$$1/(1+tg^2x)+1/(1+cotg^2x)$$ It resultet to me that it is cos^2+sin^2x=1 A I right? The other is: $$tgx/(1-tg^2x)*(cotg^2x-1)/cotgx$$. It result to me that t is sin^2x, but it is not correct

Let C={x$\in$R| x$\ge$1} and D={x$\in$R| x$\gt$0}. f(x)=lnx I know that the inverse of this is $f^-1$(x)= $e^x$ so would C be defined as $f^-1$(C)= {x$\in$R| y$\ge$1, x$\in$$e^x$} ?

I can't solve the following combinatorics problem Let G=(V,E) be a graph with |V|>=4 and with the property that for any three of its vertices u,v and w,at least two of edges uv,uw and vw are in E.Show that G is hamiltonian.

Let $(a,b,c)$ is a Pythagorean triple, which means $c^2=a^2+b^2$. If $c$ is odd and $a$ & $b$ are relatively prime, then there exist integers $m$ and $n$ such that $c=m^2+n^2, ~a=m^2-n^2, ~b=2mn$. One can easily check the above by proving $\gcd(\frac{a+c}{2},\frac{a-c}{2})=1$. My question is w...

Please help in computing the limit of ((-1)^n n^(1/n) cos(n^n))/(n+1) as n goes to infinity.

im confused about how to solve this question i managed to solve the ones before this but these two questions seem tricky any help appreciated http://i.stack.imgur.com/KnCrH.jpg

Function f:[0,inf)->R continuous on [0,inf) and differentiable at (0,inf). f(0)=0. f' is strictly increasing.

So far, I'm stuck on this problem of converting a series to a polynomial and showing that it exhibits certain properties. PROBLEM Show that the polynomial formula for $P_k(n) = \sum_{j=1}^n j^k$ is characterized by the following two properties: $P_k(0) = 0$ for all $k$ $P_k(x)-P_k(...

Let $G = A_5$ and $H=\langle(1,2,3,4,5)\rangle$. Let $g \in N_G(H)$ an element of order $5$. Compute the order of $H \langle g \rangle$ (already done) and $g \in H$. I already know that both of the subgroups $H$ and $<g>$ are of order 5 which is prime, so either they are equal, or their intersec...

i am studing the (Schrodinger operator) but this question is in topology: i found in a course that this set : $DS_\alpha(\lambda,\gamma)=\{\beta\in\mathbb{R}| \forall k\in\mathbb{Z}-\{0\},\forall l\in\mathbb{Z} : |\beta-k\alpha-l|\geq\frac{\lambda^{-1}}{|k|^\gamma}\}$ thank you very much

I am writing the expression and my final result. If there is any other step to add or if I have done any mistake please correct. cotgx-cotg2x=2/sinxcosx sin2x/(1+cos2x)*cosx/(1+cosx)=sinx/(x+cosx) (sinx+sin3x+sin5x)/(cosx+cos3x+cos5x)=tg3x

I came across this in a paper, and although I suspect its extremely elementary, I can't quite grasp it. Suppose we have vectors $\vec{x}$ and $\vec{x}^{\prime}$. As a given, we have \begin{equation} \tan\left[2\Theta(\vec{x}-\vec{x}^{\prime})\right] = \frac{2(x_{1}-x_{1}^{\prime})\cdot(x_{2}-x_...

Show that given seven real numbers always is possible take two of them, such that $$\left\vert\dfrac{a-b}{1+ab}\right\vert<\dfrac{1}{\sqrt{3}}$$ The "Pigeonhole principle" states that if $n$ items are put into $m$ containers, with $n > m$, then at least one container must contain more than o...

$x_{1}, ..., x_{k}$ is orthonormal. Show that for any $y \in X$: $\sum_{i=1}^{k} \left | \left \langle x_{i}, y \right \rangle \right | ^{2} \leqslant \left \|y \right \|^{2}$

So a) is simple, calculate the squares and cubes mod 7. However, for b) I know that Z7[x]/(f(x)) if f(x) is irreducible of degree 2 would work, but then how do I solve the equations ? I assume F7xF7 would also contain 7^2 elements but is it possible to solve the equations in that field ? How ...

Consider the following problem. A small taxi company has two taxis that each have been distributed to separate halves of a city. If a person orders a taxi, then the taxi that is distributed to that part of the city will come; unless that taxi is already occupied and the other is free, then the ot...

I've recently started exploring elementary number theory, and came across the book Number Theory for Beginners by André Weil, which is where I found this problem. The problem is: Prove that any integer $x>1$ has either a divisor $>1$ and $\leq > \sqrt{x}$ or no divisor $>1$ and $<x$ `` I'm...

lim\frac{1}{n}\left ( \frac{\sqrt{2}}{\sqrt{1}} \right )\left ( \frac{\sqrt{3}}{\sqrt{2}} \right )\left ( \frac{\sqrt{4}}{\sqrt{3}} \right )...\left ( \frac{\sqrt{n+1}}{\sqrt{n}} \right )= 1 I know that ( \frac{\sqrt{2}}{\sqrt{1}} \right )\left ( \frac{\sqrt{3}}{\sqrt{2}} \right )\left ( \frac{\...

Let $F:\;[a,b]\times \mathbb{R}^n\to\mathbb{R}^n$ where $0\in [a,b]$ and for some $K>0$: $$||F(t,x)-F(t,y)||\leq K||x-y||$$ for all $x,y$ and all $t\in [a,b]$. I would like to show that there is unique continuous $f:[a,b]\to\mathbb{R}^n$ solving: $$f(t)=\int_{0}^t F(s,f(s))\; ds$$ ...

Let $x>0$ and $q$ be real numbers. Show that $x^q$ is a positive real. DEF.(Exponentiation to a real exponent) Let $x>0$ be real, and let $\alpha$ be a real number. We define the quantity $x^{\alpha}$ by the formula $x^{\alpha}=lim_{n\to\infty}x^{q_n}$ where $(q_n)_{n=1}^\infty$ is any sequence ...

Is there any use of the concept of an orbifold in dynamical systems theory? Can orbifolds be applied to any problems in dynamical systems?

How do you find A in a matrix representation in the linear transformation $[Lv]_E = A[v]_E$

The equation recommended to model the signal measured by a radiometer is the Planck form of the Sakuma–Hattori equation $S(T) = \displaystyle\frac{C}{\exp\left(\frac{c_{2}}{AT+B}\right)-1}$ where $c_{2}$ is the second radiation constant, and $A$, $B$ and $C$ are related to the radiometer’s spec...

I'm having some trouble knowing where to start with this problem. Find an example of a function such that $$f(x)\neq-2x$$ such that $$\int_{0}^{1} \left[ -2x-f(x) \right] = 0$$ I'm looking for a nudge in the right direction rather than a complete solution.

As of now, I completely suck at computing confidence intervals, and I have an exam in a couple of days. The exact question is as follows: Some people are trying to measure a certain parameter $\theta$. Every measurement has an error, and this error has the normal distribution with mean 0 and st...

Placing 2x2 dominoes on an 8x8 chess board, in non-overlapping way, what is the lowest possible number of dominoes to lock (guard) the board, so that no further dominoes can be placed on the board? The aim here is to cover as little as possible and save the maximum number of unused dominoes Than...

My proof: Let $|f(z)|^2 = M$ for $z\in D$. Then $f(z) = \pm\sqrt{M}$ (not sure about this step, are there only two values for the square root of a complex number> No right? Could be more. But I don't think it would change the essence of the proof) But $f(z)$ is analytic in $D$ so it cannot be...

So I've been working through some of the suggested exercises through Rotman and I have one problem that took longer than I expected. Most of the starred exercises seem to have a short quick proof except this one,I was wondering if someone could help me see the faster way to do this one. Problem:...

For $a \in R$ we got matrix: $A = \begin{bmatrix} a & 1 & 1 &0 \\ 1 & a & 1 & 0\\ 1 & 1 & a & 0\\ 1 & 1 & 1 & 1 \end{bmatrix}$. For which $a \in R$ equation $A\overrightarrow{x} = \overrightarrow{b}$ got unambiguous solution ($\overrightarrow{x} \in R^{4}$) for every vector $\overrightarrow{b...

$$X_1,X_2,...,X_n \;i.i.d\;with\,density\;function$$ $$f(x)=e^{-(x-\theta)}\;,x\ge\theta$$ What is the distribution of $\bar X$? I gained m.g.f of $\bar X$. (by calculating $E[e^{{1/n} (X_1+...+X_n)}])$ Result is $\frac n {n-t}e^{\frac {t\theta} n}$. But I don't know distribution that has th...

If $h(x,y) = f(y)g(x)$ and $f$ is a function on $(X, \mu)$ and $g $ on $(Y, \nu)$, then $\int_{X \times Y} h d(\mu \times \nu) = \int_Y g d\nu \int_X f d\mu$ and $h$ is $\mu \times \nu $ integrable. Isn't the whole thing just $$\int_{X \times Y} h(x,y) d(\mu \times \nu) = \int_{X \times Y}...

Let (M,d) and (N,p) be metric spaces. Consider the space MxN endowed with the metric D=((x,y),(x',y'))=max{d(x,x'),p(y,y')} for (x,y),(x',y') in MxN. Let A in M and B in N be nonempty. Prove or disprove: a) P:MxN ->M, P(x,y)=x is continuous and an open mapping b) The set MxN is totally bounde...

So the original function was f(x) = cotx + x Now I thought -csc^2(pi/6) = -4. So -4+1 = -3 hence it should be negative from the interval (0,pi/2). Somehow I have the feeling I don't even know high school trig... please help me out my brain is exploding right now i can't think straight.

Is the exponential map to the identity component of the special indefinite orthogonal groups $$ \mathrm{exp} \colon so(p,q) \to SO^+(p,q)$$ surjective?

I'm trying to see the relationship between the sample variance equation Sigma(Xi-Xbar)/(n-1) and the variance estimate, Xbar(1-Xbar), in case of binary samples. I wonder if the outputs are the same, or if not, what is the relationship between the two?? I'm trying to prove their relationship ...

I cannot find the solution to this problem. It is part of a larger homework question but I can't go on until I solve this question.

I think my confusion here is just which how the question was given to me. I am having trouble decrypting this simple RSA message. Message: 0882 1090 1471 1899 2753 0309 p = 43 ; q = 71; e = 19. Someone please check my work that I have so far and I would appreciate help getting the final answer...

I understand $C^{\infty}_0 (\mathbb{R}^3)$ is dense in $H^1 (\mathbb{R}^3)$. But I don't understand the reason it is enough to proof the following Hardy inequality if you proof the case $u \in C^{\infty}_0$. (Hardy inequality) Let$ \ u \in H^1 (\mathbb{R}^3)$. Then, $$\int_{\mathbb{R}^3} \frac{...

Let A={x in R^3: |x_1| + 2|x_2| +|x_3|^3 = 1 } and let p in R^3/A ..Show that there exists a point y in A that is closest to P among all points in A. ** Assume R^3 has the euclidean metric I obviously want to do this by contradiction, but I'm unsure of what assuming there is no single point clos...

Is every $\{-1,a,1\}^{n\times n}$ matrix with $a$ occuring only on the diagonal full rank? Only condition on $a$ is that it is not in $[-1,1]$. What if $a=0$? What if $a=2$?

I know that if A and C are finite sets then |AxC|=|A||C|. This makes the problem quite simple but the sets may not be finite. I am guessing that the concept of cardinally of infinite sets and ℵ 0 are part of the solution but those are concepts that my class did not go into much and I do not und...

Let say we have $m,n \in \mathbb{Z}\ with\ m \geqslant n \geqslant 2.$ How many cycles of length equal to 4 are in $k_{m,n}?$ And how many paths of length equal to 2?

Solve for all values of x: cos2x = 2sinx

How do you represent arithmetic operations with predicate logic? For example if I want to represent 2+3 = 5 in predicate logic with plus(x,y) function is the following correct? plus(2,3) <=> 5 But ideally <=> is equivalent and not equal. So is the above representation wrong?How do you represent...

I rewrite the text of an exercise: Given $X$ and $Y$ topological spaces and a family of subsets $\{A_i \}_{i \in I}$ of $X$ s.t. $X = \cup_{i \in I} A_i$ . Let $\{ f_i : A_i \rightarrow Y \}_{i \in I}$ be a family of continous function such that $f_{i|A_i \cap A_j} = f_{j|A_i \cap A_j}$ then ...

I have tried solving this, but I'm unsure if I'm right. Any suggestion would be appreciated. Question: 15 kids arrive at camp and are assigned a place to sleep. There are 3 different cabins each of which can hold 5 kids. How many ways are there to assign kids to cabins? My answer: (15,5) (mean...

Let $X = {1, 2, ..., 10}$ How many strictly ergodic, preserving order maps are on this set?

50= 35cos(x) +25cos(y) 0= 35sin(x)+25sin(y) Thanks!

Suppose that A ⊆[0,1] is measurable set such that m(I∩A)≤m(I)/2, for all intervals I⊆[0,1]. Show that m(A)=0 .

I've been practicing on PDEs, since I haven't needed to work with them in a long time until now, when I got stumped on how to solve this problem. The problem asks to use the change of variable $$u(x,t) = w(x,t) + q(x)$$ to solve the PDE $$ \frac{\partial u}{\partial t} = 3\frac{\partial^2 u}{\...

Schur's lemma says that if $M,N$ are two irreducible representations of a group $G$, then either $Hom_G(M,N)=0$ if $M,N$ are not isomorphic, or every $\varphi\in Hom_G(M,N) $ is invertible if they are isomorphic. If we look at the case $G=S_n$ and $M=V_\lambda,\ N=V_\mu$ for partitions $\lambda,...

Consider the following recursive definition of a function: g:NN 1) Base Case: g(0)=0, 2) Recursive Case: For any x>0 we have g(x) = g(x-1) + 2 Prove each of the following properties holds for g using induction on nEN. i) g(n) = n+n;

Can someone help me with this particular question? I don't know how to proceed.

I am trying to solve all Apollonius' Problems.i solved 7 of them.but i can't solve the remains (PCC-LLC-CCL). i checked some pages like :(http://www.physics.princeton.edu/~mcdonald/papers/apollonius_051964.pdf) or(https://en.wikipedia.org/wiki/Special_cases_of_Apollonius%27_problem). and etc ,but...

How can I prove above equality using the definition of Big O? $O(2^{f(x)})=2^{O(f(x))}$

I am trying to solve the following exercise: Describe alla $2$ by $3$ matrices $A_1$ and $A_2$ with row echelon forms $R_1$ and $R_2$, such that $R_1 + R_2$ is the row echelon form of $A_1 + A_2$. Is it true that $R_1=A_1$ and $R_2=A_2$ in this case? According to the exercise: $$\left. \be...

I'm trying to read Lehmann's "Elements of Large Sample Theory" and I have the following question about the text. The classical Central Limit Theorem is stated as: Now, the author goes on to provide the following counter-example: My question is, what assumption of the CLT does this counter-...

Limit. Sqrt (x^2+x) - sqrt (x^2-x) (x -> infinity) I get 2 answers for this question 1 and 0 but 1 is the right answer. I dont know why it is like tht though. If u multiply by the conjugate ÷ conjugate (1) u take the sqrt out of top and get it in the bottom and then if u factor out x from ...

The title just about explains everything. What is the 4x4 rotation matrix from i or (1.0, 0.0, 0.0) to an arbitrary unit vector, such as (0.424, 0.565, 0.707)?

I am studying for GRE and need help with following question When the positive integer n is divided by 3, the remainder is 2 and when n is > > divided by 5, the remainder is 1. What is the least possible value of n? Answer says 11, but gives no explanation. I tried to solve this, i.e. n = ...

How can I calculate the force exerted on the sides of a trough when the trough is filled with water, in general? Meaning for a given shape and depth of water, how can i calculate the force exerted on a given side of the shape?

Show that a finite group $G$ is solvable group (in the sense there exists an $n$ such that $G^{(n)}=1$) if and only the simples factors in a decomposition sequence of $G$ are all abelians. I'm not able to solve this problem. Is anyone is able to give me a hint?

I'm a grade 12 student in Canada. I studied Multivariable Calculus (Larson), Linear Algebra (Strang), Proving (Velleman), Abstract Algebra (Fraleigh) and Differential Equations (Simmons & Krantz) and I'm currently studying Probability and Statistics (Tsitsiklis). None of these courses are too r...

Assume p is a positive integer > 1. can I take the integral of tan^(p)*x without using reduction formula ? The same applied to sec^(p)*x Thank you.

let $w = A(x,y,z)dy\wedge dz$ where $y = h(x,z)$ and $(x,z) \in D = [0,1]^2$. Let $c$ be the singular 2-cube on A, i.e. $c: [0,1]^2 \to A$ is continuous. I am trying to calculate the pullback $c^{\star}w$ from definition of pullback I have that $c^\star(A) dy \wedge dz = A(c)(c^\star(dy \wedge ...

The problem said: An airplane has 120 seats. The probability that a ticketed passenger will show up for a flight is 0.95. Assume that all passengers act independently and that the airline has sold 130 tickets for a particular flight. Using the Normal approximation to the Binomial (wi...

Let $$ \left[ \begin{array}{ccc|c} a & 0 & b & 2\\ a & a & 4 & 4\\ 0 & a & 2 & b\\ \end{array} \right] $$ be the augmented matrix for a linear system. For what values of a and b does the system have: a) A unique solution; b) Infinitely many solutions; c) No solutions. Could you plea...

How can I expand $$f(z)=\frac{1}{e^z-1}$$ into Laurent series? I know that $f$ has singularities in $2k \pi i, \ \ k \in \mathbb{Z}$. Just substituting Taylor series for $e^z$ in the denominator doesn't give me much.

At the Christmas market, a man was selling nuts in a market stall. The first person bought one nut, the next customer bought two nuts, the next bought four, and so on. That is, every new customer acquired twice as many nuts as the previous one. Last customer of the day bought 50 kg of nuts,...

I've been reading up on some Analysis for my comp exams, and I couldn't find in my texts a proof of $L^\infty$ being Banach. Someone pointed me to the following exercise in Royden & Fitzpatrick. Now, I've found other proofs of $L^\infty$ being Banach online, but this problem is now really botheri...

I have a question about the Herschel-Maxwell derivation, as described in the text here: http://www-biba.inrialpes.fr/Jaynes/cc07s.pdf Specifically, equation (7-4). I understand why the probability should not care about the angle $\theta$, that is I understand that the integral over the density...

find the basis of the null space \begin{pmatrix} 16 & 32 & -40 & 24 \\ -3 & -6 & 7 & -4 \\ 80 & 160 & -176 & 96 \end{pmatrix} the row reduced echelon form is: \begin{pmatrix} 1 & 2 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{pmatrix} $x+2y-s=0\righta...

I know how functions can be described, e.g. $y=x^2$ and in high school they teach you the general form, e.g. $x^2-y=0$ I believe so that later on they can abstract the notion of the curve into a set-theoretic notion, e.g. $S=\{ (x,y) : x^2-y=0\}$ and then the graph is just the same as coloring th...

I have this integral $$\int \frac{\sqrt{x}}{\sqrt{1-x}}dx$$ I tried integrating it with integration by parts, using $u = \sqrt t$, trigonometric substitutions, but I'm stuck. Can you help me please?

I'm a freshman taking calculus 1 currently studying for finals. I am reviewing stuff from the beginning of the semester,and I don't remember the proper way to deal with limits like this one. A ball dropped from a state of rest at time t=0 travels a distance $$s(t)=4.9t^2$$ in 't' seconds. I am t...

How would I be able to describe all units of the group ring $\mathbb{Q}(G)$ where $G$ is specifically an infinite cyclic group?

The problem said: In a certain region, blue cranes are twice as common as whooping cranes. Suppose that the number of eggs laid by a blue crane is a Poisson(! = 3) random variable and the number of eggs laid by whooping crane is a Poisson(! = 5) random variable. You find a crane’s nest ...

∫ 1/((1-x^3)^(1/3)) dx I tried substituting 1-x^3 as t^3 but I am not able to calculate it after that. Thanks!

The tires located on the front of the car wears out after 25000 km, while the tires on the back wears out after 15000 km. How far can you maximum ride with new tires if you can swap the tires during the journey?

How do I prove that the unit speed curve is a helix?

I have this integral that I need to solve: $\int_a^b \sum_{k=1}^K \max(x,a_k) \text{d}x$ for some constant $a_k\in\mathbb{R}, k=\{1,\dots,L\}$.

Are there any know values of $n$ for which there exists a surjective quadratic mapping $Q:\mathbb{R}^n \rightarrow \mathbb{R}^n$ with non-trivial zeroes?

I have a problem with this exercise, I have been thinking about to use some propierties of proyective plane and conics, but I dont know how to solve it. can you help me please? the exercise is: "proof that if a triangle is circumscribed to a conic projective nondegenerate then the lines connecti...

Let $R=K[x,y]/(f)$ where $f(x,y)=y^2-x^3$ I can show R is integral domain and Noetherian. But I have to show every non-zero prime ideal of $R$ is maximal ideal. but I can not realize form of ideal of $R$ I try to choose a prime ideal $P$ and want to show $R/P$ is (finite integral domain so is f...

Given $\sigma : G → (G/M) $ x $G/N$. Define $\sigma$. Can someone please check if I have defined it well. $\sigma(g) = ( g + M, g+ N)$ or $\sigma(g) = ( gM, gN)$ ? would either work given $M,N$ are normal .

Problem said: Urns I, II and II contain three pennies and four dimes, two pennies and five dimes and three pennies and one dime, respectively. One coin is selected at random from each urn. (a) What is the probability that all three selected coins have the same denomination? P(a:s...

It is given that $\frac{1}{2^j}>\frac{1}{2^{j+1}}$. So is it true that $\sum\limits_{n=j+1}^\infty \frac{1}{2^i}<\frac{1}{2^j}$?

Let $\varepsilon>0$. I was interested in understanding the justification of defining the following function $\phi$ via its Fourier transform, satisfying the following properties: (1) $\widehat{\phi}\in C^\infty(\mathbb{R})$ (2) $\widehat{\phi}(\xi)=1$ for $\xi\in[-\pi+\varepsilon,\pi-\varepsil...

Let $\{y,...,x\}$ be a finite subset of the natural numbers. Does this imply that $$\sum_{l=y}^{x} \prod_{ p \in \{y,...,x\} \backslash\{q\}} \prod_{q \in \{y,...,x\} \backslash \{l\}} (\delta_{q}-\delta_p)=0$$ for an arbitrary sequence of numbers $(\delta_n)_{n \in \mathbb{N}}.$ I have the f...

Let $N(T) \sim P(\lambda T)$, and $T\sim U[0,1]$. Compute $E(N^\alpha(T)|T=t)), \, E(N(T))$, where $\alpha \in \mathbb{N}$. Clearly that $\{N(T)|T=t\} \sim P(\lambda t)$, hence $E(N^\alpha(T)|T=t)$ is the $\alpha$'s moment of $P(\lambda t)$. For the second case, \begin{align} E(N(T)) &= E\lef...

page 6 in this notesays: A note on index convention: $i$,$j$, and $\bar{i}$, $\bar{j}$ index over the holomorphic and antiholomorphic components. I never knew there were holomorphic components, I only was aware of holomorphic mappings and holomorphic functions. Can someone please point ou...

Determine a constant $k$ such that the polynomial $$ P(x, y, z) = x^5 + y^5 + z^5 + k(x^3+y^3+z^3)(x^2+y^2+z^2) $$ is divisible by $x+y+z$. In this multi-part problem, we will consider this system of simultaneous equations: 3x+5y−6z5xy−10yz−6xzxyz===2,−41,6.(i)(ii)(iii) 3x+5y−6z=2,(i)5xy−10yz−6x...

Do you have an example of signed measure that take negative value ? I didn't found any example. By the way, do you have an explanation of those signed measure ? I don't see any utility.

I'm now extreamly tierd of not pulling off this equation. $\sum_{i=1}^n (y_i-\alpha)^2= \frac{2n\sum_{i=1}^n (y_i - \alpha)}{\sum_{i=1}^n (\frac{1}{y_i - \alpha})}$ Solve for $\alpha$, y is a stochastic variable.

I am reading Amann's book of Analysis I and I am trying to prove the following: For each non empty set X , the function P(X)↦{0，1}^x，A↦X_A is bijective Where P(X) according to the Book is the power set of X , X_A is the characteristic function of A defined: X_A:X↦{0,1}^x ,x↦1 if x∈A or x↦0 if ...

How do I prove this? For the Fibonnaci numbers defined by $f_1=1$, $f_2=1$, and $f_n = f_{n-1} + f_{n-2}$ for $n$ ≥ 3, prove that $f^2_{n+1} - f_{n+1}f_n - f^2_n = (-1)^n$ for all $n$ ≥ 1.

Let $G$ be a smallest grammar for a string $s$ that has two terminal rules: $A = abc; \ B = bcd$ Add the rule $C = bc$ to the grammar and also change the rules to $A = aC; \ B = Cd$. Then we still have a minimal grammar for $s$, $G'$. Clearly given $G'$ we can compute $G$ in $O(k)$ time where...

Consider the following recursive definition of a function: g:N→Ng:N→N. 1) Base Case: g(0)=0g(0)=0, 2) Recursive Case: For any x>0x>0 we have g(x)=g(x−1)+2g(x)=g(x−1)+2. Prove each of the following properties holds for gg using induction on n∈Nn∈N. i) g(2*n) = 2*g(n)

Given two integers: n and m and n is divisible by 2m, I have to write down the first n natural numbers in the following form. At first first m integers are taken and their sign negative is made negative, then next m integers are taken and their sign is made positive, the next m integers should...

Is there a way in which we can guarantee 3 reflections or fewer will map any two congruent rectangles (ABCD and WXYZ) to each other with?

How does one solve x²-3 and x²+x-1 in $F$7[x] / (x²-5) ? Not sure how do do it, I know there is a solution for both, but I don't see how to get there. Thanks.

How to show the following problem: My approach is following: From the dexcription of the problem, $g(x)$ should be non-decreasing. By construction, let $g(x)=\text{sup}_{r\leq x} g(r)$ to make sure $g(x)$ is non-decreasing. Then, how to do the following steps?

I'm trying to find out for which values of $\alpha$ and $\beta$ does the integral $\int\limits_2^{\infty}\frac{dx}{x^{\alpha}ln^{\beta}x}$ converge. I know that when $\alpha=1$ then $\beta$ must be greater than $1$. I tried to use integration by parts but It didn't work, so I would appreciate so...

This is a problem out of Logan's Applied Math book. Section 6.7, problem 2. Show that for any locally integrable function f on $\mathbb{R}$ the function $u(x,y) = f(x-y)$ is a weak solution to the equation $u_x + u_y = 0$ on $\mathbb{R}^2$. I've got a solution attempt that expresses $f_x$ ...

How to integrate double integral $$\int_{0}^{\infty}\!\int_{0}^{2\pi}\ \frac{1}{2}\left(\frac{\partial}{\partial x}-\frac{\partial}{\partial y}\right)g_m \bar{g_n} , d\theta dr$$ where $$g_a=(x+iy)^a$$ . I do not know how to differentiate first part of integral.

I'm trying to calculate the blowup of the curve y^5=z^2-3z^3+2z^4 at (0,0) We have the relation Ay=Bz, now I split it into two charts: The first chart(y,a=A/B): y^5=a^2y^2-3a^3y^3+2a^4y^2-y^3=y^2(a^2-3a^3y+2a^4y^2-y^3) The second chart (x,b=B/A) 0=z^2-3z^3+2z^4+b^5z^3=z^2(1-3z+2z^2+zb^5) Now t...

I have the solution to the exercise but have a doubt on one thing, I state the exercise: Given $$ X = \{ (x,y) \in R^2 | x = \frac{1}{n}, n \in N \}$$ and $Y = X/_{\sim}$ where the equivalence relation is $(x_1, y_1) \sim (x_2,y_2) \iff (x_1, y_1) = (x_2,y_2)$ or $ y_1 = y_2$ with $x_1,x_2 \in...

I#m trying to compute the genus of the normalization of the curve: $y^5=x(x-1)(x-2)$ Now I calculate the ramification points of the projection x: they are $(0,0),(1,0),(2,0)$ and they are of ramification order 5, and possibly ramification points at the infinite point $(1:0:0)$. Now the curve is ...

$lim_{x\to 3}(\frac{x}{x-3}\int_{3}^{x}\frac{sint}{t})$ without using Hopital

A = 6 -1 -5, -1 0 -1, 7 -1 -6

find the contour integral $$\oint _{c} \frac{\sinh z}{z-1} dz$$, where C is a square of side 3 centered at the origin I have problem both with finding the residues and doing the integral

I have a map α : $F$[x] -> $L$ , where $L$ is $F$[x]/(x²+2x-1) and $F$[x] is $Z$/5$Z$, and what it does is x -> t, where t is a square root of 2 in $L$. I need to prove it is a ring homomorphism. For that, I need to show first that α(1) = identity of L, then that α(a+b) = α(a) + α(b) and finally...

Hi there im trying to solve the following limit $\lim \limits_{n \to \infty} \frac{(4(n*3^n + 3)^n}{(3^{n+1} (n+1)+3)^{n+1}}$, but I've got literally no idea where to even start, it's just too big! I don't know wether l'hospital would be a good idea considering, well, look at those functions. I a...

can someone please help? I'm taking Calculus, but I'm really having trouble understanding the concept of related rates. A jogger runs around a circular track of radius 55 ft. Let (x,y) be her coordinates, where the origin is the center of the track. When the jogger's coordinates are (33, 44), he...

Suppose $f$ is differentiable on $\mathbb{R}$ and its derivative $f'$ is continuous on the interval $[a,b]$. What constraints on $f$ would such condition give us?

I am given a system of linear inequalities $Ax \le b$ where $A$ is a $m$ by $n$ totally unimodular matrix. Let $A_1,\dots, A_m$ be the rows of $A$ and let $H_1,\dots, H_m$ be the hyperplanes in $\mathbb{R}^n$ corresponding to $A_ix = 0$. Suppose I know that the feasible region $P$ has the same di...