Use Green’s Theorem to evaluate \int_cF⋅dr, if F(x,y)=<{Sqrtx}+y^3, x^2+{Sqrty}> and C consists of the arc of the curve sinyx= from (0, 0) to (π, 0)and the line segment from (π, 0) to (0, 0). Trying to get the hang of the coding, so forgive if their are multiple edits. There is a photo tagged t...

I know that orthogonal means meeting at right angles, but I'm unsure about how to go about computing it. Also, how do I find grad(f) for these coordinates?

I am not sure how to ask this question because it seems to me that my thoughts on this topic are not clear enough, but I will give it a try. What, really, do I want to know? Well, I would like to know is there any "measure" on how "far" is such a function from being differentiable at some point...

The Problem is: Find the equations of the two straight lines, each of which cuts all four of the lines x = 1, y = 0; y = 1, z = 0; z = 1, x = 0; x = y = 6z I parameterized the four lines as: a(q)=(1,0,q), b(r)=(r,1,0), c(s)=(0,s,1), and d(t)=(6t,6t,t). I've constructed two solution li...

I have three 4x4 matrices in this equation : A=XB And I want to find the matrix X. What actions should I do ?

I've been learning various basics of Sieve Methods in Analytic Number Theory, and I'm wondering what are some uses of these methods in current research? Not famous, unsolved problems, but areas of research currently underway? I understand sieve methods' uses in the context of enumerating primes...

http://imgur.com/a/obzIX The question is that imgur link. Basically, I have no clue how to go about this question at all. Any help would be appreciated. Thank you in advance ):

How can I show an irreducaible singular cubic hypersurface in $P^n$ is rational. Such question always seems very bizarre for me. Totally no idea. So can any one give me some help?

Tori has a wooden block shaped like a rectangular prism the area of the base of the block is 96 square centimeters the volume of the block is 1440 cubic centimeters what is the height in centimeters of Tori's wooden block

Let A be a nonempty set. Prove that there is a bijective function F : {Equivalence relations on A} → {Partitions of A}. I am completely lost on where to proceed with this question.

I "know" that you can use Green's theorem to integrate over a region bounded by a closed loop. But this is only if the vector field has continuous first partial derivatives inside the region. So I thought that Stokes theorem allowed us to avoid the discontinuity in the 2D region by "going over...

Why is $${n \choose k} ≥ 1$$ I've looked at the expansion of the binomial coefficient, but can't see why the nominator is larger or equal to the denominator.

I need help solving the following problem. I exhausted many options however I fell short. Any help is appreciated. Creative Good, a New York consulting firm, claimed that 35% of shoppers fail in their attempt to purchase merchandise on-line because Web sites are too complex. A random sample of 6...

Thank you for assistance, I'm just having issues remembering what this is called? For example, the equation would go like this |x+1| = 4 What is this type of equation called, with the two | | ? Thanks!

I have this exercise that I was doing and came to this question that I don't know how to prove so here is the question: It is question 5a) and 5b) from this image: enter image description here Recall the definition for a ≡n b and that for a ≡n b, we can write a = nq1 + r and b = nq2 + r where ...

I came across a problem: -16/4i. Everytime I put it into a calculator, it comes out as 4i, but when I try to solve it is -4i, because of the negative one in front of the 16.

this is the exact problem i know this question involves the binomial theorem but i am not sure what to do

I need to give the recursive function of 3n^2. I'm pretty sure the base case needs to be 3(0)^2 = 0, but I don't know where to go from there. Any help is appreciated.

Suppose that $f:X \to Y$ is a branched cover of Riemann surfaces and a covering map of degree one outside of the ramification points. Then is $f$ a homeomorphism?

Let $P(n)$ denote "If $n$ even integers are selected, then there must be a pair who share the same remainder under integer division by 10." Let $k$ denote the smallest value for $n$ for which $P(n)$ is true. What is $k$? Prove that $P(k)$ is true and that $P(k - 1)$ is false. I'm not sure how to...

Guide me learning Maths from very basic level,concepts and basic numbers, lines and addition, division, multiples,

Obviously, $\mathbb{C}$, the field of complex numbers is. But what other fields are? Is it possible to construct some degree of analysis on them?(just luke complex analysis on $\mathbb{C}$)

by factoring $p^2−1$, we have $(p+1)(p-1)$. I know that p=2 which gives 3 is the only solution, however how do I prove that p=2 is the only integer which gives a prime?

Let $X$ have pdf $f(x)=\frac{1}{\beta}e^{-(x-\alpha)/\beta}$, $x>\alpha$ How to determine if the function f(x) is a location-scale family

The trigonometric expression 1/2sin(x) is equivalent to: the answer says sin(x/2)cos(x/2) which I do not understand. Do you not factor out the 1/2 so there would be a one half at the front?

Now I am new to matlab, and now the basic syntax, however I am not sure how to write a function to loop for instance the following recurrence equation: $y(n+1)=\exp^(-y(n))$ .. $\text{where } y(1) = \exp^(-y(0))$ and $y(0) = x$

For a natural number $j$, let $Q_j$ denote the point $(0,j)$ in the coordinate plane. Find the maximum value of $n$ such that there are $n$ points $P_1,P_2 \ldots P_n$(not necessarily different) with integer coordinates all lying on a line parallel to the x axis such that $P_1Q_1=P_2Q_2=\ldots P_...

the problem I know that part a is a1=2 because it can be x or y and that a2=3 bc it can be xy or xx or yx. But i do not know how to find part b, the recurrence relation.

And vice versa? Can the lognormal distribution be represented exponentialy?

I was wondering if anyone can help me in figuring out average hop count in a 64 node hypercube.

Let $X$ and $Y$ be independent uniform(0,1) random variables. Find $P(XY\leq t)$ My question is when $t\leq 1$, why does it can be seperated to $P(XY\leq 1)= \int_0^t\int_0^1 dydx+\int_t^1\int_0^{t/y}dydx$

I know that xi are distributed poisson with parameter lambda. Can anyone explain why this holds? Thanks[1]

How can I prove that prime numbers beginning with 2, multiplied with the next consecutive prime plus 1, 2x3x5x7....+1, will give the form $4K+3$

Please help. Question: Addmath (Quadratic Equations) Given α and β are the roots of the quadratic equation 2x2 - 6x + 5 = 0. Form an quadratic equation with the roots α + 1 and β + 1.

I need to find the Laplace transform of sine(t), and it is proving rather difficult. I integrate by parts, then integrate by parts again so that the original integral is on both sides of the equation, then I get it wrong. $ f\left(x\right) = \left\{ \begin{array}{lr} \sin t & : 0 \l...

I could imagine filling up the 124, then filling the 45 from it, so there is 124-45=79 left in the 124, then again so there is 37. Then I'm lost!

I'm having trouble solving the following inequality problem: If $n$ is positive integer greater than $1$, and $x>y>1$, then show that: $\frac{x^{n+1}-1}{x(x^{n-1}-1)} > \frac{y^{n+1}-1}{y(y^{n-1}-1)}$ Any hints? Thanks.

Supose $f'(x)\ge M\gt 0$ for every $x \in [0,1]$. Prove that there exists an interval of length $\frac 14$ where $$| f(x) |\ge \frac M4$$

Let M be a manifold, and U,V open sub-manifolds in M. How would one use the Mayer-Vietoris theorem to show that X(M) = X(U)+X(V)-X(U intersect V), where X() is the Euler characteristic?

Let u satisfy the wave equation u_tt = u_xx, where x ∈ R, t > 0. u(0, x) = f(x), u_t(0, x) = g(x), x ∈ R, where f(x) = sin πx if 2 ≤ x ≤ 3, f(x) = 0 otherwise, and g(x) = 1, if 4 < x < 5, g(x) = −2 if 6 < x < 6.5, g(x) = 0 otherwise. Find all values of t for which u(t, 1)≠ 0. I had t...

10 people in a room, 2 cannot be near each other. Determine the number of ways the people can be placed in a line without 2 of them being next to each other.

I know that SAT goes to 3-SAT and SAT is reducible to CNF-SAT and CNF-SAT is reducible to 3-CNF-SAT but is 3-SAT reducible to 3-CNF-SAT? They are not the same thing though right because cnf makes it conjunctive normal form which is unique in itself.

I've been studying some introductory network optimization, and I noticed an interesting problem about flight networks. Suppose there are four locations $A$, $B$, $C$, and $D$ hold. One can fly from $A$ to $B$, $A$ to $C$, $B$ to $A$, $B$ to $D$, $C$ to $B$, $C$ to $D$, $D$ to $A$, and $D$ to $C.$...

Hello I am wondering about how to approach the following question; Suppose we are told that the weight of each gum ball ( in centigram) is given by the gamma distribution function, with $\alpha=25$ and $\beta=2$. We are wanting to know the probability that 100 gum balls will go over the limit o...

If one commuter train comes every $15$ minutes and another comes every $40$ minutes, what is the average amount of time one would have to wait before getting on a train? Suppose that they are not synchronized. The answer by the way is $6.5$ minutes. I don't see how the trains are not synchroni...

I know this probably has a simple answer, but I am having trouble understanding the steps to find the identity for this problem. My guess is some kind of $8sin^2(x)cos^2(x) = 2sin^2(2x)$ The closest Identity I can find is: $sin(x)cos(y) = 1/2[sin(x+y) + sin(x-y)]$ Which would give $1/2[sin(...

Prove that for $n\in \mathbb N, n\geq 2$: $\forall a\in \mathbb N, \forall b \in\mathbb N, \forall k\in \mathbb N, a\equiv_n b \rightarrow a^k \equiv_n b^k$. the definition for $a \equiv_n b$ and that for $a \equiv_n b$, we can write $a = nq_1 + r$ and $b = nq_2 + r$ where the remainders $r$ ...

Suppose that in a group of people that any two people are either friends, enemies of strangers. Show that in a group of seventeen people, there exists a trio who are either three mutual friends, three mutual enemies, or three mutual strangers. I know I need to use the Generalized Pigeonhole Prin...

Can anyone rigorously prove this? $$\int dx\delta\left(x-\alpha\right)\delta^{\prime}\left(x-\beta\right)=\delta^{\prime}\left(\alpha-\beta\right)$$.

$x_{n+1}=\frac{1}{x_n}$, $x_1>1$ Find if the function converges and diverges and then prove it. If we try and find the limit we get 1 or -1. 1 does not work because then $x_1$ and $x_n$ contradict each other. -1 works for showing its the lowest bound. I'm not sure about this proof for proving i...

I was wondering what example can I come up with to illustrate the valid use of Accept / Reject Algorithm.

I have difficulty understanding the solution to this problem and would appreciate if someone can explain why my solution is wrong: You meet a person who either always lies or always tells the truth. He flips a standard coin and makes a the following statement: The toss is head if and only if I a...

Compute the entropy of the density function $\frac{b}{{\pi ({b^2} + {x^2})}}$. I think the entropy of a density function $f(x)$ is given by $H = -\int f(x) \ln f(x) ~dx$ My calculation is $H = - \int_{ - \infty }^{ + \infty } {\frac{b}{{\pi ({b^2} + {x^2})}}(\log \frac{b}{\pi } - \log ({b^2} +...

We have PARTITION3 = {(S)| S is a multi-set of positive integers that can be partitioned into 3 sets where the sum of each set is equal} I want to prove that this is NP-complete. To show that it is in NP is pretty simple. You just guess 3 partitions nondeterministically and check if the sum is...

I'm currently learning about spherical potentials in the context of quantum mechanics (ex. hydrogen and hydrogen-like systems) and am trying to work through the problem of a generic spherical potential well. I'm comfortable with the separation of variables and how the radial equation is obtained...

Show that \int_{0}^{\infty} 1/(1+x^n)dx=(\pi /n)/sin(\pi /n) where n is a positive integer. He wants us to perform a contour integration on a "pie slice" containing a pole. We're supposed to find the singularities and then find the residue to solve it. I'm not ever sure how to approach it becau...

I am really just unsure of how to start this problem, any kind of help/hint would be much appreciated. Continuos Stirred Tank Reactor

Could you help me prove this? I've gotten stuck, need some help.. sin^2Θ + tan^2Θ = sec^2Θ - cos^2Θ Here's what I've done so far: Left Side: sin^2Θ + sin^2Θ/cos^2Θ ((sin^2Θcos^2Θ)+sin^2Θ)/cos^2Θ Right Side: 1/cos^2Θ - cos^2Θ (1-cos^2Θ(cos^2Θ))/cos^2Θ Thanks in advance. Note: I have trie...

I had a question regarding blackjack. In the game, the odds of winning are about 48 percent or something. I think the limit of your profit as the number of games played approaches infinity would be 48 percent. Based on this, the odds of earning money would be zero after playing an infinite amount...

Prove that, $\int_0^1\int_0^1 \frac{x^2dxdy}{\sqrt{(1-x^4)(1+y^4)}}=\frac{\pi}{4\sqrt{2}}$

I am unsure how to proceed with the current equation to determine a three-term outer expansion and three-term inner expansion due to the nature of the equation. Equation: $\epsilon \frac{d^2y}{dx^2}+\frac{dy}{dx} = 2x$, $y(0) = 0$, $y(1) = 2$ Whilst I only have lecture notes with equations in...

I was able to get the last row of A by replacing the asterisks with x and y, doing (0-λ)(x-λ)-(x)=0 and plugging in 4 and 7 for λ. I got x=-28 and y=11. However, I tried to do the same for matrix C and could not get the values for the last row. My work resulted in (z-1)=(4z-8)=(9z-27), which does...

The function: $$f(x)=4x\sin\left(\frac{1}{x}\right)-2\cos\left(\frac{1}{x}\right)+1 ,\text{ if }x\neq0$$ takes both positive and negative values on any open interval around $x=0$, i.e. on interval (-c,c). There's hint that any such interval contains zeroes of both cos(1/x) and sin(1/x), but I'm ...

I have a following matrix related problem: Let $F$ be a $n \times n$ discrete Fourier matrix defined as $F_{j,k} = \frac{1}{\sqrt{n}}exp{(\frac{i 2\pi jk}{n})}$, for $0 \leq j,k \leq n$, where $i^2 =-1$. For $x \in C^n$, let $l(x)$ denote the number of non-zero components of $x.$ Show that for...

I'm working on a problem from my textbook and found that (1/2, 1/2, 1) is an eigenvector for a particular eigenvalue of 4. The textbook solution says that the answer is (1, 1, 2) which is just 2 * (1/2, 1/2, 1). Thanks in advance.

It is given that $triangle ART$, $triangle BPT$, and $triangle CQT$ have an area of one. I have no idea how to approach showing that all of the triangles have an equal area of one.

If you have Part = {(S)| S is a set that can be partitioned into 3 parts whose sums are equal} Prove that Part is NP Complete. So to show Part is in NP is easy. Just guess 3 partitions non deterministically and check sums. But to show Part can be reduced from Subset Sum is a little tricky. I ...

I'm using Sipser's book and after several attempts I still don't quite understand how to show $\overline{E_{TM}}$ is NP-hard.

What axioms of set theory are needed for Cantor's diagonalization argument to work and why? What happens if we do away with some of these axioms (for instance Axiom of Choice)?

I have a definition of a Hypergeometric distribution as follows: Definition: the Hypergeometric distribution is a discrete probability distribution that describes the probability of $k$ successes in $n$ draws, without replacement, from a finite population of size $N$ that contains exactly $K$...

Show that the only way these ratios can be equal is if both ratios equal one, and show that in that case, $\overline{XY}$ is parallel to $\overline{BC}$.

Let $X_1 \sim U[0,1]$ and $X_i \sim U[X_{i - 1}, 1]$, $i = 2, 3,...$. What is the expectation of $X_1 X_2 \cdots X_n$ as $n \rightarrow \infty$? Thank you very much!

laplace residual distribuiton 을 갖는 location model 일때, Q2 가 mle라서 \bar{X} 보다 efficient 한데, ARE 를 구해보면 2가 나와서, 보통 2배 efficient합니다. 건 그렇고, theta 에 대한 CI를 만들때, Q2나 \bat{X} 를 가지고 만들수 있는데, Q2는 MLE라서 mle 정규근사에 의해 정규분포에 근거해서 만들 수 있고, \bar{X}는 sample mean이라서 CLT에 의해 역시 정규분포에 근거해서 만들수 있습니다. (1-\...

I was not able to prove these identities.I was wondering if anyone can answer them since I couldn't. b) sin2x-tan2x = -sin2xtan2x c) (cos2x-1)(tan2x+1)=-tan2x d) cos^4x-sin^4x = cos^2x-sin^2x

Consider a differential equation $y'' + y = F(x)$, where $F$ is continuous on $[a,b]$. If $x_0 \in (a,b)$, I'm supposed to show that the solution to the IVP $$y'' + y = F(x), y(x_0) = y_0, y'(x_0) = y_1$$ is $$y(x) = y_0 \cos(x-x_0) + y_1 \sin(x-x_0) + \int_{x_0}^x F(t) \sin(x-t)dt$$ What I did ...

If we called the perimeter of some triangle b. Prove that if you added the lengths of any two of its medians (i) It would not be not larger than (3b)/4 (ii) It would be smaller than (3b)/8 This came up in a math competition some time ago and I think that using the triangle inequality will be n...

GIVEN: Trapezoid $\partial R$, in the $xy$-plane with vertices $A, B, C, D$ given by $A = (1,0), B = (2, 0), C = (0, 2), D = (0, 1)$ Let $R$ be the (finite) region of the $xy$-plane enclosed by $\partial R$ and $f: R \to R, f(x,y) = e^\frac{-x+y}{x+y}$ FIND: $\int\limits_Rf(x,y)dxdy$ HINT: Con...

If $G$ is a group with a subgroup $H$ of finite index $n$, then $G$ has a normal subgroup $N$ whose index in $G$ is finite. I found a proof of the question here: How to prove that if $G$ is a group with a subgroup $H$ of index $n$, then $G$ has a normal subgroup $K\subset H$ whose index in $...

I've found this $\pi$ formula: $$ \pi =\text{Limit}_{n\to \infty }\underset{k=1}{\overset{n}{4\text{ }\sum }} \frac{2 n^3 (1-2 k)^2 \left((k-1) k-n^2\right)}{\left(k^2+n^2\right)^2\left((k-1)^2+n^2\right)} $$ What is interesting is that the formula has a geometric origin. So it could have bee...

I need your help in deriving a formula for calulating three taxes with a comination of cumulative and incusive tax. Here are the examples with expected result. Please let me know the formula on how to get the given result. Example1: Total Product Value (V) = 1000 (with inclusive of all 3 taxes ...

Just playing around with the modulus definition doesn't really confirm that thought... Is it true? If |z| = |x+iy| < 1, is $$\lim_{n\to \infty} z^n = 0 ?$$ Thanks,

I want to solve the following coupled ODEs using Green's function. $\begin{eqnarray} m_1x''+k_{12}x+c_{12}x'-k_2y-c_2y'=f_1(\xi)\quad \text{for}\quad a\leq\xi\leq b\\m_2y''+k_{23}y+c_{23}y'-k_2x-c_2x'=f_2(\xi)\quad \text{for}\quad a\leq\xi\leq b\end{eqnarray}$ where, $x=x(\xi)$ and $y=y(\xi)$. ...

Let $G$ be the group of order 20 defined in terms of generators and relations: $$G:=<x,y|x^5=y^4=1\text{ and }yx=x^2y>.$$ Can anyone help me to derive the character table. According to gap the conjugacy classes can be represented by $1, x, y, y^2$, and $y^3$. I can't determine the sizes of these...

I have the following question I have answer for the first one as S->C1S C1->C2A C2->a S->aA S->C2A S->a Hope im right?

I found the following question in a paper I was trying to solve: The following figure shows a $3^2 \times 3^2$ grid divided into $3^2$ subgrids of size $3 \times 3$. This grid has $81$ cells, $9$ in each subgrid. Now consider an $n^2 \times n^2$ grid divided into $n^2$ subgrids of size $n \ti...

A {\em completion} of a metric space $(X,d)$ is a metric space $( \widetilde{X}, \widetilde{d} )$ such that $X \subseteq \widetilde{X}$ and such that: (I) $\widetilde{d}$ extends $d$. That is, we have $\widetilde{d} (x_{1},x_{2}) = d(x_{1},x_{2})$ for every $x_{1},x_{2} \in X$. (II) $X$ is d...

I am learning Fourier series and I have a problem which has me confused and would like to here others take on it. $$F(t)=\begin{cases}v_0 & \ \ 0 \le t\le T\\ 0 & \ \ T\le t \le \ 2T\end{cases}$$ The problem doesn't have any period given. I have solved it for period $2L=2T$ $=>$ $L=T $ ...

A, B, and C are all integers. B is divisible by A however C is not. How do I prove the sum of B and C is not divisible by A?

f(x) = ceiling(x) has a domain of set of real numbers and has a range of set of integer values. So f(x) is everywhere defined? Is it a bijective? Why?

If $f:R\to R$ and $g:R\to R$ be functions such that $g(x)$ is onto and $fog(x)$ is injective then prove that $g$ must be injective. I dont know how to prove it.I only know that composition of two injective functions is an injective function and composition of two surjective functions is surjec...

The number of sick days taken during a year by an employee follows a Poisson distribution with mean 3. Let us observe 5 such employees. Assuming independence, compute the probability that their total number of sick days exceeds 9.

I wonder why under assumption that w>>$\frac{1}{T}$ then $\int_{0}^{T} sin(wt)dt$ is zero? Since the integral should be like- $\frac{cos(wt)}{w}$ from $0$ to $T$ and after plugging the valued we will end up with : $\frac{-cos(wT)+1}{w}$

I am completely stuck on this, I want to say it's true and do a proof by contrapositive, since if g is not surjective, then $\exists b \in B $ such that for $c \in C, f(b)\neq f(c)$, but I'm not sure where to go with this. Thanks!

for some reason my mind is drawing a blank when I'm trying to figure out a certain model. For example, if I had a model Xt=Yt+Wt, where t=1,..K, where Y is an exponential(P) distribution and W is a Poisson(Q) distribution, would my likelihood function for l(P,Q;x,y) simply be: l(P,Q;x y) is p...

A syndetic set $S$ is a subset of the natural numbers $\mathbb{N}$ or integers $\mathbb{Z}$, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded. That is to say, $S$ is a syndetic set, if there exist a positive integer $l$, such that for...

For any $x > 0$, show that $$\lim_{x\to \infty}\cfrac{x^n}{n!}=0$$

Let $(X,Y)$ be asymptotically normal with their means, variances, and a covariance. Then, I would like to show $X/Y$ is also asymptotically normal. I think there should be some references related to this topic. However, I can't find any documents about ratio distribution in terms of asymptotic pr...

let T be a linear operator on vector space v with characteristic polynomial λ^4(λ-4)^5 and minimal polynomial λ(λ-4) then rank of T=?

Does the convergence of the left hand side imply the convergence of the right hand side and the equality of the two sides? Do we need further requirement for the function $f(j,k)$? $$ \sum_{k=0}^{\infty} \sum_{j=0}^{k} f(j,k) = \sum_{j=0}^\infty \sum_{k=j}^\infty f(j,k)$$

$a ≡_n b$ can be written as $a = nq_1 + r$ and $b = nq_2 + r$ where the remainders $r$ are the same. Prove that for n bigger or equal to 2 ∀a ∈ N, ∀b ∈ N, $a ≡_n b → a^2 ≡_n b^2$ and therefore $a ≡_n b → a^m ≡_n b^m$ where ∀m ∈ N Where do I begin to prove this? Do I use induction?

$w,v\in R$ and $f\in C^1(R)$ . How to compute $$ \lim\limits_{t\rightarrow0}\frac{\int_w^{w+tv}f(y)dy}{t} $$ I feel it should be $f(w)$, but I don't know how to prove it

If F:R^n×R^n->R is the function F(x,y)=, where <,> is the standard inner product of R^n and A is a n×n real matrix.Here D denotes the total derivative.which of the following statements are correct? (A)DF(x,y)=+. (B)DF(x,y)=0. (C)DF(x,y) may not exist for some (x,y)€R^n×R^n. (D)DF(x,y) does not ex...

I know that $\sum_{n=0}^\infty 1/n$ diverges whereas $\sum_{n=0}^\infty 1/n^2$ converges. Intuitively, I do not see the difference. If $n \to \infty$, the denominators in both fractions will be so big that the fraction approaches zero. So why doesn't both the series converge against zero? I hav...

This is hopefully a better and clearer version of a deleted question. Let $P(s),C(s),\zeta(s)$ be the prime zeta function, the analogous composite zeta function, and the classical zeta function. I do not know whether it is known that there are infinitely many zeros of C, or whether there exist...

This is problem 1.D.4 in Isaacs, Finite Group Theory. I think I have a proof, but it's a rather grungy element-pushing argument (very un-Isaacs in style). My questions are: Is there a cleaner, more "group theoretic" way I can prove this? Unlike most problems in the book so far, I don't really h...

I'm having a hard time solving this. I'm taking a proof-based undergraduate linear algebra course that has no assigned textbook and this has been making things a little hard since the only resource I have are my notes from lecture -- he's doing things his own way (starts with set theory, then gro...

I think to show f' is continuous, I need to show that lim(x->x0) f'(x)=f'(x0) exists. But i don't know how to prove that.

it is known that any prime > 2 is odd, however how do I show the combinations of all primes > 2 is indeed odd, or 2K+1? I tried using induction however, what is appropriate for $prime_n$? $3 * 5 * 7 * 11...2k + 1$ = $2k + 1$ ? can we use $2k + 1$ to replace $prime_n$ since we know it will be odd?

In the game of blackjack, the odds of winning each hand are slightly less than 50 percent. As you play an infinite amount of hands, you would always lose money because you would win less than 50 percent of the time. By this rationale, wouldn't your highest odds of winning be if you only played on...

let $G$ denote the group of all the automorphisms of the field $F_{3^{100}}$ that consists of $3^{100}$ elements.Then what is the no of distinct subgroups of G?

this is some lecture slides from my school. I don't understand why we need to sum up all from m=0 to m=+oo, I think m is a fixed number which represents the number of type 2 event? plz help me! many thanks in advance!

I'm trying to find an answer to this question. Let $K(k)$ be the elliptic integral of the first kind and $K'=K(\sqrt{1-k^2})$. According to Abel's theorem (see this link) we know that if $\frac{K'}{K}=\frac{a+b\sqrt{n}}{c+d\sqrt{n}}$ where $a,b,c,d,n$ are integers, then $k$ is the root of an alg...

I'm stack on the following problem. I have to show that the minimum cost max-flow problem can be solved by reducing it to a minimum cost flow problem. Any hint ?

I've been working on a 3D game for a bit now and simply can't figure out a proper formula for limiting the users YAW rotation rate. I decided to post on this forum with hopes of a possible solution to my issue, so i'm sorry if this is the wrong section. Basically the user ingame should only be ...

On a previous Cryptography exam I'm working through, there is the following problem: Given $$f(x) = x^{134}+x^{127}+x^{7}+1$$ and the field $\mathbb{F}_{2^{n}}$ where $n=1463$, how many roots does the polynomial have? The polynomial has the form $f(x) = (x^{7}+1)(x^{127}+1)$, this is clear. It'...

I am on my way to solve the hydrogen-atom in 3D, and now I am stucked with the Laplace's equation, where there occurs a "=", which I can't understand.

The base axioms of a matroid state that A collection $B\subseteq 2^E$ is a set of bases of a matroid M(E,I) if and only if the following hold B1: $B\neq \emptyset$ B2: If $B_1,B_2\in B$ and $x\in B_1 - B_2$ then $\exists y\in B_2-B_1$ such that $(B_1 - \lbrace{x\rbrace})\cup \lbrace{y\rbrace}\...

$ f(kx$ $mod p)$ $\equiv$ $k.f(x)$ $mod p$ for some function f defined as $f: \{0,1,....p-1\} \rightarrow \{0,1,....p-1\} $, where p is an odd prime number and k lies in the range $[0,p-1]$ (both p and k are fixed integers). We have to find the number of distinct functions, f, for which this re...

I'm pretty new to the world of fuzzy set theory, and I am trying to understand implications. So, I am wondering if someone help tell me if the following is correct. I am trying to find the minimum of: $$ H \rightarrow \lnot G $$ and any advice or comments would be fantastic. My attempt is:$$ H ...

I'd like to calculate the following: $$ \int y\partial_{y_2} [(y_1+y_2)p(y)] dy,$$ where $p$ is a probability density and $y=(y_1,y_2,y_3)$ is a vector. I need integration by parts here so I need the integral $$ \int \partial_{y_2} [(y_1+y_2)p(y)] dy.$$ The first moment is the centre of mass...

Okay, so I was reading the concept of "Increment and Differential of a Function". It says: Let us consider a function $y=f(x)$ which has a derivative. The increment of this function $$\Delta y=f(x+\Delta x)-f(x)$$ corresponding to the increment $\Delta x$, has the property that the ratio $\frac{...

(1) In a contest,Harvard sent 3 participants.If there are 10 participants all in all,what is the probability that all three participants are in top 3? (2) In a contest,Harvard sent 3 participants.If there are 10 participants all in all,what is the probability that they will win first,second,thir...

I'm familiar with basic high school trig. The answer is 2sin(Theta)cos(Theta) Id appreciate it if someone could give me an explanation.

How do I prove this knowing that f(x) = tan((pix)2) is a bijection between (0,1) and (0, ∞). We also have a bijection between (-1,1) and (0,1)

We have got two sets $A = \{1, \ldots, k \}$ and $B = \{k, \ldots, 2 k - 1\}$. Let $\sigma$ be random permutation of $\{1, \ldots, n\}$, $n \ge 2k - 1$. Let $A_{max} := max\ \sigma(A)$, $B_{min} := min\ \sigma(B)$. What is the probability that $A_{max} = B_{min}$?

Let G be a noncyclic group. Show that G has at least 5 subgroup. Anyone help me?

Take two basic differentiation rules (from Baby Rudin) for example: Theorem 5.3 (c). (The quotient rule) Suppose $f$ and $g$ are defined on $[a,b]$ and are differentiable at a point $x \in [a,b]$, and assume that $g(x) \neq 0$, then $(f/g)$ is differentiable at $x$, and $(f/g)'(x) = \frac{g(x)f'...

Can someone explain to me how to find the domain and range of a piece wise function using this example? Thanks

Like the D'Alembert test case for convergence of series I have seen something similar for sequences in my math homework which required proof for this: If we have a sequence $a_{n}$ such that $\space\ lim_{n\rightarrow \infty}|\frac{a_{n+1}}{a_{n}}|<1$ then $\space\ lim_{n\rightarrow \infty}(a_n)...

$∀x∃y:R(x,y)∧∀x∀y:(R(x,y)⟹¬R(y,x))∧∀x∀y∀z:(R(x,y)∧R(y,z)⟹R(x,z))∧∀x:¬R(x,x)$ Does it have finite models? Is it satisfiable? If so, give a countable model for it. Genuinely, I'm unable to understand this question. Somewhere, it explained as : broken in $4$ parts, i.e. $A, B, C, D$. It explain...

I'm solving a problem.I have to prove $a^2+b^2+1 <6ab$ to complete the proof. Well, $a$ & $b$ are natural numbers.Is there an elementary way to prove this? I tried plotting the graph fixing the value of $a$.I can't prove it though

$$\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$$ Can someone show why this estimate holds true? I tried quite a bit but couldn't reall find a way to approach this. WolframAlpha says it is true but I don't know what the gamma function is. $$ \sum_{k=n}^\infty{\frac{1}{k!}} = \frac{1}{n!} +...

so i failed a math exam, and my professor gave me this question about number sequence for which if i could answer it, i will get a B on my paper. what are the next five sequence of this. 7394, 2263, 8766, 5154, 8021, 6193, ----, ----, ----, ----, ----.?

I have a time series data with $52$ observations and I would like to check for the independence between observations. The ACF for correlation and covariance of my data look I am aware that $covariance = 0$ does not imply independence, except for Gaussian process. I wonder if I can use ACF to...

I have a black-box polynomial function, with $n$ inputs. The only things I know for sure about this polynomial are that (a) it has no constant term, and (b) all coefficients and exponents are integers with magnitude below some $k$. (Exponents are positive, of course.) Let's say I get to evaluate...

I'm trying to solve the follow problem: Suppose that we are on the point $P=(1/\sqrt{2},1/2,1/2)$ over $z=\sqrt{1-x^2-y^2},$ $z\geq 0, x^2+y^2<1.$ In which direction we have to move over the surface such as: a) rate of change of z is zero? b) rate of change of z is maximum? My attempt is based ...

Let $X_1,...,X_n,...$ be independent variable satisfying $P(X_i=0)=P(X_i=1)=\frac{1}{2}$ for all i then denote $Z_i=X_iX_{i+1}$ for all i .I want to show that $lim_{n\to \infty}\frac{Z_1+Z_2+...+Z_n}{n}=\frac{1}{4}$ a.s by using law of large number. I'm new to this area and struck immediately sin...

Given the matrix A $$A:=\left(\begin{array}{cc} 2&1\\ -1&2 \end{array}\right)$$ What is the spectral radius $p(A)$ $||A||_2$ $||A||_\infty$ I got for the spectral radius $p(A)=\sqrt(5)$ for $||A||_2 = \sqrt(5)$ and for $||A||_\infty = 3$ Is this correct?

How to derive that $$\Delta z=\frac{\partial y}{\partial x}\Delta x+\frac{\partial f}{\partial y}\Delta y+\alpha\sqrt{\Delta x^2+\Delta y^2}$$ Is it a definition? Or there is a proof?

Let $P$ be a $p$-group. For a subgroup $K$ of P containing $Z(P)$ of G. By $\overline{K}$ we denote the quotient $K/Z(P)$. Let $H$ be a proper subgroup of $P$ containing $Z(P)$. The proof of the first theorem in section 6.1 from the book Abstract Algebra by Dummit and Foote uses the following e...

How to show the system have solution ? $R_{ij}$ is ricci tensor, $R$ is scalar curvature. I feel this is complex question, because I have little knowledge about PDE. So, if it is complex, just tell me what I should read ? I try to find answer in Evans' PDE book, but I am not sure. Whether the 11 ...

I have to find equation and starting condition to solve nonlinear Schrodinger Equation with periodic edge condition. This method should control the propagation of fiber optical signal. In details I need to find a valid case as it was already found for the semilinear wave equation (see pages 26-27...

The equation for the volume of a cone is 1/3 x Pi x r^2 x h starting from the base. However, I wanna calculate the height of the cone for a particular volume from the tip of the cone. Could you please help me with a formula? Thanks.

How to convert this equation $$\frac{\left(\frac{a^2b^2x}{a^2y^2+b^2x^2}-p\right)^2}{A^2}+\frac{\left(\frac{a^2b^2y}{a^2y^2+b^2x^2}-q\right)^2}{B^2}=1$$ to the standard form an ellipse? $$\frac{\left(x-p\right)^2}{a^2}+\frac{\left(y-q\right)^2}{b^2}$$ The algebra is really annoying :( Thank...

In the case of $f(x)=ln(x+\sqrt(1+x^2)$ in the derivative we multiply $f'(x)=\frac{1}{x+\sqrt(1+x^2)}*(1+\frac{2x}{2\sqrt{1+x^2}})$ when the expression multiply the numerator?

We have to answer the following question, and I can make some progress on the first couple of parts but get stuck in finishing them off. The third part I'm not sure where to start! *Consider the following Bayesian game. Nature chooses the type of player 1 from the set {1, 2, 3, 4} where each typ...

Let $X_n$ be a DTMC, with transition matrix P and state-space I. Let $Y_m=X_{T_m}$ for $m \in \mathbb{N}$. Define $T_0=\inf\{n\geq0:X_n\in J\subset I\}$ and $T_{m+1}=\inf\{n\geq T_{m}:X_n\in J\subset I\}$. These are stopping times, so we can use the strong markov property when conditioning on th...

Let $N(x^3 + y^3 = 1)$ be the set of pairs $(x, y) \in \mathbb{F}_p \times \mathbb{F}_p$ such that $x^3 + y^3 = 1$. How do I see that$$\left|N(x^3 + y^3 = 1) - p + 2\right| \le 2\sqrt{p}?$$

Let $B \subset [0,1]$ be a nowhere dense closed set. $a)$ Show that there exists $s \in [0,1]$ such that for no natural number the point $(s, \frac{s}{n})$ is in the circle $S((0,0),r)$ where $r \in B$. $b)$ Show there is a irrational number satisfying the condition from a). By Baire Theorem, ...

Is Sum[Binomial[(k-1)i,i]((k-1)/k)^((k-2)i))(1/k)^i,{i,1,Infinity}] convergent for any integer k? What is the exact result?

I have the following problem I think that solution is wrong x1=b and y1=b3.They do not match,So how is this solution possible?

Well, similar questions have already been asked. But they still were not identical and the solution methods offered there were not the same. Anyway, i want to solve functional equation $f(m) + f(n)) = f(m) - n$ where $f:\mathbb{Z}\rightarrow \mathbb{Z}$. Somehow i showed that such function doesn'...

let $X\sim N(0,1)$ be a standard normally distributed stochastic variable and let $Y=X^2$. Show that $Y\sim\Gamma(\frac{1}{2},\frac{1}{2})$, ie. $Y\sim\chi^2(1)$.

I have this question here that I'm really confused about can someone please tell me how to prove this question the full question is here: I need help with question 7 from this image

I have a little mathematic problem. I have bar code with 3 types of black (x, y, z) lines and 2 types of white lines (w, v). There are 12 black lines and 11 white lines. And black and white lines alternate like B, W, B, W, B.. 2 black lines are outside. Now should I find how many codes I can mak...

Let say we have a Coast Guard center wich have a emergency center hold only one rescuer. accidents comes according to a Poisson process. There is usually 16 casualties on 8 hours. Time passed to healed sufferers is 15 minutes for each of them. Casualties are examined according to an exponential l...

A school-wide survey revealed that 40% if the students like to eat pizza. If 10 students are randomly selected from the population, what is the probability that 6 likes to eat pizza?

Let $\mathbf H=\begin{pmatrix}a&b\\0&a\end{pmatrix}$ Show that $e^{\mathbf Ht}=e^{at}\begin{pmatrix}1&bt\\0&1\end{pmatrix}$ I have $$\begin{aligned} &Ht=\begin{pmatrix}at&bt\\0&at\end{pmatrix} \\ & \iff e^{Ht}=e^{\begin{pmatrix}at&bt\\0&at\end{pmatrix}}=e^{at \begin{pmatrix}1&0\\0&1\end{pm...

I have coded a reader for MPS files in MatLab, which yields A, b and c as in min c'*x s.t. Ax=b x>=0 The reader transforms the problem from the file to standard form. In the next step I was using a self coded dual and primal simplex to solve this problem. Some small problems from a lectur...

The inverse from $R =_{def} \{ (x,y) | x^2 -1 =y \} \subseteq \mathbb{N} \times \mathbb{R} $ is $R^{-1}: y \mapsto (y+1)^{\frac{1}{2}} $ but is this a left-total function?

R is a relation of real number . $xRy <-> x+y = 0 $ . is it equivalence relation ? so my answer is no proof : -(Reflexive) let $x = a$ , $ aRa <-> 2a=0$ . so since 2a doesnt hold for every real number a , R isnot reflexive. since R isnt reflexive R isnt equivalence relation. so is my reason...

Please help me solve this problem. Show that lim (x,y)-->(0,1) ye^x =1 using epsilon delta method.

Assume that $| z + 1 | > 2$. Show that $|z^3 + 1| > 1$. My try was: $$|z^3 + 1| = |z + 1| |z^2 + z + 1| > 2 |z^2 + z + 1| $$ but I'm stuck proving that $|z^2 + z + 1| > \frac 1 2$

Can someone help me with this integration question $$\int_0^1 \frac{dx} {x+\sqrt{1-x^2}} $$ I tried substitution with x=sinx and also tries times both denominator and numerator by $$ x- \sqrt {1-x^2} $$ but it becomes even more complicated. Can someone gives me a hint how to solve this questi...

I am following a book which has a part on numerical optimization techniques. In order to elaborate Karush-Kuhn-Tucker theorem, they gave the following example: When the unconstrained solution $x=A^+ b$ does not lie in the feasible region, they apply an iterative approach to find the solution wh...

function ex2a12(a,x,n) y=zeros(n,1); y(1)=x; for j=1:n y(j)=a*y(j-1)*exp(-y(j-1)); end end I am getting Undefined function 'ex2a12' for input arguments of type 'double'. any ideas how to fix this"?

(b) Can anyone shed some light on the method for solving this problem?

The question: Let {$X_n$} be i.i.d random variables. $EX_1=0$. Then $\sum_{i=1}^{n}X_i\over n$ converges almost surely to zero. I know that when the sequence $\{X_n\}$ satisfies $\sum_{i=1}^{\infty}{Var(X_i)\over i^2} \lt \infty$, the conclusion holds. So I try to cut $X_n$ to make it satisfy th...

Let $f:R\to R$ be an one-one function,$g:R\to R$ be a many one function,$h:R\to R$ be an onto function and $l:R\to R$ be an into function. $(1)$I know that one one function composition one one function is a one one function. $(2)$What is one one function composition many one function. $(3)$What ...

Let $a$ and $b$ be elements of a group, with $a^2=e, b^6=e$ and $ab=b^4a.$ Find the order of $ab$ and express the inverse in each of the terms $a^mb^n$ and $b^ma^n.$ Just want to cross check my solution. Would be very grateful for the complete solution.

I'm using this article for the proof. I thought some parts are extra and tried to make a new shorter proof. Here goes: Let $\Delta(x)$ be a set of formulas (with one free-variable $x$) in the language $\mathcal L$. It suffices to show that if each finite subset of $\Delta(x)$ is realized in the...

Find general inverse of $A$ where $ A = \left( \begin{array}{ccccc} 1 & -1 & 0 & 0 \\ 0 & 3& 2&1 \\ -1 & 1 & 1&1\\ 2&1&0&4\\ 0&1&0&1 \end{array} \right) $ can you help me?

Let $f_i, \ i=1,2,...n$ be the frequencies of class intervals. Let $x_i, \ i=1,2,...n$ be the class marks. And $\bar x$ is the mean. What is the value of $\Sigma(f_ix_i-\bar x)$? I got this question in a test. I was confused, whether, a solution even existed or not? Because inserting real values...

My professor pitched me a problem like so, 2∅(25) mod 25 ≡ ? The answer is 1. However, after much browsing of the internet, I couldn't find anything to justify this answer. I was wondering exactly how she came to this conclusion.

If $f: [a,b] → R$ is a continuous function which is differentiable on $(a,b)$ If $f(a) < f(p)$ and $f(p) > f(b)$ for some $p ∈ (a, b)$. Show that there exists $d ∈ (a, b)$ such that $f'(d) = 0$ I know I need to use Rolle's Theorem and the intermediate value theorem to do this, however I am unsu...

Random variable X has uniform distribution on a line segment $[-3, 3]$, $Y=X^2$ Find cdf of the variable Y. $$ Fy= \begin{cases} \sqrt{t}/3&for &t\in[0,9] \\ 1 &for & t\ge9 \end{cases} $$ I mean why do we change those "boundaries" why we have distribution on $[-3,3]$ and then we change it into $[...

if any one can help me please if we consifer two continuous random variable X and Y ; we suppose that enter image description here is that any relation betwwen the Expected value of X and Y or ather mathematical expressions that related X and Y . thank you

(A’B’C) + (A’BC) + (ABC’) I'm struggling with this concept..

The cafeteria offers discount to students who opt to buy two meals.As part of the plan,the students may chooses exactly two among the meals A,B,and C.If they opt to buy only one meal,then they cannot avail the discount.The proportions of the students who chose meals A,B ,and C are 1/6,1/3,and 5/1...