Let ATS be a fixed acute-angled triangle, i.e., all the three angles of the triangle are less than 90 degrees. Let E and F be two points on the sides AS and AT, respectively, such that the segment EF is parallel to the segment TS. Let M be the midpoint of EF and let MN be the perpendicular bisect...

In other words, I'd like to express y in terms of x instead of having x expressed in terms of y! I just don't know how to proceed!

Evaluate lim k-->infinity (1+1/k)^k Is there another approach to this solution other than l'hoptital. Here's my working till now ln(1+1/k)^k=kln(1+1/k) As k-->infinity ln(1+1/k)-->0 due to the fact that ln(1)=0

At what rate do Ramsey Numbers grow? Please include a proof of this for k colors.

prove that if 2a+5b is multiple of 7, then 5a+2b is also multiple of 7.

Can someone check if this is good? y=-10x^5 is injective y=10|x| is surjectiv e^(4x) is bijectiv for A = R B = N C = [0, infinite) E = (-pi/2, pi/2) F = (-1,1) J = (-10,10)

How to show that the dual space of $R^n$ with Taxicab norm is $R^n$ with sup norm? Thanks!

If we have a sequence of random variables $X_1, X_2, \cdots$ of IID random variables, is the sequence of random variables $X_1^2, X_2^2, \cdots$ also a sequence of IID RVs?

I am currently revising for my Additional Maths test and cannot figure out a way to get the solutions for the following questions. I know I have to use the change of base rule, but when I try it doesn't work. a) log_3(x) = 4log_x(3) b) log_2(x) - 4 = 5log_x(2) c) 2log_4(x) + 3log_x(4) = 7 I hav...

Sketch the region of the triple integral $$\int_{0}^1 \int_{0}^{y} \int_0^{x} x^2yz dzdxdy$$

φ is valid if and only if for every structure A, A 􏰀 φ. We write this as 􏰀 φ. Definition: φ is satisfiable if and only if there is a variable assignment s such that s satisfies φ, i.e., A 􏰀 φ[s]. Prove that if 􏰀 φ, then ∼φ is not satisfiable

A computer engineer is twice as old as his wife was when he was as old as his wife is now. He is 24. How old is his wife?

Can someone please show me how to factor 6q(7$(6q)^2 +5) to show that it is a multiple of 6? I'm working on a division algorithm problem, and I understand concepts of div alg but I really don't have much experience factoring and I can't find any good sources on learning this type of thing. Thank ...

Let say I have a cone located on z-axis. I have this equation: (x^2+y^2)/r^2=(z-z0)^2/h^2 I made a matlab code to make this figure, but I want it pointing to Z plane. here is the image: (http://i.imgur.com/RBD2r7y.png?1) I cannot rotate it on matlab, so I think I have to modify the equation. ...

(M-1)x^2+2(m+3)x+(m-7)=0 Please find the double roots of this equation . Please solve!!

A parking lot contains 100 cars, k of which happen to be lemons. We select m of these cars at random and take them for a test drive. Find the probability that n of the cars tested turn out to be lemons. Here is a picture of my work I understand what the sample space of the question is (100 choo...

Linear models are $$Y_1= 2\theta_1+3\theta_2 +\epsilon_1$$ $$Y_2= -2\theta_1+\theta_2 +\epsilon_2$$ and $\epsilon_1=3z_1-z_2$ and $\epsilon_2=4z_1+z_2$, where $z_1, z_2$ are two random variance such that its independent mean is 0 and variance is $\sigma^2$ I have written this model as $Y=X\be...

Hello I am wondering if my approach and understanding of the epsilon delta definition is correct. For example, if I wanted to use it to show that $$\lim_{x \to c} \frac{x}{1+x}=\frac{a}{1+a}$$ for $a \in \mathbb{R}, a \neq -1$ Then here are my thoughts; I know that we want that for all $\epsil...

$$\lim _{ x\rightarrow \infty }{ \frac { 1-2x }{ 3-4x } =\frac { 1 }{ 2 } } $$ Proof: Let $\epsilon > 0$ Then, $$\left| \frac { 1-2x }{ 3-4x } -\frac { 1 }{ 2 } \right| <\epsilon$$ $$\Longleftrightarrow \left| \frac { 2-4x }{ 6-8x } -\frac { 3-4x }{ 6-8x } \right| <\epsilon $$ $$\Longleft...

Let $A$ be a $2\times 2$ matrix which is diagonalizable and $V=\{B\in M_{2\times 2}\mid AB=BA\}$. What's the $dim V$?

I am trying to test if the given number N is a power of 2: public class Solution { public boolean isPowerOfTwo(int n) { if(n ==1) return true; if(n%2 != 0) return false; if(n%2 ==0) { isPowerOfTwo(n/2); } return true ; } }

For simple random sampling, I have calculated somemean square errors for ratio-type estimators such as Isaki estimator, and Prasad Singh estimator. But, Mean Square Errors i obtained are negative. Does MSE eqaul to a negative value? I there a mistake ?

Let A be a non-empty set and $f:\emptyset \rightarrow A$, why f must be 1-1 ? This is a part of my Sets Numbers and Functions' mid-term test, I'm still confusing about it...

How would I go about proving the following statement? "Let A be an $n x n$ matrix. $Col(A^2)=R^n$ if and only if $Col(A)=R^n$" I started off by proving that A is linearly independent if and only if $Ax=0$ which implies that $x=0$. I showed that the columns of A span $R^n$ if and only if Ax=b ha...

How to write the SDP(Semidefinete program) of the following optimization problem \begin{multline} \max_{Z,f ,g} \ trace(KZ) − f^Td \\ subject to \\ trace(W^{−1}Z) = k \\ Z_{ij} ≥ 0 \\ Z = Z^T \\ Ze = Wf \\ e^Tf = (1 + α)n \\ e^T g ≥ (1 − β)n \\ f \geq g \\ 0 \leq g \leq 1 \end{multline} I have ...

Is the following true or false: Suppose that X and Y are two discrete random variables defined on the same probability space. If E[X] = E[Y] = 0 and E[X | Y=y] = 0 for all y∈S Y, then X and Y are uncorrelated. How does this tell me E[XY] = 0?

Pretty lost on this one, partially because I don't understand the relationship between the determinant of a matrix and the determinant of its inverse. Thanks.

This answer explains that any elementary plane geometry problem can be reduced to the existence of a solution of a polynomial system (called the analytic reformulation). Question: Is there a problem of elementary plane geometry whose analytic reformulation gives a polynomial non-solvable by ra...

I am trying to perform a cellular edge detection algorithm. The pseudo-code of this algorithm : The CED algorithm The probablity vector for the learning automaton to be updated : where a and b are the reward and penalty parameters. Also, the local rule for CLA is : Bi,j=F(N)= 1 if Ai,j =A(N) ...

I just want to clarify some of my intuitions about (square) matrices...

This isn't strictly a maths question but it is a questions about maths. Should it be in meta? I don't know. I have a deep problem in that I had not expected how deep my disliking for mathematics goes. I do genuinely detest the subject with a passion. Sadly, it's also a crucial part of my degree...

$G$ is a group, if ∀a,b∈G, $a^2b=ba^2$, how to make a counterexample show that $G$ is NOT Abelian? What's the counterexample when $a^nb=ba^n$?

I am really confused trying to prove the following statement formally although it makes perfect intuitive sense. I want to show that if $Y$ is conditionally independent on $W$ given $X$, then: $E[W|X]=E[W|X,Y]$ I know I have to use the Law of Iterated expectations but I am failing to see how ...

such that T(5v⃗1+3v⃗2)=−5v⃗ 1+5v⃗ 2 and T(3v⃗ 1+2v⃗ 2)=−5v⃗ 1+2v⃗ 2. Then T(V1) = 5V1 + 4V2 T(V2) = ?V1 + -5V2 T(4V1 - 4V2) = ?V1 + ?V2 Can someone give me a hand on how to solve these problems, I would appreciate it. Thank you.

Urgently need help on this! Let's say I have 2 matrices, both 2x2, B = [2,6,8,16] and C = [6,12,24,1] How would I calculate the matrix for 3BC?

Actually, I do not have mathematica. Currently I tried to use another famous software to obtain inverse Laplace for below equation. Unfortunately, the software is unable to do it. So, I am wandering if Mathematica can do it. If Mathematica do, I will join you. Obtain the inverse laplace for bel...

How many ways are there of seating six people at a round table so that two specific people sit together? I think it is a permuation question but not 100% sure. Thanks

M= 4 -4 2 -8 7 -6 -30+k 22 -16 I keep getting -6 but its marked incorrect Thanks

This is from the book stochastic calculus: a practical introduction. I don't understand the last sentence in the picture. Could anyone help explain why the first one is a local martingale? Thanks so much!

I could use some help solving these 2 problems. I've finished the rest, I just need help on these two. I know I'm supposed to use natural log and l'hopitals rule. As the limit goes to 0, find tan(x)^sin(x). As the limit goes to infinity, find (e^x+x)^1/x I tried to use natural log for both o...

How many five-letter palindromes are there (using ordinary 26 letter alphabet)? I was thinking it was either using permutation with first three leters or using 26 cubed. Thanks

I have the following problem : pretty version sqrt(x+8) - sqrt(x-4) = -2 So far my thinking is to isolate sqrt(x+8) by changing our equation to -2 + sqrt(x-4) = sqrt(x+8) When we square all of it, i'm running into my issue. I don't understand what I should be doing with my binomial left over. A...

The first part is trivial. How to go about the second part?

I need to get local coords from global coords. I did not find ready-made examples. OPEN IMAGE Please help me how to do it on the example image. Thank you!