If not prove otherwise. Please give counterexamples if not true.

How many weighings of a balance are necessary to determine if a coin is counterfeit among eight coins. The counterfeit coin is either heavier or lighter than the other coins. I understand the reasoning behind this problem when you know how the weight of the counterfeit coin compares to the r...

How do you find a formula for the nth term of the sequence defined by: $x_n$$_+1$=$x_n$+$2$x_n$$_-1$ Where $x_0$=4 and $x_1$=-1 ? n=1,2,3,...

From textbook definition of vector spaces, we know that it is a set V together with an operation of addition, x+y for any x,y $\epsilon$ V, and an operation of scalar multiplication, sx for any x$\epsilon$R. This is straight-forward, but how would you know if x+y of x,y $\epsilon$ V, would actua...

Question: Suppose the order of a group $G$ is prime. Does $G$ have subgroups? If so how many? Attempt: Yes, G has subgroups. Let $G$ be a group of order $p$, where $p$ is prime. Let $H$ be a subgroup of $G$. By Lagrange's Theorem $\mid H \mid \vert \mid G \mid$. But since the only (positive) div...

The Poincare Theorem is, "If a limit cycle exists in the second order autonomous system \dot{x}=f(x), then N=S+1". Does this theorem gives only a necessary condition for existence of a limit cycle or it is also sufficient? If the condition is not sufficient, is there any counterexample? Thanks fo...

I need to know how to define formally an array of length n and composed by ones and zeros depending on certain condition Maybe something like this? $\vec{A}= (a_1,a_2,a_3...)\mbox{ with } a_i=1 \mbox{ where } X_i=Y\mbox{ or } a_i=0 \mbox{ where }X_i\neq Y\hspace{0.5cm}; 1 \le i \le n$

I am trying to calculate $$\lim\limits_{x \to 0} \frac{(x-\sin x)^{70}}{1-\cos x^{105}}$$ Here is my attempt: $ $ write $\cos$ and $\sin$ as Taylor series, and plug back into the original expression yields:$$\lim\limits_{x \to 0} \frac{(x^3/3!-x^5/5!+x^7/7!-...)^{70}}{x^{210}/2!-x^{420}/4!+x^{63...

Let $A ∈ R^{2×2}$. Show that the Jacobi iteration converges for all starting guesses if and only if the Gauss-Seidel iteration converges for all starting guesses.

I need help with this problem, I keep solving it my own way but am not getting the same series as a result.

I'm curios whether or not the following implication is true: If $x_{n} \notin \ell^2{(\mathbb{N})}$, is there necessarily a sequence $y_{n} \in \ell^{2}(\mathbb{N})$ such that $x_{n}y_{n} \notin \ell^{2}(\mathbb{N})$?

Hey i have tried root and alternant but couldn't make it, how u guys solve this, if u r in an exam. which test should i tried? hel please thank you to everybody bye good night and god bless u.

A permutation $s$ of $n$ elements has order $2016$ ). What is the minimal value of $n$? Give an example of an $s$ for the smallest $n$. How do I solve such a problem? I know I must consider the cycle decomposition of $n$.

I have tried evaluating $|\psi\rangle^2$, both using the tensor product notation and the Kronecker product. Why do the results look so different? Are they even right?

I am looking for some direction on the following problem. I am sure there are many variations with specific distributions, constraints, ect. but I will phrase it in a fairly general form. Suppose we have a set of $N$ real random variables $S=\{X_1, ..., X_N\}$ and we know (or can estimate with r...

If the ratio of cosines of 2 angles are bounded from above, can we conclude that the ratio of the angles are bounded as well ?

$$\sum_{n=1}^m \frac{k \cdot k! \cdot \binom{m}{n}}{m^n} = ?$$ My attempts on the problem: I tried writing out the summation. $$1+\frac{2(m+1)}{n} + \frac{3(m-1)(m-2)}{m^2} + \cdots + \dfrac{m\cdot m!}{m^m}$$ I saw that the ratio between each of the terms is $\dfrac{\dfrac{n}{n-1} (m-n+1)}{m...

According to Axiom 3.5.5 of the HoTT book, we have propositional resizing if there is a function f such that f : Prop$_u$$_{_i}$ $\rightarrow$ Prop$_u$$_{_{i+1}}$ isequiv(f) The idea is that p and (f p) are in some sense 'equivalent' for any p : Prop$_u$$_{_i}$, and similarly for q and...

Please consider Example 4.8 on http://www.probabilitycourse.com/chapter4/4_1_3_functions_continuous_var.php Would someone mind explaining how $P(X^2≤y)=P(−\sqrt{y}≤X≤\sqrt{y})$ and how $R_y=[0,1]$ (essentially if $-1<x<1$ how does it become $0<y<1$)? Thanks.

A curve is given by the equation: 3(x+1)^2-9(y-1)^2=32 (A) Find the coordinates of the two points on the curve at which x = 3 (B) Find the slope of the curve at each of these two points.

What is a comparatively elementary way to see that $S^n$ cannot be an $H$-space if $n$ is even that does not invoke, say, Adams's full solution to the Hopf invariant one problem?

For real numbers $a,b,c$ and $d$, define $\mathcal{O}=\{x|cx+d\neq 0\}$. Then define $$f(x)=\frac{ax+b}{cx+d}\;\;\;\text{for all $x\in\mathcal{O}$}$$Show that if $f:\mathcal{O}\rightarrow\mathbb{R}$ is not constant, then it fails to have any local maximizer or local minimizer. Sketch the graph...

Is it possible that there exist a set U that has the property U=UC?

If we used the first 2015 terms to estimate the sum of the convergent series $\sum\limits_{i=1}^n$ $(-1)^(n+1)/(n^2)$ would it be an overestimate or an underestimate? I really have no idea where to start with this so could someone help me out please? P.S. I'm having a bit of trouble with the Ma...

Consider the function f: (Z_10, +_10) --> (Z_20, +_20) defined by f(n)=2n. Show that f is a homomorphism and calculate the image. [NOTE: Z_m = {0,1,2,...,m-1} is a group under the operation +_m defined via i +_m j equals the remainder of i+j when divided by m.] I understand that to prove homom...

I have shown: $$\bigg(\int_{0}^{1}f(t)g(t)dt\bigg)^{2} \leq \int_{0}^{1}g(t)^{2}dt\int_{0}^{1}f(t)^{2}dt$$ and now I'd like to use this to show the Minkowski inequality for $p=2$, i.e. $$\Bigg(\int_{0}^{1}(f(t) + g(t))^{2}dt\Bigg)^{\frac{1}{2}} \leq \Bigg(\int_{0}^{1}f(t)^{2} dt\Bigg)^{\frac{...

Is "$\bar {int(A)}$ for all closed sets $A \subseteq R$" true? How can I prove it?

Consider the closed curve in the xy plane given by x^2+2x+y^4+4y=5 (A) Show that dy/dx = - x+1/2(y^3+1) (B) Write an equation for the line tangent to the curve at the point (-2,1)

Suppose $H_1= g \cdot H_2 = gH_2g^{-1}$. Is there an easy way to see $|H_1|=|H_2|$? Thanks!

I need some hint to solve this problem. Let U be a unitary matrix. Prove that if λ is an eigenvalue of U, then |λ| = 1.

At a seaport, the depth of the water $h$ metres at a time $t$ hours during a certain day is given by this formula; $$h=1.8sin[2\pi{t-4.00\over12.4}]+4.3$$ What is the maximum depth of the water? When does it occur? I know the maximum depth is 4.9 metres but what I don't know is how to solve the ...

Can anyone tell me how can I solve this?? $$\frac{\theta^n}{\Gamma(n)}\int_1^{\infty} \left(\frac{x-1}{x}\right)^{2n-2}x^{n-1}e^{-\theta x} $$ Where $\theta > 0$

discuss the convergence of the series for all p. 2/n^(p-1/2)= 2*sqrt(n)/n^p < (sqrt(n+1)+sqrt(n))/n^p < 3sqrt(n)/n^p < 3/n^(p-1/2) so ~ (sqrt(n+1)+sqrt(n))/n^p ~ K/n^(p-1/2) converges when p>3/2, diverges when p<=3/2 Intuition: I there exist M, an Is my reasoning right?

I have been stuck on this for some time I am not sure how to approach this type of problem: Let $n \ge 1$ be an integer, and for each $i=1,2,...n$ let $p_i$ be a real number such that $0<p_i<1$ $\sum_{i=1}^n p_i$$\prod_{j=i+1}^n (1-p_j)$$=1-\prod_{i=1}^n (1-p_i)$ Example: for $n=1$ the equat...

This is exercise 1.C.4 in Isaacs, Finite Group Theory. I think I have a proof, but would like to verify the proof and also inquire whether it can be shortened significantly. Let $|G| = 120 = 2^3 \cdot 3 \cdot 5$. Show that $G$ has a subgroup of index $3$ or a subgroup of index $5$ (or both). ...

Let $M = p_1 *p_2 * ...*p_n$ be a positive integer with prime factorization. Let $gcd(a,M) =1$. Prove that $a^ {(p_1 -1)*(p_2 -1)...*(p_n -1)} = 1 (mod M)$, by induction. The base case is just Fermat's Little Theorem but I am unsure how to proceed.

Are two finite groups isomorphic iff their subgroups are? Are two finite graphs isomorphic iff their subgraphs are?

A fair die is tossed twice. Let $X$ = the ssm of the faces, $Y$= the maximum of the two faces, and $Z$=|face 1 - face 2|. write down the value of $X,Y,$ and $W=XZ$ for each outcome $w\in\ S$ I already found the value and range of $X,Y$ but I'm not sure how to find $W=XZ$. I saw someone post a s...

I would appreciate it if you could verify my reasoning is correct or inform me of where the flaws in my reasoning are for the following problem: "Find the Galois Group of $x^3-3x+3$ over (a)$\mathbb{Q}$ (b)$\mathbb{Q}(i\sqrt{15})$ (a) Since the discriminant of $X^3-3x+3$ is negative, I know t...

I am sure this is a duplicate, but sometimes a question is so basic and silly that it gets ignored / deleted by experts so a newbie to the community is missing basic information because "everyone knows that" (I think this is some famous paradox but this is another topic) The question is dead si...

In the symmetric group of degree 4, S_4. Find a 3-subgroup and a 4-subsgroup of Slow. Can anyone help me? Thanks for all your help!

I'm teaching advanced level mathematics. And I need good question banks on quadratic equations . More likely good question bank on advanced level maths.

Suppose S and T are two disjoint and compact sets in a normed vector space W. Then there are $s \in S$ and $t \in T$ s.t. the infimum of $||s-t||=||s-t||$? Is this true? How can I prove that?

In this webpage Computing the Digits in π there is a proof of the Euler Transform. The proof there relies on measure theory and Lebesgue integration, I haven't studied that yet. In page 22 there is the following statement: Euler didn’t actually prove any general theorems about this transfo...

I want to find the radius of covergence of $$\sum_{k = 0}^{\infty}\frac{ k^{2 k + 5} \ln^{10} k \ln \ln k}{(k!)^2} x^k$$ Thank you for any help!

Is Free Functor $Set ---> Ab$ continuous? Free Functor sends a set X into the free Abelian group FX generated by X.

How many 4-elements subset of a set containing numbers 1 to 9 have atleast two odd number as elements?

Can someone please explain to me what a combined function is and how it is different from a composite function. From my understanding a combined function would be (f+g)(x) = f(x)+g(x) and a composite function is (f+g)(x)=f[g(x)] Also what factors affect a combined functions domain?

http://imgur.com/J8QUiAt In this problem, I have to find the area of that blob. pretty much I have to see if Nx-My is equal to 1. For the first choice, it IS equal to 1, yet the answer key says it is 4?

What are the colorings of a Cuboctahedron for up to $j$ colors? The colorings are the same if they can be rotated to be the same. I know I must use Burnsides Lemma, how? The Cuboctahedron is not fully symmetric

The statement from the title. Thank you.

Suppose exist a $[n , k , d ]$ liner block code like $C$ can you get hint an idea to prove exist a $ [ n , k , d-1 ] $ liner block code like $C' $?

How do i check convergence of series whoose general term is given by $(\frac {n}{n+1})^{n}$. I have tried ratio and root tests but they don't furnish any results. Thanks

Hi can someone please explain to me what Causal process means and how to make sure a ARMA process is causal? For example, I have this equation: $X_t = 7X_{t-1} + aX_{t-2} + 3Z_t +9Z_{t-1}$ Thank you.

Two sets S and T are disjoint and compact in a normed vector space. Define $f(S,T)=inf\{||s-t||:s \in S, t \in T\}$. Are there elements $s \in S$ and $t \in T$ s.t. $f(S,T)=||s-t||$?

A fair coin is tossed three times. Let X be the number of heads among the first two tosses and Y be the number of heads among the last two tosses. What is the joint probability mass function of X and Y? What are the marginal probability mass function of X and Y? (i.e. p_x (x)and p_Y (y)?) Find...

This is my LMI constraint : (P1) + (beta^2) * R * R' > 0 where P1 = A' * (I + F*C)' * P + P*(I+F*C)*A + A'*C' * G'*(Py) + (Py) * G * C * A - C' * (Pk)' - (Pk) * C + 2I R = P + P * F * C + (Py) * G * C beta = some constant A' means the transpose of A matrix. A * B means matrix multiplicatio...

How many diagonalizable matrices are there under the field $p$, where $p$ is a prime?

I have an example from class notes that I do not understand and would appreciate some clarification. Particularly, I haven't found a direct explanation online or in my text with regards to the limits of integration for the following example. If $X$ is a nonnegative random variable, then \begin{...

I have the following ODE: $$y^{''} + 4y^{'} + 3y = 65cos(2x)$$ I first used the method of undetermined coefficients to find the solution to this equation, which was relatively easy, and found that the solution was. $$C_1e^{-3t}+C_2e^t+8sin(2x)-cos(2x)$$ where to the my $y_h =C_1e^{-3t}+C_2e^t$, I...

Given $$ AX^2+2X-1=0 $$ What value of A would make the absolute value of both roots bigger than 1? I found the roots using quadratic formula but not sure what to do from there. Thanks!

Let's say I have two data sets (x,y) and (p,q) and two approximation trendlines: Logarithmic: y = b*ln(x) + a Linear: y = bx + a Let's say I applied logarithmic approximation to both data sets, so I have this: y = b1*ln(x) + a1 p = b2*ln(q) + a2 where: b1 > b2 After I've applied linear ap...

Find the general solution of $f''(x)+f(x)=0$ I know it has to be cosine, sine or the exponential, but I was wondering if there was a general form for the solution before applying the initial conditions. Thanks.

I have a few questions in regard to the following example given and corresponding solution in Wackerlys Mathematical Statistics; Suppose $Y_{1}$ and $Y_{2}$ have jdf given by $f(y_{1},y_{2})=k(1-y_{2})$ if $0 \le y_{1} \le y_{2} \le 1.$ , $0$ elsewhere Then it asks to find (1) the value of k ...

Suppose that $X$ is a Banach space and $\mathcal{F}(X) = \overline{span}(\{ \delta_x : x \in X \})$, where $\delta_X : Lip(X) \rightarrow X$ and $Lip(X)$ is the set of real-valued Lipschitz functions on $X$. Define $\beta : \mathcal{F}(X) \rightarrow X$ given by $$\beta(\mu) = \int_X id_X du$$...

I need help approximating the following integral to four decimal places using Taylor Series, I also need help approximating the error. Thanks. \int_{0}^{2}(1+2x)^{-1/5}

I get that I have to show that it is injective and surjective. But I'm confused as to how to show that with an ordered pair to a single element.

By what trigonometric trick does \begin{align} \sin\alpha\Bigg[\cos(\omega t + \varphi)+\frac{\cos\alpha\sin(\omega t + \varphi)}{\sin\alpha}-\bigg(\cos(\varphi)+\frac{\cos\alpha\sin(\varphi)}{\sin\alpha}\bigg)e^{-\omega t/x}\Bigg]\\ \end{align} reduce to \begin{align} \bigg[\sin(\alpha+\omega...

Part 1 Show that the equation for the tangent plane to the ellipse $x^2/a^2+y^2/b^2+z^2/c^2=1$ at the point $(x_0,y_0,z_0)$ is given by $xx_0/a^2+yy_0b^2+zz_0c^2=1$ (*). Part 2 Let $V(x_0,y_0,z_0)$ denote the volume bounded by the planes $x=0,y=0,z=0$ and the plane (*) for $x_0>0,y_0>0,z_0>0$....

Second order logic implies categoricity in peano arithmetic. But why are the models isomorphic to the standard model of aritmhetic and not to another non standard model, for example?

How do I solve the following simultaneous congruence: 25x congruent 18 (mod 48) x congruent 11 (mod 35). *I don't know how to make the congruence symbol on my keyboard

My teacher did this problem in class however he did not go over the steps. He gave the answer of .1808. Can someone guide me through the steps? I am really struggling with Beta Distribution. Suppose the proportion of new new businesses in the city of Barryville that fail within one year is a be...

If: y=f(x) and y=0 when x-> ∞ Is it possible that: d/dx(y) is not equal to zero when x-> ∞ And prove it!

I am confuse in the following two scenerio In a bag of 3 apples and 3 oranges. You pick 2 items from the bag. 1) what is the probability that you will have 2 apples? 2) what is the probability that you will have 1 apple and 1 orange? My attempt: 1) P(2 apples)=P(1st apple)xP(2nd apple)=3...

I define the function $d_{\mathrm{avg}} : [0, 1]\to [0, 1]$ such that for $0.x_1x_2x_3\cdots$ the decimal expansion of $x$ (defined such that $\nexists N : x_k = 9$ for all $k \geq N$), $$d_{\mathrm{avg}} : x\mapsto \lim_{n\to \infty} \frac{1}{n}\sum_{i=1}^{\infty} x_i$$ How would I show that the...

I have to prove that determinant of skew- symmetric matrix of odd order is zero and also that its adjoint doesnt exist. I am sorry if the question is duplicate or already exists.I am not getting any start.I study in Class 11 so plase give tge proof accordingly. Thanks!

So I have a function g that maps from some subspace, S, of R^n to R. g is concave such that g(x) > 0 for all x in this subspace, S, of R^n. f(x) is defined as f(x) = 1 / g(x) and the question is to show that f(x) is convex. How would I do this by the definition of convexity? My try: As g is ...

math problem on here Have been trying this problem for 4 hours still can't figure it out.

Is categorical ZFC2U?? Is expressable the cardinality of a set A from ZFC2U from a sentence in second order logic?

Let the sum of 3 sides of a reactant le be "a" find the max area possible with these dimensions My identity I derived is "a"square divided by 8 2x+y=a Area =x(a-2x) X(a-2x)-area= 0 Xa-2x2-area=0 Put in derivatives cause area is the max when slope is 0 A-4x=0 X=a/4 Y= a-2(a/4) Y is a/2 A/4*a/...

Consider the equation x^2+y^2=n 1. find all solution to x^2+y^2=2 2. suppose that n is a positive integer congruent to 3 modulo 4. Prove that the equation x^2 + y^2 = n has no such integer solutions.

Whether the following double series is convergent or divergent?

What is your favorite proof that $$\prod_{n\in P} n=4\pi^2\text{?}$$

Can someone help me with this textbook question: can you devise a nonzero matrix whose row echelon form is the same as the row echelon form of its transpose? Thank you.

Let $N_m$ denote the number of objects from a collection of $N$ objects that possess exactly m of the properties $a_1,a_2,\ldots,a_r$. Generalize the principle of inclusion-exclusion by computing $N_m$ as the following form and please explicitly give the $s_k$. $N_m= \sum\limits_{k=m}^r(-1)^{k-m...

Is it at all possible to rearrange this equation to make c the subject. My dad and I believe it is possible but I am unable to correctly rearrange for c.

Часть 1.Хотим ли мы видеть любое упоминание "и более" в тексте?Нет А теперь продолжение: Часть 2Хотим ли мы акцентировать внимание на слове "Ответ", поставив его на первое место? Первое слово в описании знака акцентирует внимание на неком предмете или действии. Вероятно, в случае достижения от...

I have a quick question. Are all the subgroups of $S_n$ symmetric as well?

Suppose I have a function $f(z)$ defined for every complex $z$ with real part strictly greater than $0$, and that I have been able to show $\frac{d^kf}{d(g(q))^k}(k)=k!h(q)$ for every positive integer $k$ and expressions $g$ and $h$ ($q$ is a fixed parameter). Are there any general results or met...

Let $X$ be a CW complex. How do I see that the cohomology ring $H^*(X)$ is commutative in the graded sense:$$x \cup y = (-1)^{pq} y \cup x \text{ if }\deg x = p \text{ and }\deg y = q?$$

Compute $\displaystyle\int^\infty_{-\infty} dx\displaystyle\int^\infty_{-\infty} dy\displaystyle\int^\infty_{-\infty} dz \delta(\sqrt{x^2 +y^2+z^2} - R)$.

Let $D$ be a domain in $ C$\ {0} such that the annulus {z$\in$$ C$ such that 1<|z|<2} is contained in $D$.Prove that there is no branch of the logarithm defined in $D$. My Attempt: Select a point Z'$\in$ {z$\in$$ C$ such that 1<|z|<2} such that Z'=-x where 1 < x< 2. Then Log(Z')=lnx +i$ \pi$ . ...

I need the find at which branch of the logarithm the $log(z - z_0)$ defined on, where $z_0 \in \mathbb{C}$. I need it for a problem that I am solving.

I wonder what the formulas are for ARMA(p,q), ARMA(p,0), ARMA(0,q)? I was asked to calibrate for a ARMA(3,0). I found 3 parameters using matlab but I am not sure how to formulate the equation...

Find all real polynomials $P(x)$ having only real zeros and which satisfy the equation $$P(x)P(-x)=P(x^2-1)$$ Please explain me the process and refer some books to learn polynomials. Thanks in advance !

Let $G=\{$ \begin{bmatrix} a & b \\0 & a^{-1}\end{bmatrix} :$a,b\in \mathbb R; a>0\}$ $N=\{$ \begin{bmatrix} 1 & b \\ 0 & 1 \end{bmatrix} $:b\in \mathbb R\}$ Prove that $G/N$ is isomorphic to $(\mathbb R,+)$ and $G/N$ is isomorphic to $(\mathbb R^{+},*)$. I have to get a onto homomorphism fro...

There are integers $m, n > 0$, and a collection $S$ of distinct subsets of some ambient set $A$, each of size at most $m$. Assume $|S| > (n − 1)mm!$. Prove that there exist $n$ sets $A_1, . . . , A_n ∈ S$ such that the intersections $A_i ∩ A_j$ are the same for all pairs $(i, j)$ where $i$ is not...

(a) The space of all linear trnasformations from V->V has dimension n^2. Consider I,A,A^2,A^3,.....A^(n^2). Then this collection forms a linearly dependent set.So we have a non trivial relation amongst these which will give us a polynomial such that p(A)=0. (b) Don't know how to proceed.

I have a given set (0<size<10^6) of numbers (up to 10^9). How do i calculate Nth (0<N<10^6) subset when sorted by sum

I recently found myself rather delighted after I had sort of proved the arc length formula for a curve $x \mapsto f(x)$, $x \in [a;b]$. However, I realized that I didn't really know how to argue rigorously why the limit yields the definite integral (I just need a good argument). What I have on my...

Let X be the sample from the uniform distribution $U(0,\theta)$ where $\theta \in (0,\infty)$ Show that the statistic $T(\textbf{X})=X_{n:n}$ is complete. Thanks a lot for all your help.

Identify the correct translation into logical notation of the following assertion. Some boys in the class are taller than all the girls Note: taller $(x, y)$ is true if $x$ is taller than $y$. $(∃x)($boy$(x)→(∀y)($girl$(y)∧$taller$(x,y)))$ $(∃x)($boy$(x)∧(∀y)($girl$(y)∧$taller$(x,y)))$ $(∃x)...

Suppose I am given the problem $$\min c^Tx\\ s.t. Gx \leq b\\ Ax = b$$ The Lagrangian is: $$L(x,\lambda,\nu) = c^Tx + \lambda^T(Gx -b) + \nu^T(Ax-b) = (c^T + \lambda^TG + \nu^TA)x-\lambda^Tb -\nu^Tb$$ What is the correct way of thinking to formulate the dual function at this point? Two alte...

For me I (x^2+ 1)/(x+1) by x^2 since that's the term with the highest exponent making it: (1+(1/x^2))/((1/x)+(1/x^2)) but I'm not sure where to go from there. I looked at textbook solution and they have: (X^2+1)/(x+1) = (x-1+2)/(x+1) but I'm not sure how they came up with that.

What will be the units digit of: 8^97 I know I will have to start like: 8^1, then 8^2, 8^3 etc... is there a shorter method to finding the units/digit?

Basically the question goes something like this: A floor of length 16m and width 4m is covered with tiles. The tiles are square tiles. You are required to find the area of the tiles and the number of tiles used.Sometimes they might ask you to find the greatest area of the tiles. So far this is ho...

I need solve the follow equation: a) A man to be building a table of 6 hours , while another man to do the same table needs 12 hours. For how many hours the two will perform the same work. b)A sportsman distance between countries A and B passed for a limited time. If moving with speed 35km / h w...

Given the probability density function $f(x) = \frac{1}{x^2}$ for $x > 1$. The expected value of this function is: $$E[X] = \int_1^\infty \frac{1}{x} dx = \infty$$ Can someone please explain this? Is $E[X]$ only finite under certain condition? Is $\infty$ really the expected value for this probab...

I am looking at $f(z)=\sqrt{1-z^2}$ and a branch cut on the real axis from $z=-1$ to $z=1$. Is it correct to say that $f(x+i\epsilon) = -f(x-i\epsilon)$ when $x\in(-1,1)$, $\epsilon\in\mathbf{R}$ and $\epsilon\to 0$? If this is true, is it also true that $f(x+iy)=-f(x-iy)$ when $y>0$? (The prob...

While I was reading the paper entitled (http://dx.doi.org/10.1109/ISCAS.2003.1204947) Kocarev, Ljupco, and Zarko Tasev. "Public-key encryption based on Chebyshev maps." Circuits and Systems, 2003. ISCAS'03. Proceedings of the 2003 International Symposium on. 2003. I encountered a table,...

in a lemma of sobolev spaces, i found this integral: for $x\in \mathbb{R}^k$ and $t\in\mathbb{R} $ this intgral (verify somme properties) $$\int_{\mathbb{R}^k}\int_\mathbb{R}|{\frac{u(x+te_i)-u(x)}{t}}|^pdxdt$$ is there any problem here of $dxdt$ i mean the first integral is normally mast be i...

I have a sample data set that looks like this: x y w 1 1 5 1 2 1 6 2 3 1 7 3 4 2 8 4 5 2 7 5 6 3 5 6 7 4 6 7 8 4 5 8 x and y represent indices from datax and datay. w represents a score from comparing datax[x] with datay[y]. I want to maximize the total score (or w) from d, where each value ...

I'm having troubles with the Levi-Civita symbol. I understand what the normal epsilon-tensor means and how it works. But how do I interpret this: $\mathrm{{\epsilon_{i}}^{jk}}$ or $\mathrm{{\epsilon_{ik}}^{k}}$ ? Thank you!

Question: There are two batteries. A Battery's life is following the Exp(1/20) distribution. The other one's life is following the Exp(1/40) distribution. One day, a person randomly chose one battery. When the battery was used for 20 hours, find the probability that the battery can be alive 10 ...

I need help proving why RSA requires the use of distinct primes. What I have done so far: I figured since I need to prove why RSA requires the use if distinct primes I am going to disprove the following: if p and q are the same number, n = pq c and are positive integers such that : cd is con...

Given that xn includes interval [a,b] and limit of xn = z. Prove z includes interval [a,b]. a<=xn<=b is given and I want to show a<=z<=b. Since limit of xn converges to z, abs(xn-z)<ε and z-ε

What is 72,000 km/hour in Système Relativistique (SR) units? I am getting 1/14490 or 1/14490th of the speed of light.

Can anyone help me in finding solution of the LTV equation given by t\ddot{x}-\dot{x}+tx=0 or equivalently the LTV system \dot{x}_1 =-tx_2, \dot{x}_2 =-(1/)tx_1. Thanks for your time and help! Shah

I understand it intuitively but please help me with its mathematical proof. Thanks

In "Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions" and many other books is described a lifting of triangulations for branched covers between surfaces. Lifting process May we generalise this branched covers (open, discrete maps) bet...

Hi i got a quick question because im trying to figure out exactly how much money i will get when i take out money from the ATM in Prague (visiting from Sweden). The last time i took out money it was 4000 Czech Koruna and the amount it took from my swedish bank is 1.529,36. The currency exchange...

Solve $\dfrac{dy}{dx}=\dfrac{y-3}{y^2+x^2}$ given that it passes through $(0,1)$. Right now I do not yet know how to solve differential equations with both $x$ and $y$ that you cannot separate. Please help. Thanks.

I'm trying to understand why \begin{equation*} \begin{split} E(R(\hat{f}'')) & = \frac{1}{nh^6} \int \int K'' \left( \frac{x-y}{h} \right)^2 f(y) dy dx \\ & \quad + \frac{n(n-1)}{n^2 h^6} \int \int \int K'' \left( \frac{x-y}{h} \right) K'' \left( \frac{x-z}{h} \right) f(z) f(y) dz dy dx \\ & =...

Suppose M = p1*p2*...pn, where p1,...,pn are distinct primes Prove by induction that, for all positive integers n, if gcd( a, M) = 1, then a^(p1-1)(p2-1)(pn-1) is congruent to 1 (mod M): I have no clue where to even start I.E base case. please help!

3/8= (0.125*3) = 0.375 = 37.5% is easy to calculate mentally but is there a better way to find the percent of these divisions fast and mentally? 3.5/8 4.5/7

Q What is the remainder when 1!+2!+3!+4!+5!+.......+50! is divided by 5! MyApproach $1$+$2$+$6$+$24$+$5$!/$5$!+$6 . 5$!/$5$!+$7$ .$6$ . $5$!/$5$!....so on $33$+$1$+$6$+$42$+...... I am not getting correct Ans as the solution is getting complex. Can anyone guide me how to approach the ...

Let $m$ be Lebesgue measure on $[0,1]$, and define $\|f\|_{p}$ with respect to m.Find all functions $\Phi$ on $[0,\infty)$ such that the relation $$ \Phi(\lim_{p\to 0^{+}}\|f\|_{p})=\int_{0}^{1}\Phi\circ f dm $$ holds for every bounded,measurable,positive $f$.Show first that $$ c\Phi(x)+(1-c)\Phi...

Show that P is the taylor polynomial for f at c, for example: $$P(x)=\sum_{k=0}^n \frac{f^{(k)}(c)}{k!}(x-c)^k$$ Sorry the whole problem did not fit in the title. Other Important information: f is an element of $C^n(I)$ where I is an interval, and c is an element of I. My Work: I plan to comp...

I have a question about infinity. There was a thought coming to my mind after watching a video about infinity. Could it be that infinity is somekind of movement? Now let me explain this thought. There is this paradox with the Grand Hotel. It is full. But due to its infinity someone can come and c...

I see that the below formula is the explicit formula of the Stirling numbers of the second kind. I know that the Stirling number of the second kind is the number of ways to partition set of $n$ objects into $k$ non-empty subsets. But, I don't at all see from where the below formula comes from. Cl...

I roll 10 initial dices. Whenever I roll a $6$ I roll 5 more dices. What's the expected number of dices I'm going to throw? I was thinking about it when trying to create a die system, but can't solve the problem.

I recently came across this problem. the problem The hypothenuse of the triangle has a length of 4, is base (highlighted in green) a length of x and the last side (highlighted in red) a length of y I found this two equations to solve the problem. First of all, x²+y²=16 (considering Pythagore'...

I try minimizing the following expression : $ V(x)=\sum_{i=1}^n|x_i - u|w_i $ I need to find u that minimize this expression. I know how to do it for w=1 (which is is the median). Any ideas for the general case ?

So I have the following equation: $\sin(x-\frac{\pi}{6}) + \cos(x+\frac{\pi}{4})=0$ It should be solved using the fact that Sin is an odd function, I can not really get the gripp of how and what I need to do? Any sugestions?

Prove or Disprove: if $\lim\limits_{n \to \infty} (a_{2n}-a_n)=0,$ then $\lim\limits_{n \to \infty} a_n=0.$ I don't think that this is true. and I'm trying to think about counterexample, but couldn't figure out a mathematical form of the sequence that I thought about, its a sequence where certai...

We got set some work to find some interesting facts or proofs regarding rational and irrational numbers. I wonder if anyone could offer some insight or recommend a good book/ website to look at.

The question says to prove that $$\sum_{n_1+n_2+n_3=n} \binom{n}{n_1*n_2*n_3}*(-1)^{n_1-n_2+n_3} = 1$$ where the summation extends over all nonnegative integral solutions of $$n_1+n_2+n_3=n.$$ I used the multinomial theorem to get: $$(x_1+x_2+x_3)^n = \sum_{n_1+n_2+n_3=n} \binom{n}{n_1*n_2*n_3...

What is the result of evaluating the following two expressions using three-digit floating point arithmetic with rounding? $(113. + -111.) + 7.51$ $113. + (-111. + 7.51)$ $9.51$ and $10.0$ respectively $10.0$ and $9.51$ respectively $9.51$ and $9.51$ respectively $10.0$ and $10.0$ respectively...

How can I estimate the volume of the region inside an open bounded set in dimension $n$ at distance less than $1/n$ from its boundary?

Find $\lim\limits_{n\to\infty}\frac{x_{n+1}}{x_n}! $ where $x_n=x_{n-1}+x_{n-2} ,(n>2),x_1=1,x_2=2$ $x_n=x_{n-1}+x_{n-2}$ $x_{n+1}=x_{n}+x_{n-1}$ From the first recurrence relation, $$x_n=\frac{3+\sqrt{5}}{5+\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^n+\frac{3-\sqrt{5}}{5-\sqrt{5}}\left(\frac{...

I have a problem where an amount, lets say $x$, are reduced by some percentage $p$ over a duration of $n$ descrete iterations. I want to calculate a factor $f$ such that $x_i = f_i \cdot x$, and the $n$th factor reduces $x$ to $x_n$ by $p$ percent: $f_n = (1-p)$. This gives a sequence of factors ...

How to calculate $\lim _{n\rightarrow \infty }\dfrac {n!}{n^{n}}=0$, $\lim _{n\rightarrow \infty }\sqrt [n] {n}$, and $\lim _{n\rightarrow \infty }\sqrt [n] {\dfrac {1}{n}}$?

So that's the question. But I am unable to find a counter example if it's false. Or how should I proceed to prove if it is true? I am confused please help. Many thanks.

show that $x$ is a power of 2 is equivalent in $N$ to $\delta_{0}$formula.(use the facts that $y $divides $x$ and 2 divides$y$are are equivalent in $N$ to $\delta_{0}$formulas.

Suppose, for two positive random variable $X$ and $Y$, and a convex function $g$, $$Cov(X,Y)=Cov(X,g(Y)).$$ Then how to show that $g(Y)$ is a linear convex function of $Y$?

Problem A man bought three tables or rs. $2500$. He sell the first at $5 $ % loss, second at $5 $ % profitable third at $10$% profit. Find the cost price of each table if on the whole he neither gains nor losses. Solution Let cost price of first table be x Let cost price of second table be y...

How to prove this statement where f is a mapping between topologial spaces X and Y and f(Cl(A)) is contained in Cl(f(A))

Consider the label sequences obtained by the following pairs of traversals on a labeled binary tree. Which of these pairs identify a tree uniquely? preorder and postorder inorder and postorder preorder and inorder level order and postorder I've read that inorder is necessary to draw unique ...

From a exercise list: Let $f:[a,+\infty)\rightarrow \mathbb{R}$ with continuous second derivative and such that $\lim\limits_{x\rightarrow\infty}f(x)=0$ and $\lim\limits_{x\rightarrow\infty}f''(x)=0$. Prove that $\lim\limits_{x\rightarrow\infty}f'(x)=0$. Does anybody has a hint or a solution? I...

We have been set some work to find out some detail proofs or facts about the number one. Can anyone help or tell me where to look? This is at university level.

$\Omega\subset R^n$ is a open set, how to show $\Omega=\bigcup\limits_{n=1}^\infty T_n$,$T_n$ is n-cube.

This expression is written under a radical and I need to take it out. So it needs to be in the form of $n^2$ to come out. ($\sqrt{n^2}=|n|$) $$(a+b+c)(a+b-c)(a-b+c)(-a+b+c)$$

Prove that $1/2^2$+$1/3^2$.......+$1/2015^2$ is ~ 1

I need help solving this maximization assignment. I am to find maximum $\rho_{k}$. All other variables are constants. \begin{equation} \begin{aligned} & \underset{ \rho_{1} \ldots \rho_{k}}{\text{maximise}} \frac{P_T\sum^{R}_{k = 1}\left[ \frac{|h_{k}g_{k}|\rho_{k}^2 (1-\rho_{k})(\beta\rho_{k} P...

How do you integrate: $$\int \underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_jx^j}{b_j}dx=\int x\cfrac{a_1}{b_1+\cfrac{a_2x}{b_2+\cfrac{a_3x^2}{b_3+\ddots}}}dx$$ Can you use closed form?

Let $A\in M_n $, we define $A^{(−1)} = [a^{−1}_{i j} ]$. Suppose $A=XX^*$ and $A$ be positive semidefinite. Why are, all eigenvalues of $A^{(-1)}$ positive?

Let $s = \sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + ...}}}}$. $$st = t\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + ...}}}} = \sqrt{t^{2} + \sqrt{2t^{4} + \sqrt{4t^{8} + \sqrt{8t^{16} + ...}}}}$$ Let $2t^{4} = t^{2}$: $$2t^{2} = 1$$ $$t^{2} = \frac{1}{2}$$ $$t = \frac{1}{\sqrt{2}}$$ $$\frac{s}{\sqrt{2}...

Limit $ a,b\in \mathbb {N} \ lim_{x\to\infty} ((n+1)^k + (-n)^l) /((n-1)^k -n^l)$ I am really stuck, I do not know where to start. I tried to find different case when k is bigger than l or when k is smaller, but it seems it is getting me nowhere. Thanks for help

If you have derived a power series expansion for an ordinary point, is there any way to check whether your answer is correct?

Prove that any (simple, undirected) graph G = (V, E), with m = |E| edges has chromatic number χ(G) ≤ √2m + 1.

In the link here http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/specgeom.pdf, p.4, it says that, given a fundamental 2-from $\mathcal{K}$ $$\mathcal{K}=\frac{i}{2\pi}g_{i\bar{j}}dz^i\wedge d\bar{z}^{\bar{j}}$$, a manifold is said to be Kahler if this form is closed, i.e., $$d\mathcal{K} = i\pa...

Let $X$ and $Y$ are two continuous random variable and $$P(X<Y)=P(X<g(Y)),$$ for some convex function $g$. Then is it true that $g$ is a linear function?

Let $f \in L^p$, where $p \ge 1$, then $|f(x)| < \infty$ almost everywhere. Does anyone know how to prove this by contradition?

If X and Y are both random variables with continuous uniform distribution with bounds [0,1], what is the distribution of XY? X~U[0,1] Y~U[0,1] XY~???

Example I'm trying to understand this example and can't figure out how it partition the covariance matrix. In this case, what is {Sigma_22}inverse ? Thank you for your help.

11 letters are selected with replacement from the word ‘BRACED’. i) How many different combinations of letters are possible? ii) How many distinguishable permutations of ‘ABRACADABRA’ are there? I do not understand how 11 letters are selected, with replacement, from a 6-letter word. However, her...

Is the statement ∃x(S(x)∧∀y(F(y)→A(x,y))), equivalent to ∃x∀y(S(x)∧F(y)→A(x,y))? Since from ∃x∀y(S(x)∧F(y)→A(x,y)), you can distribute ∀y across the statement, obtaining, ∃x(∀yS(x)∧∀yF(y)→∀yA(x,y)). Since x and y are not dependent, we obtain ∃x(S(x)∧∀yF(y)→∀yA(x,y)), which can be shortened to ∃...

I am taking a course in algebra and this problem was on my problem set, and I had no idea how to solve it. Suppose we have a sequence $s_n$ such that $5*s_{n+1}-s_{n}-3*s_{n}s_{n+1}=1$ for $1 \leq n \leq 42$ and $s_1=s_{43}$. What is $s_1+s_2+ \ldots + s_{42}$?

enter image description here I have solved the first two parts, but i didn't know who to find the rest, can someone help ?

How to prove that $$\int_0^\infty \frac{x\sin x}{x^2+1}=\frac{\pi}{2e}$$ I've tried several basic approaches like substitution and IBP but can't move forward.

How would I prove that for all elements of a set {n | n ∈ ℕ ∧ n > 7} (all natural numbers greater than 7), there are multiples of 3 and 5 which, when added up, are equal to that element of the set? A := {n | n ∈ ℕ ∧ n > 7} ∀ n ∈ A: n = x*3 + y*5 I do understand why this is true: 8 = 1*3 + 1*...

Given a graph, there are numerous algorithms out there to find a set of 3 vertices such that a triangle exists, but why is this topic studied? What are the applications? The advantages? The problems encountered?

I've got a question. As you know if we connect the dots on the medians , it makes another triangle inside the main triangle, so here's my questions : 1. Are the line flashes drew rightly? [As the picture] 2. If we make the second triangle (Red one) bigger , does it have similarity with bigger o...

Let $G=\{A\in M_{2x2}(\Bbb{F})\colon \det A\not=0\}$ be a group and $Z(G)\subset G$. I am working on a project(lin.alg.) and I am curious if there is a mapping(bijectively) that maps $G$ to $Alt(5)$? Is this a known problem?

The question is self explanatory

What happens to the unit circle and unit disc under the Mobius Transformation w = (z+i)/(z-i)? I don't know how to go about setting this out...i.e. how do i represent the disc as a complex equation?

Knowing that the function $$\frac{1}{\sqrt (1-x^2)}\ $$ is defined only for -1 < x <1 Are the limits of integration below allowed? $$\int_{-1}^1 \frac{1}{\sqrt (1-x^2)} \,dx= \pi$$ PS: As stated above, I can calculate the result as pi, I'm just questioning the reasoning behind using -1 and ...

Given a compact, self-adjoint operator $T$ on a Hilbert space $\mathcal{H}$ I know that we can write it as $T=T_++T_-$ where $T_+$ and $T_-$ are, respectively, positive and negative and $T_+T_-=0$. However, all of the proofs that I can find of this fact seem to rely on using continuous functional...

I am wondering if there is a practical interpretation of a principal component analysis: Consider you have a data matrix $X\in\mathbb{R}^{N\times p}$ and you perform a principal component analysis where you typically receive certain directions $v_1,...,v_q$, $q<p$, in $\mathbb{R}^N$ that explain ...