane R2 and A intersects any line in at most 2 points, prove that R2/A is radially open. Ok, so I have to prove that in R2/A there is an open line segment i...

As we know, cplex has the capabilities to solve LP, MIP, QCP, MIQCP, SOCP, and MISOCP. But, cloud cplex solve MISDP?

I went through the concept of Kuratowski monoid in the paper by " B. J. Gardner and M. Jackson, The Kuratowski closure-complement theorem, New Zealand J. Math. 38 (2008), 9--44". They consider the set of all distinct operators on the topological space $( X,\tau )$ produced by composition of c...

Using 'coneprog' in R and 'cvxopt' in Python I'm getting slightly different results. Is that to be expected? How stable are the various algorithms compared to each other?

function r=isssdd(A,s) [m,n]= size(A); sbar=1:m; sbar(s)=[]; if m==n a=diag(abs(A)); B=abs(A-diag(diag(A))); rs= sum(B(:,s)')'; rsbar =sum(B(:,sbar)')'; c= (a(s)-rs(s))*(a(sbar)-rsbar(sbar))' > rsbar(s)*rs(sbar)'; if all(a(s)> rs(s)) & all(all(c-diag(diag(c))+eye(m))); ...

I need to perform the parameter sensitivity analysis (SA) of the steady-states of my ODE model. Can I use the extended Fourier Amplitude Sensitivity Testing (eFAST) for this purpose? Should I use the time step long enough to reach the steady-state?

Hi Everyone, This is my first post here. I have no knowledge of queuing theory, so forgive me if the question is quite basic. I have a poisson distribution that calculates the number of events at a given time interval. I have split the distribution into 5 minute time periods for a day, meanin...

I have a differential equation x'(t) = tx + t^4, with initial condition x(5)=3. I am asked to find the estimates using the taylor series method from o < t < 5 with h=0.01 steps. I get that you have to use the formula x1 = x0 + hx0' + h^2/2! x0'' .... but this is a recursion. If I was given initia...

Consider the following versions: $$dX_t=x_0+sgn(X_t)dW_t \tag1$$ $$dX_t=x_0+1_{(0,+\infty)}(X_t)dW_t \tag2$$ $$dX_t=x_0+1_{(-\infty,0]}(X_t)dW_t \tag3$$ SDE (1) is a classical example. SDE (2) has a strong solution $X_t=x_0+W_{t\wedge \sigma}$, where $\sigma=\inf\{t\ge 0 : x_0+W_t\le 0\}$. Ho...

I'm stuck on the following homework question: Given the triangle $PQR$, with $X$ placed on $PR$ dividing it into a ratio of $2:3$, and $Y$ the midpoint of $PQ$, prove that if $Z$ is the midpoint of $QX$ and $YR$, then it divides $YR$ into a ratio of $1:3$. Note that this is to be solved pur...

If I were to construct a certain set of complex numbers such that it is comprised of complex numbers whose norm is irrational, how can I show that this set is open? I was thinking about using the definition of an open set. I would show that each member of the set is an interior point. However I ...

If $G^{'}$ is Commutator subgroup of $G$ and $G=G{'} $. Can I show that $Z(G)= \{e \} $? I think its not True but i can not find example.

function r=isssdd(A,s) [m,n]= size(A); sbar=1:m; sbar(s)=[]; if m==n a=diag(abs(A)); B=abs(A-diag(diag(A))); rs= sum(B(:,s)')'; rsbar =sum(B(:,sbar)')'; c= (a(s)-rs(s))*(a(sbar)-rsbar(sbar))' > rsbar(s)*rs(sbar)'; if all(a(s)> rs(s)) & all(all(c-diag(diag(c))+eye(m))); ...

I have an ordinary differential equation in which I substitute $\xi = \xi(x), \eta = \eta(x,y)$ and I wish to express $y''_{xx}$ in terms of $\eta''_{\xi\xi}$. So I get an equation $$d^2\eta=\frac{\partial^2 \eta}{\partial x^2}+2\frac{\partial \eta}{\partial x\partial y}dxdy+\frac{\partial^2 \et...

Let $A=\{\left(\begin{array}{c} 18 \\ 6 \\ -4 \\ 12 \end{array}\right),\left(\begin{array}{c} 6 \\ 2 \\ 2 \\ -6 \end{array}\right)\}$ find the vectos to be added so A will span $\mathbb{R}^4$? So what I did is : \begin{pmatrix} 18 & 6 & -4 & 12 \\ 6 & 2 & 2 & -6 \\ \end{pm...

The oriented Oberwolfach problem (with one table) and its solution are the following. In a meeting of $n$ people during $n-1$ days (combinatorists at Oberwolfach for concreteness), they all have diner around one table. As they only speak to their right neighbour, everybody wants to be seated eac...

given the graph defined in this post: A binary sequence graph how would one go about finding its chromatic number?

Abc is a triangle in which l is the midpoint of ab and n is a point on AV such that an =2CN Aline thought l parallel to BN meets AC at M prove that AM=CN

My book says that $f_U(u)=\Phi'(u)+\Phi'(-u)=\phi(u)+\phi(-u)$ because of the chain rule. Why is that?

If $f(x)=\frac{\arcsin(1-\left\{x\right\})\times\arccos(1-\left\{x\right\})}{\sqrt{2\left\{x\right\}}\times(1-\left\{x\right\})}$ Find $\lim_{x\to 0^+}f(x)$ and $\lim_{x\to 0^-}f(x)$.Where $\left\{x\right\} $is a fractional part function. As $x\to 0^+,\left\{x\right\}\to 0$,so $\lim_{x\to 0^+}...

Let f be a continuous function on each sequence of compact sets {$K_i$}. Does it have to be continuous in the finite union of these sets $K= \bigcup\limits_{i=1}^n K_i $ ?

Prepping for an exam and wondering whether I correctly calculated the time complexity. Function is given as: $function XYZ(n:integer)\\ begin for\ i:=1 \ do \ 2*n^2 \ do;\\ if \ n = 1\ then\ return (2);\\ else \ return(2*XYZ(\lfloor(XYZ(\lfloor n/2\rf...

Let $(X,d)$ be metric space and f be a one to one and continuous function, now we consider $ k(x,y)=d(f(x),f(y))$, is k(x,y) and d(x,y) equivalent ? In especial case we can consider $ k(x,y)= | \frac{x}{1+|x|} -\frac{y}{1+|y|} |$ and $d(x,y)=|x-y|$, now $f(x)=\frac{x}{1+|x|}$ and with definitio...

Let $\Omega \subset \Bbb R^n$ and $L$ be a linear differential operator 1 $$L:V\to W, V:=\mathcal C^1_0(\Omega;\Bbb R), W:=\mathcal C^0_0(\Omega; \Bbb R^{n*n})$$ $$L(u):=(\partial_j u_j - \partial_i u_j)_{i,j=1,...n}$$ equipped with the following inner products $$\lt u,v\gt_V:=\int_\Omega\sum_{i=...

I want to prove that the following problem admits a unique solution $A_I\in H(curl;\Omega_I)\cap H(div;\Omega_I)$. $$ \begin{cases} curl(\varepsilon_I^{-1}curl A_I)=curl v_I\;\;\text{in}\,\,\Omega_I\\ div A_I=0\;\;\text{in}\,\,\Omega_I\\ A_I\cdot n_I=0\;\;\text{on}\,\,\Gamma\cup\partial\Omega\\ c...

can you help me in this question? I would like to know if I am doing it right. A group of monkeys are given two tasks to do, the second of which is harder than the first. A proportion (3/5) get the first one right and (7/15) get the second one right. If a monkey gets the first one right, it h...

How to prove the following statement: If $n\in\mathbb{N}$ satisfies: $$ \left(\sum_{k=1}^{n}[k,n]^{(k,n)}, n^3\right)=1 $$ where $[k,n]$ and $(k,n)$ are the least common multiple resp. the greatest common divisor of $k$ and $n$, then $n$ is square-free. I started with tackling the sum: $$ \sum_...

They are both Galois fields of order 8. I'm not exactly sure what the question means - how does one determine/describe an isomorphism?

?? ¿¿????????????????????????!!!!!!!!!???????????????????????!!!!!!????? 0<1? 1<2?

How to solve the following equation? (2x-y+4)dy+(x-2y+5dx)=0 It is necessary to determine the type and total solution.

I have variables made of vectors of numbers that I need to transform. Calling these X Y Z, the arithmetic means of the numbers are M(X) Cheers

Let $\pi(k)$ denote the number of primes $p$ such that $p\le k$. For $k=2,3,4,6,8,\cdots{},$ we have $\pi(k)|k$. Does there exist infinitely many integers $k$ such that $\pi(k)|k$ ?

Is there an open set $S$ with a bounded function $f$ such that $\int_\overline{S}f$ exists but $\int_Sf$ does not?

I had a question in regards to solving a Big-Theta problem. Our professor wanted us to prove that $n^3 - 47n^2 + 18 = \Theta(n^3)$ and to do so rigorously, meaning he does not want us to use the below method: $\lim\limits_{n \to \infty} \dfrac{f(n)}{g(n)} = c$ $f(n) = \Theta(g(n))$ iff $0 < c <...

Suppose that $f(z)$ is an analytic function on complex plane. For all $z\in D_f$ We have $|f(z)^2-1|<1$. Prove That the sign of $Re f(z)$ is fixed for all $z$?

A circle rests in the interior of the parabola with equation $y=x^2$ so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency ? I've seen the solution of the problem here on Aops site and it left me a bit skeptical. I a...

I am not getting understand Inversion method to solve linear Equation 2x1 + 4x2 = 4 9x1 + 3x2 = 6 How to solve this please clarify steps.

A rise breeder is trying to cross yellow and red roses. He finds, on average, that 1 in every 4 of his test seeds produce a yellow rose. If he plants 12 seeds, determine the prob that: Exactly 2 roses are purely yellow. Less than 3 of the 12 are purely yellow. My attempt a: n=2 p=0.25 12C2= 66 6...

Assume that $A,B \subset R^2$ and $A,B$ are compact and connected is $A^c$ and $B^c$ are homeomorphic ? I don't have a good idea to solve that.

Problem: $$ y=\ln(3x-2)^2 $$ State the domain and the coordinates of the point where the curve crosses the x-axis At first sight, you say that the domain is $x>\frac23$ because $\ln$ is undefined for negative numbers, so you just rearrange $3x-2>0$. But the input of $\ln$ is squared, whic...

I need to prove those but i am really confused because i think i have a counter example for the first two : for the first : f(x) = {if x < 0 then 0. if 0 <= x < 4 then 1. if x >= 4 then x^3} in that way, if x -> 4, then the limit of f(x) is 64, but the limit of sqrt(4) = 2 is 1, and not sqrt(6...

function question2a(a,x) y=zeros(1000,1); y(1)=x; hold on; for j=2:1000 y(j) = a *y(j-1)*exp(-y(j-1)); plot(j, y); end hold off; end Is my code for a certain exercise, however I am not getting a line instead of a line I am getting a lot of dots. Can someon...

Suppose $X$ and $Y$ are Banach spaces. Denote a function $F:X \rightarrow Y$ and $U \subset X$ is an open set. Gateaux differential $F$ at $u \in U$ in the direction $\phi \in X$ is given by $$\lim_{t \rightarrow 0}\dfrac{F(u+t \phi) - F(u)}{t}.$$ In Wiki, there is this sentence in the sectio...

The book proves the statement as follow: If $E$ is finite over $F$, then $E=F(\alpha_1,...,\alpha_n)$, where $\alpha_i \in \bar F$. Let $irr(\alpha_i,F(\alpha_1,..,\alpha_{i-1}))$ have $\alpha_i$ as one of $n_i$ distinct zeros that are all of a common multiplicity $v_i$, then $$[F(\alpha_1...

Hankel transform is defined by $F_{\nu}(k) = \int_0^{\infty}f(r)J_{\nu}(kr)rdr$, and the inverse transform by $f(r) = \int_0^{\infty}F_{\nu}(k)J_{\nu}(kr)kdk$, In my problem, r is a complex variable. Is it still possible to define the hankel transform by integrating on a complex path? Best...

I recently encountered some polynomial inversion in some physics literature, the simplified version would be the following: $$f(x)=\sum_{n=0}^{\infty} f_nx^n=\frac{g(x)}{(x-x_1)(x-x_2)}$$ where $g(x)$ is some power series in $x$. People may naturally think that $x_1$ and $x_2$ should just be two...

Let f be a $2\pi$ periodic function. Assume that $f$ is quadratic integrable in the interval $[0,2\pi]$. Consider $f$ as a vector in the Hilbert space $L^{2}([0,2Pi])$. Give based on the Fourier coefficients of $f$ the best approximation of $f$ in $L^{2}([0,2Pi])$ as a linear combination of $sin(...

Recently I started to study Operads. My reference is Algebraic Operads of Jean-Louis Loday and Bruno Vallette. In this book they define augmented algebra of the following form: an $\mathbb{K}$-algebra $A$ is augmented when there is a morphism of algebras $\epsilon: A\rightarrow \mathbb{K}$ called...

I want to solve an example like this : $\int_{0}^{4}\sqrt{4^2-x^2}\ dx$ according to this equation :$$\int \sqrt{a^2-x^2}\ dx= \frac{1}{2}\left(x\sqrt{a^2-x^2}+a^2\sin^{-1}\left(\frac{x}{a}\right)\right)$$ I have problem with that sine . I want to see the solving step by step.

Which of the following regular expression identities is/are TRUE? $r(^∗)=r^∗$ $(r^∗s^∗)=(r+s)^∗$ $(r+s)^∗=r^∗+s^∗$ $ r^∗s^∗=r^∗+s^∗$ AFAIK : I can't say anything, but it should be valid. $LHS\subset RHS$ $RHS\subset LHS$ $RHS\subset LHS$ My question is : Is $r(^∗)=r^∗$ valid regul...

I am trying to simplify an integration expression like this:$\int_0^{D}\frac{1}{x^{a}+1}dx$. I know that for a is 2, I can have it as arctan. But what about the general case? Thanks for your help!

I am solving a system of nonlinear equations in Matlab. The solution G returns a struct object and I can pull out the solution(s) x, y as S1=[G.x, G.y]. I cannot evaluate the values though. If I examine S1(1, 1) it contains a multiple of RootOf(cubic in a dummy variable z1) and if I try float(S1...

I know almost nothing about transcendental numbers, I know the definition of them and maybe few results about them and that is all. But the question in the title somehow naturally arises when thinking about transcendental numbers. I think that it is okay to state it once more in the body of the...

Suppose $f\colon [0,\infty) \rightarrow \mathbb R$ is measurable, and suppose there exists $M>0$ s.t. $\forall h\in\mathbb L_2[0,\infty)$ We have $ |\int _{[0,\infty)}f(x)h(x)d\mu (x)|\leq M\cdot \|h\| _ {L_2}$. Show that $f\in L_2$. I got stuck at the end (My attempt is at the bottom). I would...

Why is $\csc\theta=\frac{1}{\sin\theta}$ instead of $\csc\theta=\frac{1}{\cos\theta}$ and the same thing for $\sec\theta$? I wonder if there is an interesting reasoning for this.

If in a ring $R$, $x^2=x$ for all $x$ then show that $2x=0$ and $x+y=0 \Rightarrow x=y.$ I am unable to proceed. Plz help.

What does XY -> Z mean? Does it mean X -> Z and Y -> Z? Or does it just mean XY -> Z?

The probability that a seed will germinate is 0.37. Suppose 124 seeds are planted. Use the Central Limit Theorem to determine the probability that at most 42 seeds germinate.

Two circles have an external tangent with length $36$cm .The shortest distance between these circles is $14$cm.If the radius of the longer circle is $4$ times the radius of the smaller circle then find the radius of the larger circle. I dont know how to solve this problem.I tried but i failed....

I'm using this article for the proof. everything sounds well, but I don't think I have a proper comprehension on some (specially final) parts. For example, what is the function $f$ doing at the last stages and how is it working? please give me some intuition on what is going on in the proof.

I would like to proof the following statement : $$[ (\neg p \implies q) \land (\neg p\implies \neg q)] \implies p$$ My Proof: Methode $1$: Let $ R: [ (\neg p \implies q) \land (\neg p\implies \neg q)]$ Suppose that $R$ is true \begin{align} R &\iff [ (\neg p \implies q) \land (\ne...

Let $f \in \mathbb R[x,y]$ be such that for some open set $U \subseteq \mathbb R^2$ , $f(x,y)=0 , \forall (x,y) \in U$ ; then is it true that $f(x,y)=0 , \forall (x,y) \in \mathbb R^2$ ?

How can we find the number of numbers between 1 and 1000 which are divisible by 7 but not 2,3,5?

where w is 3x1 matrix, J is symmetric matrix. I don't know how this equation is obtained..by mathematically. Please help me!!