Find the sum of integers $a,b,c,d,$ and $e$ if $\dfrac{2011}{1990} = a+\dfrac{1}{b+\dfrac{1}{c+\dfrac{1}{d+\dfrac{1}{e}}}}$.

Find the sum of integers $a,b,c,d,$ and $e$ if $\dfrac{2011}{1990} = a+\dfrac{1}{b+\dfrac{1}{c+\dfrac{1}{d+\dfrac{1}{e}}}}$. I could simply the big fraction on the RHS, but I don't see how that would help. Also, there are infinitely many solutions to this equation so how should I find the intege...

" I(X;Y|Z) is interpreted as `` the reduction in uncertainty of X due to the knowledge of Y when Z is given." Would it make sense when talking in a geographical sense to say : When information flow in the EU (Z) occurs the flow of interaction between the countries (I) increases due to labour f...

Prove the following equation has a unique solution: x^101 + x^99 + 5x + 2 = 0 (Use continuity for existence and Roll Theorem for uniqueness) I have no idea where to start and how to prove.

I'm working on the following problem in the realm of stochastic calculus, and don't really know how to proceed. I have a stochastic process, call it $X(t)$. I have a function defined as: $V = 3H^2 X^2$, where $H$ is some non-random real variable. The question is I have to show that for: $...

I would like to show the convolution of two functions is smooth. $\newcommand{\Rn}{\mathbb{R}^N}$ $\newcommand{\dxi}[2]{\frac{\partial #1}{\partial x_{#2 } } } $ $\newcommand{\conv}[2]{#1 \star #2}$ $\newcommand{\ball}{B(0, {1 \over n})}$ $\phi_n \in C^\infty_c( \Rn )$ which is a mollifier. $f\...

I am aware of the "definition" of the total differential as follows: $$\mathrm{d}f = \frac{\partial f}{\partial x} \mathrm{d}x + \frac{\partial f}{\partial y} \mathrm{d} y.$$ Now, assume we wished to show that: $$\frac{\mathrm{d}y}{\mathrm{d}x} = - \frac{\partial f / \partial x}{\partial f / \...

Q: Assume that $S$={(x,y): x-y=0}. Find the projection formula $P_S$. I'm a bit stuck on where to even begin? Any help would be great!

'To select optimal floor.' is the quiz. Suppose you have 10th floor flat, and you have only the elevator that can asecending not go down. You have girls in every floor (1 per each floors). Which way I can select the floor that optimal in probability? I simply can do the answer because prf say...

I was asked to show, given an isometry U from a finite dimensional inner product space V to itself that $$ A_n(x)=\frac{1}{n}\sum_{0}^{n}U^{n}(x) \to 0 \text{ as } n\to \infty \text{ for} x\in Im(I-U) $$ so letting $ y-U(y)$ $$ A_n(x)=\frac{1}{n}\sum_{0}^{n}U^{n}(y-U(y))=\frac{1}{n}\sum_{0}^{n}U...

I am stuck with the following exercise: let $\alpha$ be a complex number which is not integer. Prove $$\frac{\sin\pi(z+\alpha)}{\sin\pi\alpha}=e^{\pi z\cot\pi\alpha}\prod_{n=-\infty}^\infty \left(1-\frac{1}{n-\alpha}\right)e^{\frac{z}{n-\alpha}}$$ I tried to prove this in the following way: us...

I want to demonstrate: Finite sum of sigma-finite measures is a sigma finite measure. I try this: Suppose that are m sigma measures in a space ($\mathcal{X}$,$\mathcal{A}$), denoted by $\mu_k$ with $k \in {1,...,m} $. Then exist $A_n^k$ for each k,and $\mathcal{X}=\bigcup\limits_{n \in \mathbb{...

I'm computing the eigenvectors of a real non-symmetric Matrix. I know that complex eigenvectors would come in conjugate. However, does anybody know a algorithm to rotate such complex eigenvectors that i) its real and imaginary parts are orthogonal and ii) real part has norm 1 while imaginary par...

When except for gcd$(0,0)$ does a greatest common divisor equal $0$? gcd$(x,0)$, for example, = $x$, not $0$.

Given two tuples $A = \langle a_1, \dots, a_n \rangle$ and $B = \langle b_1, \dots, b_m \rangle$, let $A\oplus B \stackrel{def}{=} \langle (a_1+b_1), \dots, (a_1+b_m), (a_2+b_1), \dots, (a_2+b_m), \dots, (a_n+b_m) \rangle$, where $+$ is just the addition (e.g. over $N$). Is there a known name for...

I recently asked this question: Roll a fair 6 sided die twice, What is the probability that one or both rolls are 6? And that makes sense, however, the reason I was confused in the first place is because we went over the following question during section: Suppose the six-sided die you used in t...

I've been struggling with the following problem. I suspect it's actually not so hard, but I'm missing something relatively obvious about the proof. The attached image will help. Suppose $K$ and $L$ are two curves inside the solid torus $\mathbb S^1 \times D$, which are each on one side of the ho...

As far as I know if you have p(x) = x ^ a + x ^ b + x ^ c + .... + x ^ d; and then you want to know p(x + Z). p(x + Z) = x ^ a + x ^ b + x ^ c + .... + x ^ d + Z; IN other words p(x + Z) = p(x) + Z. how to prove it?

Show that the function $$f(x) = \begin{cases} \frac{x^2y^4}{x^4+y^8} ,& \text{if } (x,y)≠ (0,0) \\ 0, &\text{if } (x,y)= (0,0)\end{cases}$$ is Gateaux differentiable at $(0,0)$ but not continuous at $(0,0)$. Not too sure how to start this. I know there is the Gateaux differentiability forumula...

When Amy and John go out for dinner it is either just the two of them, or the two of them together with one or both of their two closest friends. This month they have gone out with each of these friends a total of 7 times, and with both of them together 3 times. If, over the month, Amy and John h...

I came up with this problem: The manufacturer of a company plans to produce and sell 8000 units per year. Each year, 10% of the units become inoperative. So basically, I need a definition for this series, recursive or explicit it does not matter. I know that A1=8000 + .9(8000), A2=8000 + .9(A1)....

G(x) := integral(f(t)) dt from x to x^2 Calculate G'(x). I've made some progress by integrating by parts with f(t) = 1(f(t)) but I'm stuck now and don't know where to go.

It $|z+1|\le 1 \text{and} |z^2+1|\le 1$, then we have $$ |z|\le 1.$$ I wrote $z=x+iy, x,y\in \mathbb{R}$ and the inequalities from hypothesis become \begin{equation} (x+1)^2+y^2\le 1 \text{ and } (x^2-y^2+1)^2+4x^2y^2\le 1 \end{equation}...and I don't see how to deduce from here $$x^2+y^2\le 1.$...

Show that if a ∈ Q is positive and if 0 < x < y then x^a < y^a. I was told to use the difference theorem for this question, but the difference theorem is only for natural numbers.

I'm having trouble understanding this example. Why is $\mathbb{Z}$G considered a ring? How would this be shown? Any advice would be helpful.

Is the following proof correct? Let’s say we find integers $x$ and $y$ such that $x^2 ≡ y^2($mod $n)$ and $n$ has at least 2 distinct factors not equal to 0 or n. I intend to show that there is at least $50$% chance that gcd$(n, x - y)$ is a non-trivial factor of $n$. (Assume already proven tha...

I'm reading this proof (page 7), and get confused by the ending part , quoted here: Take $c$ a constant less than $\sqrt{2}$. Then Since $var(x) \leq n^{3-2c}+n^{2-c^{2}}, E(x) \leq n^{2-c^{2}}$, and $E^{2}(x) \leq n^{4-2c^{2}}$, it follows that $var(x) \leq E^{2}(x)$. What's the reason ...

In probability, an experiment consisting of throwing a die until an uneven number occurs. How do I solve this ? Please Help T^T

The series whose nth term is $\frac{n}{(n+1)(n+2)}$, I have to check the convergence or divergence of it. So $$a_n=\frac{n}{(n+1)(n+2)}=\frac{2}{n+2}-\frac{1}{n+1}$$ Hence as $n\to \infty$, $a_n\to 0$ so the series is convergent. Am I correct?

Prove mathematical induction $\sum_{i=0}^{k-1} 2^i = 2^n-1$

It is well known that open interval $(0,1)$ is not compact on real line. But I somehow remembered that if we take the open interval $(0,1)$ itself as a metric space, then it is a compact metric space. Is this true? Sorry if this question is naive...

In graph G, there is a Hamilton Path from vertex v to vertex m, with v and m not adjacent. Suppose that G is a simple graph with n vertices, n >= 3, and deg(x) + deg(y) >= n whenever x and y are nonadjacent vertices in G. To prove or disprove that G has a Hamilton Circuit. My teacher gave us an ...

I have to show whether or not the set $[0,1] ∪ [2,3]$ has an interior of (0,1). I'm still really confused how to determine the interior of a set. Can someone please explain?

Can anybody verify that the below equation equals $0$? $\prod\limits_{k=2}^{10} (\sum\limits_{i=1}^{k-1}(2(i-1)))$ Here is my work, I believe it's correct:

Let $H$ be a subgroup of $G$. Let $a$, $b \in G$. Prove that if $b \in aH$, then $aH = bH$.

I have to integrate the following : $$\int_0^\infty x^{s-1}e^{(-ax^p-bx^{-q})}$$ Please help me I am not able to solve this .. However there is this identity available in a book $$\int_0^\infty x^{s-1}e^{(-ax^p-bx^{-p})} = 2 p^{-1}(\frac{b}{a})^{(\frac{h}{2s})}K_{(\frac{s}{h})}(2a^{(\frac{1}{...

How does one prove that $\alpha (1+\alpha/2)^{-1} > \log(1+\alpha) $ for $\alpha > 0$?

Let $T\in \mathcal{L}(\mathcal{P_3}(\mathbb{C})$ be the operator $$T:f(x)\to f(x-1)+x^3f'''(x)/3$$ Find the Jordan normal form and a Jordan basis for $T$.

I'm having trouble with the laplace transform: $\mathcal{L} \lbrace \sqrt{\frac{t}{\pi}}\cos(2 t) \rbrace$ The problem gives me the transform identity $\mathcal{L} \lbrace \frac{\cos(2 t)}{\sqrt{\pi t}} \rbrace = \frac{e^{-2/s}}{\sqrt{s}}$ but i'm not sure/confused as to why that would help me

Use singular perturbation techniques to find the leading order uniform ap- proximation to the solution to the boundary value problem $$\epsilon y'' + \frac{2 \epsilon}{t} y'-y=0$$ $0<t<1$ and $y(0)=0 , y'(1)=1$ This has a boundary layer near $t=1$ I am having trouble figuring out how to co...

For points x , y in $\mathbb{R}^k$, let $$d_1(\boldsymbol{x},\boldsymbol{y}) = max\left \{ \left | x_j - y_j \right | : j = 1,2,...,k\right \}$$ $$d_2(\boldsymbol{x},\boldsymbol{y}) = \sum_{j = 1}^{k}\left | x_j - y_j \right |$$ Let $\vec{x}$ , $\vec{y}$ $\in$ $\mathbb{R}^n$. prove that for a...

So i decided to do this using normal induction. P(12) true since 12=4*3 $P(k)=3a+7b$ $P(k+1)=3(a-2)+7(b+1)$ So i think it is proven but i cannot see why we have to assume n is larger than 11. I mean i know it is impossible for 11 but lets say 9. P(9) true since 9=3*3 $P(k)=3a+7b$ $P(k+1)=3(a-2)+7...

I don't know how to express the problem to the form $f(x|\theta)=h(x)c(\theta)(\sum_{i=1}^{k}\omega_i(\theta)t_i(x))$ Let $X$ have pdf $f(x)=\frac{1}{\beta}e^{-(x-\alpha)/\beta}$, $x>\alpha$ Determine whether $f(x)$ is an exponential a. if both $\alpha$ and $\beta$ are unknown. b. if only $\b...

Let $M$ be a $3 \times 3$ matrix with strictly positive real entries. Let $S \ = \ \{ \left. \ \left(\begin{array}{c} x \\ y \\ z\end{array}\right) \in \mathbb{R}^3 \ \right| \ x \geq 0, \ y \geq 0, \ z\geq 0, \ x^2+y^2+z^2=1 \}.$ By considering the map $f:S \to S$ defined by $\mathbf{x} \...

I'm trying to understand what the quotation ring is. I know that $\mathbb{Z}/4\mathbb{Z} = \mathbb{Z}_4$, but I can't get the same result by myself. Having used the definition of the quotation ring that $R/I = \left \{x + I | x \in R \right \}$ I've got: $$ 4\mathbb{Z} = \left \{4x | x \in \math...

Suppose $A \subset \mathbb{R}$ is closed. How can we show that there exists a continuous real-valued function $f$ such that $ker(f)=A$?

I have a factorial series as shown below: \begin{equation} (2n+1)!~\text{for all $n \geq 0$} \end{equation} And I would like to know if the recursive definition that I wrote is accurate: \begin{equation} Factorial(n)=\begin{cases} n, & \text{if $n<0$}.\\ (2n+1)\cdot Factorial((2n+1)-...

Assume we have $p + 1$ vectors $a_1, a_2, \ldots ,a_p, b \in \left\{0,1\right\}^n$. How to find $\alpha_1, \alpha_2, \ldots ,\alpha_p\in\left\{0,1\right\}$ that vector $y$: $$y =\sum_{i=1}^p \alpha_i a_i + b$$ has the minimum numbers of $1$ ? All operations are on based $2$.

I want to create my own function for labeling and regional properties of binary in MATLAB. Please help in coding mean radial distance of shapes in binary images.

Let f be a bounded linear functional on a subspace Z of a separable normed space X. Theb there exist a bounded linear functional l on X which is an extension of f to X and has same norm.

Prove that if c is algebraic over $F$, so are $c + k$ and $kc$ for $k\in F$

I encountered a very interesting implicit function: $ z \exp \left[ (x-0.5-e^{z-y})^2+y^2-0.2z+3 \right] = \sin \left[ (xz-0.5)^2+2xy^2-0.1z \right]$ I wonder if there is any ways to find a parametric representation for it.

A flight starts from Bangalore at 7 am local time and reaches Dubai at 12:30 am local time. Another flight starts from Dubai at 3 pm local time and reaches Bangalore at 8:40 pm local time. Both flights travel at a constant speed of 800 km/hr. Find the distance between Bangalore and Dubai.

So I saw this question. Office wants to create a new car licensing system in which each license plate will have seven characters: three letters (from the 26 letter alphabet) followed by four digits What is the probability that a license plate under the proposed system will contain no duplicate ...

I have a problem set due soon and I have been trying to fix my questions but I am stuck on one question in particular: "Are there any points on surface $x^2 - y^2 - z^2 = 1$ where the tangent plane is parallel to the plane $z=x+y$? [Hint: Remember that the normal vector to the tangent plane to a...

I have a pdf for continuous random variables X and Y such that $f_{xy}(x,y) = 2(x+y)$ for all $0<x<y<1$ and 0 elsewhere Does this mean for $0<x<1$ and $ 0<y<1$ or $0<x<y$ and $x<y<1$? I have to find $E[X^2Y]$

enter image description here I'm trying to solve the problem of the above picture.Can anyone help.

I'm working on Atiyah-Macdonaldand in problem 26's Introduction to Commutative Algebra, and in chapter 1, problem 26, the goal is to show that for a compact Hausdorff space X, the space is homeomorphic to the subspace of Spec(C(X)), the zarinski topology on the continuous funtions of X, that ...

I am teaching linear algebra this semester, and I would really like to recommend my students some cool youtube videos visualizing some simple stuff like the span of a set of vectors, linear dependence, subspaces (and their intersection and sum) and how does a basis give you a "coordinate system"....

I want to calculate the integral $$I = \int_0^{2 \pi} \cos^2 \theta\ \sin^4 \theta\ d \theta$$ by converting it into a complex integral around the unit circle. I use the identities $$\cos \theta = {1 \over 2} (e^{i \theta} + e^{-i \theta}),$$ $$\sin \theta = {1 \over 2 i} (e^{i \theta} - e^{-i ...

Is there already a established rule on how to get the function (say cubic function) given a table of x-y values?

I just read the following theorem: If a sequence of real numbers is increasing and bounded above, then its supremum is the limit. How is it possible to have an increasing but bounded above sequence? Can you give me an example please?

Sorry for this simple question, I'm a first year student, it's really basic but I don't know how to answer The question is: A random experiment is conducted which has sample space Ω {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20} . Assume that all elementary events are equally likely to occu...

U1 and U2 are IID U(0,1). I want to know why max{U1,U2} ~ RT(0,1) The cdf of RT(0,1) is as follows. enter image description here

This is from a real analysis module. I'm supposed to determine whether this integral is positive or negative without evaluating it. I have no idea how. Obviously x*cos(x) is positive over (0,pi/2) and negative over (pi/2,pi), so whichever partial integral is larger obviously determines the sign o...

How can I find the poles and the order of $1/sin(z)$?

What would be the Fourier transform of $\nabla^2 f(x)^2 $ ? (Where $y$ is some function of a space variable $x$)

I want to know how to use composition to generate random variates whose cdf and pdf are as follows. CDF and PDF of the aimed distribution

Why is that a contradiction? I thought we assumed $a \neq 0$, so what are they talking about?

I am calculating the Shannon entropy of $\left|\Psi_{+}\left(x_{+}\right)\right|^{2}=\frac{1}{\sigma^{3}_{+}\sqrt{2\pi}}x^{2}_{+}\exp\left\{-\frac{x^{2}_{+}}{2\sigma^{2}_{+}}\right\}$, which is given by $$ H=-\int\left|\Psi_{+}\left(x_{+}\right)\right|^{2}\ln\left|\Psi_{+}\left(x_{+}\right)\right...

let $T=(V,E)$ be a tree with $n$ vertices , which his degree sequence is $d_1 \le d_2 \le ... \le d_n$ if and only if $d_1\gt 0$ and $d_1+d_2+...+d_n=2(n-1)$ $$$$ i managed to prove it one direction with indaction... but for days im not able to prove its a tree...(in the other direction). Please ...

Establish the following test for primes. If $n$ is odd, greater than $5$, and there exist relatively prime integers $a$ and $b$ such that $a — b = n$ and $a + b = p_1\cdot p_2\cdot... p_k$ (where $p_1, p_2 , . . . , p_k$ are the odd primes less than $\sqrt n$ ), then $n$ is prime.

How do I go about a proof to show that both the set of rational numbers and the set of irrational numbers are dense in R(real number).

Let $T:V\rightarrow V$ be a linear, normal transformation (meaning $TT^*=T^*T$) where it is known that $T^{-1} = -T$. Can it be proved that $T$ is unitary (e.g. $TT^*=T^*T=I$)?

David's truck can load 28tons and 70m³ of cargo and it costs $19600 to move from point A to point B. Mike wants to send his washing machine from point A to point B by David's truck. Washing machine's weight is 300kg, and it's volume is 1m³. How much should David charge Mike? Cost for every kg= ...

Find a formula for the least-squares solution of Ax = b when the columns of A are orthonormal.

How to prove that $A_n$ is normal in $S_n$? Note that $A_n$ is a group of even permutations on a set of length $n$. $S_n$ is the group of all permutations on $n$ symbols.

How i evaluate this $\iiint_{D}\sqrt{x^{2}+z^{2}}dxdydz$ where domain D is restricted by $y=x^{2}+z^{2}$ and plane $y=4$ ? I've already known that $y=x^{2}+z^{2}$ is paraboloid and but i'm not sure about the domain. Help me please!

I recently encountered an equality with the condition that $a,b,c$ are positive reals and $\sum\frac{1}{1+a}\le 1.$ The solution says that this condition is equivalent to $a+b+c+2\le abc.$ But I have a hard time figuring out why. Can anyone show why the two conditions are the same?

How can i solve something like that? $$\int\frac{x+1}{(x^2+7x-3)^3}dx$$ How should I start? Should I try rewrite it in partial fractions?

I was asked this question in my statistic class. I thought the way to do this was (1/6)^3 x (5/6)^7, because that is getting six 3 times and not getting it 7 times. However, that's wrong, I figured that it's because that (1/6)^3 would be getting 6 three times in a row. Could you explain how to d...

Let $[x]$ denote the rounded (nearest integer) value of $x$. How can I find a value of $n$ which yields the minimal difference between $\dfrac{2^n}{88}$ and $\Big[\dfrac{2^n}{88}\Big]$? Empirically, I have noticed that the minimal value of $\dfrac{1}{11}$ can be achieved with any $n=5k+3$: $\...

I'm not looking for an answer to the actual problem, just what this notation could plausibly mean. Let $G$=$R$ modular addition $R$, and fix positive real numbers $a$ and $b$. Let $H$=<<$a$,$b$>>. Give a geometric interpretation of the cosets of $H$. What is that <>? I've never seen that notati...

How to use composition algorithm to generate random variates whose cdf and pdf is as follows. CDF and PDF

Please help me. Suppose we have the group $\frac{\mathbb{Z}_m\times \mathbb{Z}_n}{\langle (a,b)\rangle}$. We need to justify that (i) There exist $c, d$ such that $\langle (a,b)\rangle$ is isomorphic to the group $\mathbb{Z}_c\times \mathbb{Z}_d$ with $c|m, d|n$. (ii) $\frac{\mathbb{Z}_m\times ...

I am currently using Rouche's Theorem from complex analysis but am working on an upper bound and want to show $$\sum_{k=0}^{n-1}\frac{(2n)^k}{k!}< \frac{(2n)^n}{n!}$$ Any suggestions are welcome. Thanks,

14 balls total, 4 red, 4 blue, 3 green, 3 yellow. What is the probability of drawing at least 2 yellows? My first take on this is to calculate 1 - the chance of not drawing a yellow in 4 pulls. 1 - (11/14 * 10/13 * 9/12 * 8/11) However, I realized that this doesn't cover the case in that you c...

I have a question regarding the proof of Erdos-Kac central limit theorem in on p.117 of Probability: theory and example. 4th ed. Durrett used a probability model $$ S_n = \sum_{p\leq \alpha_n} X_p, $$ where he claims that $$ \frac{S_n - b_n}{a_n}, $$ where $ a_n^2 $ is the variance of the series ...

Suppose t₁,t₂,...,t_{k},... is a sequence from a compact metric space T. Consider a sequence of constrained optimization problems V(t₁) = max_{x∈X}{f(x):g(x,t₁)≥0} V(t₁,t₂) = max_{x∈X}{f(x):{g(x,t_{i})≥0}_{1≤i≤2}} .... V(t₁,t₂,...,t_{k}) = max_{x∈X}{f(x):{g(x,t_{i})≥0}_{1≤i≤k}}, where bot...

I have the following question I have minimized the DFA as the following since the states can only be partitioned to [S0][S1 S2]

What is the equation of a curve with jelly bean shape? I have found a quartic equation for bean shaped curves, but nothing for jelly beans.

As you see, I did a finite semigroup and then try to find its possible ideals in a very basic way (with the hand of a friend): > f:=FreeSemigroup("a","b");; a:=f.1;; b:=f.2;; s:=f/[[a^3,a],[b^2,a^2],[b*a,a*b^3]];; e:=Elements(s);; w1:=List([1..Size(e)],i->Elements(Semigroup...

Never was much of a math student but I am brushing up on my arithmetic and algebra for college. I am using sample questions from Accuplacer and then using video lectures and practice on Khan Academy. It is going well but sometimes the examples in the lecture are very trivial and I am currently ...

I have $12$ players which one player can choose an arbitrary number in the set $\mathcal{A}=\{1,2,3 \}$. I get stuck in how to define and display all possible combinations by using Matlab. Do you have any idea for this?

On page 432 (pdf-page: 455) of Kato's book perturbation theory of linear operators, I do not understand why in Theorem 1.15 $$H_n = \int dE_n(\lambda)$$ instead of the ususal thing $$H_n=\int \lambda dE_n(\lambda).$$ Is this a misprint?

Consider the following steady state problem $$\Delta T = 0,\,\,\,\, (x,y) \in \Omega, \space \space 0 \leq x \leq 4 ,\space \space \space\space 0 \leq y \leq 2 $$ $$ T(0,y) = 300, \space \space T(4,y) = 600$$ $$ \frac{\partial T}{\partial y}(x,0) = 0, \space \space \frac{\partial T}{\partial y...

We say that a matrix $P $ = $(P_{ij} : i, j \in I)$ where $I$ is a countable set, is stochastic if every row $(P_{ij} : j \in I)$ is a distribution. What do we mean by $distribution$ here?

1) I didn't understand really what is a maximal atlas. Is it as set of compatible chart maximal in the sens that adding one more chart will yield the atlas not compatible ? 2) Let two atlas $\mathcal A$ and $\mathcal A'$. So if they are compatible, they are both in a maximal atlas $\hat{\mathcal...

Let V=[2015]. Show that there is a graph G=(V, E) such that for any A, B $\subseteq$ V where A $\cap$ B= $\emptyset$ and |A $\cup$ B|=5, there exists $x \in V-(A\cup B)$ such that x is adjacent to all vertices in A but none vertices in B.

If we have $p\implies q$, then the only case the logical value of this implication is false is when $p$ is true, but $q$ false. So suppose I have a broken soda machine - it will never give me any can of coke, no matter if I insert some coins in it or not. Let $p$ be 'I insert a coin', and $q$ -...

I'm studying the RSK correspondence,but I can't proof the following theorem.Can anyone help me? enter image description here

Integrate $f(x)$ from 0 to 1 where $f(x) = \frac{x^3-1}{lnx}$ I received this problem and a variety of others in an advanced mathematics exam. I tried a classical trigonometric substituition approach with $x=sec^\frac{2}{3}t$ but it gets really long and still gives no answer. I tried multiplying...

I recently had a course on functional analysis. I was thinking of studying the applications of functional analysis. I came to know it had some applications on calculus of variations. Can anyone give a brief on what are the applications of functional analysis? Also, please suggest some good book...

Consider the following PDE: $$ u_t=u_{xx}+f(u)-w,~~~~~w_t=\varepsilon (u-\gamma w),~~~~~~~~~(1) $$ where $f(u)=u(u-a)(1-u), 0<a<\frac{1}{2}, \varepsilon,\gamma >0, \varepsilon\ll 1,\gamma\ll 1$. A travelling wave for (1) is a solution that is a function of the single variable $\xi=x-ct$, i.e., $...

I guess the following is true but i do not know how to prove it: Let $(G_i, i\in \mathbb{N}^+)$ be family of connected $d$-reuglar graph (same $d$ for all $i$), such that $v(G_i)\rightarrow \infty$ as $i \rightarrow \infty$. Then diameter$(G_i)\rightarrow \infty$ as $i\rightarrow \in...

Are there any good books, that shows some advanced math models in genetics particularly? The problem is i don't see any good math model in genetics interactions exept combinatorics. I'd like to know what else math structures and rules are there in genetics. I'd like create strong math backbone f...

We are given a distribution with mean u, and variance sigma square. An unknown random variable X exists whose values take on belong to a range of numbers. This range of numbers from which X takes values depends on an integer k. We define P{X(w):w€(some range on the borel line)} = p For what what ...

I have difficulties understanding the proof of a contraction mapping in context of ordinary differential equations. For example: We have a banach space $(Y,||\cdot ||)$, where $||y||=\sup\{\frac{|y(x)|}{x}:0<x\leq 1\}<\infty$ Also, we have the map: $T:Y\rightarrow Y$, where $Y$ is the set of ...

Hello. How can I show that $$\sum_{k = 0}^{\infty} \frac{ \log(t)^k}{k!}h^k ≤1 $$ is right?

The problem I'm fighting with right now originally comes from signal noise detection: Given a prob. space $(\Omega, \mathcal{A}, \mu)$ and random variables $X, Y : \Omega \to \mathbb{R}$ (where $X$ encaptures a 'true' signal and $Y$ contains its noisy variant $Y = r(X) + \text{error}$ that was ob...

can you help me on this? An urn contains 7 balls, 4 of which are black and the rest white. 1) Starting with all 7 balls in the urn, balls are drawn randomly with replacement. Find the probability that among the first 5 balls which are drawn a) 2 of them are white. For 1 a) I used Binomial...

Lebesgue-Radon-Nikodym Theorem shows that: If $\mathbb{M}$ is a $\sigma$-algbra on the set X. $\mu,\lambda$ are a $\sigma$-finite positive measure and $\sigma$-finite signed measure on $\mathbb{M}$ respectively. then $\lambda$ has a decomposition. I wander why we do not consider the case when $...

Is there any totally ordered field with cardinality larger than the continuum? If such field exist, please give an example (an simple one if possible).

simplex algorithm Attaching coves $10$ reinforcement wire length $50$ m and $30$ m. The armature must have a size of $8$ m and $6$ m and $8$-m two times more than $6$-m cove cut so as to obtain the largest number of sets of armatures I can not understand how to do it

proof or disprove : a) A ∩ (B \ C) = (A ∩ B) \ (A ∩ C) b) A ∩ (B \ C) = (A ∩ B) \ C c) A \ (B ∩ C) = (A \ B) ∪ (A \ C) d) (A \ B) ∪ (B \ C) ∪ (C \ A) = (A ∪ B ∪ C) \ (A ∩ B ∩ C) can someone help this please?

For every k Part of the natural numbers show that we have n So that: 2^n+3^n-1,2^n+3^n-2,...,2^n+3^n-k Not all prime numbers

I know that Legendre Symbol is unique non-trivial group homomorphism from $U(Z_n)$ to ${-1,1}$. Is the same true for Jacobi symbol? I can prove it for the case when $n$ is power of odd prime. In this case the kernel of non-trivial homomorphism must be subgroup of size $\frac{\phi(n)}{2}$ and $U...

Show that the number of $\lambda$, eigenvalue of $$\int_a^b \int_a^b K(s,t)^2 ds dt $$ is finite.

The Dirichlet Kernel on the real numbers is: $\mathcal{D}_\mathbb{R}(x) = \frac{sin(2 \pi R x)}{\pi x}$. Let $D^*_N(x) = \sum_{|n| \leq N - 1} e^{2 \pi i nx} + \frac{1}{2}(e^{-2 \pi iNx} + e^{2 \pi iNx})$ denote the Dirichlet kernel for functions of period 1. Show that $\sum_{n = -\infty}^{\inf...

Let $a_1,a_2,a_3$ be a non constant arithmetic Progression of integers with common difference $p$ and $b_1,b_2,b_3$ be a geometric Progression with common ratio $r$. Consider $3$ polynomials $P_1(x)=x^2+a_1x+b_1, P_2(x)=x^2+a_2x+b_2,P_3(x)=x^2+a_3x+b_3$. Suppose there exist integers $m$ and $n$ s...

Why the winding number in the image is different for these different points? Should not it be +3 for all? I really don't understand this... http://s8.postimg.org/sda6gvdh1/windingnumberclosedarch.png

I have a transportation problem with complication. Essense of complication -- shipping is carried out with a trucks. Each truck has limited capacity. Each truck has the same capacity. Let's call capacity of eack truck -- K. How to reduce this problem to the linear programming problem?

Please refer this link for some background material http://www.docdroid.net/161p6/curve.pdf.html So i propose a statement to a online tutor, the answer at the below link is the proof of the statement? I couldn't get it. http://postimg.org/image/j26erz5k3/

I encountered the following limit while doing calculation $$\lim_{x\to\infty}\frac{\text{Ei(x)}}{e^x}=0$$ which is equivalent to $$\lim_{x \to \infty }e^{-x}\sum_{n=1}^{\infty}\frac{x^n}{n·n!}=0$$ and $$\lim_{x\to\infty}\int_0^\infty \frac{e^{-t}}{t-x}dt=0$$ where the integral is undestood as the...

If a set A is open, and you take the Minkowski sum with any other set B (open or closed), will A+B then be open? In my head it makes sense if it is open but I can't prove it on paper.

A rope is wrapped $M$ whole turns round a cylindrical post, the two ends of the rope going in opposite directions. The coefficient of friction between rope and post is $0.25$. It is desired that by pulling with a force of $1N$ on one end of the rope, I can prevent the rope from moving away from m...

I was reviewing this question and got motivated to solve this general problem: Find all functions $f:\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$, $$f(x+y)-f(x)-f(y)=\alpha(f(xy)-f(x)f(y))\tag0$$ where $\alpha$ is a nonzero real constant. I found out that it can be ...

The symmetric difference can be expressed as the union of the two sets, minus their intersection- how to prove that ?

I encountered a problem: Every finite integral domain is isomorphic to $ \mathbb{ Z }_{p} $. I know that finite integral domain is isomorphic to a field, but I have no idea on how to construct a homomorphism to $\mathbb{Z}_{p}$ (or maybe it is wrong, but I haven come up with a conterexample).

Good afternoon, I am currently having issues solving a problem: Let A and B have position vectors a = 2i – j + 3k and b = 5i – 2j + k respectively. The force F = 3i – 5j – k passes through the point A. Show that the vector moment of force, F about the point B is equal to 3(3i + j + 4k...

If two sets (A,B) in R^n are closed. Will A+B then be closed or open? In my head it makes sense if it is closed but I can't prove it on paper.

if $c>1$ then $\sup\limits_{0 < r < 1}\left(\frac{1}{2\pi}\int_0^{2\pi}|\dfrac{1}{1-re^{it}}‎ ‎\left( ‎\dfrac{1}{re^{it}}‎ \log \dfrac{1}{1-re^{it}} ‎\right)‎^{-c}|~ dt \right) ~<~ \infty$‎

I was wondering about how to solve a simple linear Fokker-Planck equation using space-time laplace transform on space interval $(0 \ \infty)$. $\frac{\partial f(x,t)}{\partial t}= k_1 \frac{\partial f(x,t)}{\partial x} + k_2 \frac{\partial^2 f(x,t)}{\partial x^2}$. The usual method is to do lap...

I am in the process of learning Category Theory with the purpose of being able to create a game that will help explain it to others in a simple way. I have read many texts and articles about it. While I have learnt some of the definitions, I have found it hard to be able to create concrete exampl...

Differentiate the following using the first principle f(x)=x^3/2 f(x)=1/square root of x i have tried the questions but i am stuck in the second step

I’m designing a 3d game and am having trouble computing where the mouse pointer is actually pointing on the flat map when taking into account perspective which skews the view. Let’s assume that the flat plane is part of the map, and the tilted plane is the viewport. Player’s camera is rotated ...

We define a new function: Psum(N)=Sum[1<=a<=N, 1<=b<=N, phi(a*b)] For example, PSum(2)=phi(1*1)+phi(1*2)+phi(2*1)+phi(2*2)=5 You are given that PSum(10)=1271, PSum(100)=10813139, and PSum(1000)=107260466665 Find PSum(10000000) Give the rightmost 9 digits as your answer I can find the small val...

Consider the Delta Method as stated in van der Vaart Theorem 3.1 at page 26 (you can find the page here https://books.google.co.uk/books?id=UEuQEM5RjWgC&pg=PA32&lpg=PA32&dq=van+der+vaart+theorem+3.1+because+the+sequence+converges+in+distribution&source=bl&ots=mnRJLD8XLC&sig=inIMmSPvWDfrPc6r4U7dnu...

This problem has been extreme confusing for me since 2 weeks ago. I've tried many times with Maxima but still i got no results. Following is the equation system: equations with related functions. To be computed are b1[n], b2[n], b3[n], a4[k] and a5[k]; a3[k] is already obtained before. R1,R2,R...

Prove that a finite group G is abelian if G has an automorphism $\sigma$ such that $\sigma(g) = g \iff g = 1$ and $\sigma^{2}(x)=x$ for all $x \in G$. Show that every element of G can be written in the form $x^{-1}\sigma(x)$ and apply $\sigma$ to such an expression.

We have approximation of $f(x)=\sqrt(x)$ such $P(x)$ in interval $[a,b]$ . How we can approximate the accuracy of my approximation in this interval?

I would like to calculate the following: There are 3 points in space of which the coordinates are known: C(a,b,c) M(d,e,f) A(g,h,i) Each point is a starting point of a line with a certain length. These lengths are known: |CB|=l1 |MB|=l2 |AB|=l3 There exist 2 possible points (B1 and B2) ...

I need to write a regular expertion for the language of all the binary words that not contains continuum of more then $3$ zeros, for example $0011110100\in L,\,\,\,\,\,\,\,\,\,11000001100\notin L $ My try: $[(0+1)^*(00+1^*)^*(000+1^*)]^*$ My attempt is correct?

Prove that $N_G(H)=\{g \in G| gHg^{-1}=H\}$ is the largest subgroup of $G$ such that $H \unlhd N_G(H)$. I have an idea of the proof that, if we assume $S \leq G$ with $H \unlhd S$ then $$\forall s \in S, \ sHs^{-1}=H$$ We know that for $S$ to be the largest subgroup of $G$ with this propert...

Can someone help me to prove this? is the base case 1 here?

Let n>=2 be an integer and consider two fixed integers a and b with 1<=a<=b<=n.  Use the Product Rule to determine the number of permutations of {1, 2,...,n} in which a is to the left of b.  Consider a uniformly random permutation of the set {1,2,....,n}, and denote the event A = "in this pe...

Assume that f is a-strongly convex and g is b-strongly convex, then I want to know that f+g is ?-strongly convex?

I'm stuck at the ending part of a math exercise on congruences. I must solve the following system of congruences $S$: $x \equiv 2\ (3)$ $x \equiv 3\ (5)$ I was first asked to give the remainders of the division of $3y +2$ by 5, with knowing the remainders of the division of $y$ by 5. Here's ...

A,B squares exp(t(A+B))=exp(tA)+$\int_0^t exp((t-s)A)*B*exp(s(A+B))*ds$ I found it from problem sets. It seems to difine a new kind of matrix operation. I can check its truth when A,B are scalars (n=1), but fail to prove it for general cases. The matrices do not commute and I cannot think of ...

What is exactly semaphore in java?Can someone explain it in simple words by citing a short example.

for arbitrary dimension, what are the convex polytopes such that all vertices share a facet of some dimension, which is not the top facet (the entire polytope), with all other vertices? One example is simplexes of any dimension, as all vertices share an edge with all other vertices. Are there oth...

Is Triangle AOB congruence to Triangle BOC?

For any i where 2 <= i <= n, use the product rule to determine how many permutations of the set {1,2,…,n} are there such that i is the leftmost value in an increasing subsequence.

"In a class, 40% of students want to study physics, while less than 33 1/3% of them want to study Chinese History. If the number of students who want to study physics is 3 more than that of students who want to study Chinese History, at most how many students are there in the class?" Don't know...

$$f\left ( z \right )=\frac{e^{az}}{1+e^{z}} ,\left ( a\in\left ( 0,1 \right ) \right )$$ The point $z=i\pi$ is one of the nonremovable singularities of this function. In order to expand it about that point I introduced the change of variables $z=\omega +i\pi$ and expanded the function $f\left(\o...

i tried to connect two ideals: I = {x^3 + 1} J = {x^3 - 1} so I + J = ? i tried like this: I + J = = = ... ? tnx

The old Greek did not consider 1 a number. Nevertheless Euclid called 6 = 1+2+3 a perfect number. How could he use 1 which was not a number?

enter image description here I am very unsure how to do this, As the sign changes I would seperate them in two series, one positive and one negative - 1/((2n-1)(2)^(2n-1)) and -1/((2n)(2)^(2n)) However I can't evaluate them right.

to expand $1/p$, I tried first letting $1/p = a+b*p+c*p^2+d*p^3+...$ and it is $1=a*p+b*p^2+...$ but I guess there's no way to make the equality hold. it's somewhat similar to dividing by 0. is it possible? I'm new to p-adic theory, everything looks just confusing.

I have no idea where to start or what kind of concept I should use for the following problem: Suppose $f \in L^1(\mathbb{R})$. Show for almost all $x \in \mathbb{R} $ we have $$\lim_{h\to 0} \frac{1}{h} \int_0^h |f(x+h)-f(x)| dt = 0$$ Could someone help me with this problem? Thanks!

4. Consider the following simplified version of a game of poker. There are two players and three cards in a deck: Jack, Queen and King of spades. Player 1 always gets a Queen. Player 2 gets a Jack with probability 0.5 or a King with probability 0.5. In the first round, player 2 can either Raise o...

If f ∈ C[a,b] is not a polynomial, then show that for any sequence of polynomials {p_n} that converges to f uniformly, one must have that m_n = degree of p_n → ∞.

Could someone please verify the following proof for this statement: If $E$ is a finite dimensional subspace of a normed space $X$, then $E$ is a closed subspace. Proof: If $E$ is a finite dimensional subspace of normed space $X$, then $E$ is complete. Let $\{x_n\}_{n=1}^\infty$ be a Cau...

John Harsayni showed that a mixed strategy equilibrium of a perfect information game can be thought of as an approximation to an equilibrium (i.e. Bayesian NE) of a game where each player has a slight amount of incomplete information about the exact preferences of the other players. Consider the ...

I am working on a VBA algorithm that will solve simple versions (single stock length, <1000 patterns) of the Cutting Stock problem, and after a lot of research I have managed to write a VBA program that correctly calculates the patterns available. As the Cutting Stock problem is a minimisation p...

Let $f$ be locally integrable and positive on [0,$+\infty$) such that $L \;:= \;lim_{x\rightarrow \infty} \dfrac{-ln f(x)}{ln x}\;$ exists in $\bar{\mathbb{R}}$. Prove that $\int^\infty_0 f$ converges if $L > 1$ and diverges if $L < 1$. The idea seems close to the Root Test for Integrals, but I...

There is a famous low pass filter $[1,2,1]$ in signal processing which can be factored in the sense of a convolution product : $[1,1] * [1,1]$. Today when I was bored, I investigated some complex valued filters and came to the conclusion that for $k = 3$ and $k = 5$ and $z_0 = e^{2\pi i/k}$, i.e....