Does there exist a sequence $\{n_k\}$ of positive integers such that $$\sum\limits_{k=1}^{\infty}\frac{1}{n_k}<\infty,\space \sum\limits_{k=1}^{\infty}\frac{1}{n_{n_k}}=\infty\space$$

This shows up as problem 2.2.26 in Hatcher's Algebraic Topology. Given a pair $(X,A)$ let $X\cup CA$ be $X$ with a cone $CA$ attached at $A$. Suppose that $A$ contractible in $X$. I want to show $H_n(X,A)\cong \tilde{H}_n(X)\oplus \tilde{H}_{n-1}(A)$. What I know: If $A$ contractible in $X$ th...

What is the definition of an evaluation oracle in complexity theory?

Given symmetric $A,B\in\Bbb R^{n\times n}$ what is a good description of colection of $X\in\Bbb R^{n\times n}$ such that $$XX'=I$$ $$AX=XB$$ holds?

Express the indefinite integral $\int e^{-x^2}dx$ using function Φ(x). Φ(x) is the following special function : Φ(x) = $1/2$ +($1/\sqrt{2π}$)$\int_0^x e^{-t^2/2}dt$

I posted a question earlier Ring of polynomial functions on unit hyperbola is PID but nobody responded to it and it seems like an interesting question. Maybe somebody could take a look?

I cannot understand why, if $\limsup_{n\to\infty} \ x_{n} =-\infty$ then $\lim_{n\to\infty} \ x_{n} =-\infty$? Can anybody explain it? What's the relationship between $\limsup$ and $\lim$?

I just want to be sure if i'm right so i have: For 4 digits: C(10,4) = 210 - to get number of combinations of 4 different digits 1 digit repeat 2 more times - C(4,1) = 4 number of other combinations - P*(3,1,1,1) = 120 2 digit repeats 1 more time - C(4,2) = 6 number of other combinations - ...

I've been thinking lately about the pedagogical differences in learning things informally/intuitively first vs rigorously first. It makes sense to take a few calculus courses before tackling analysis, but assuming one has a certain level of maturity (at least mid-upper level undergrad) do you thi...

How can I show using natural deduction that: $a = b \vdash f(a) = f(b)$?

> If $A$ is a self-adjoint linear transformation on the finite-dimensional inner product space such that $A^{n}=I$ for some positive integer $n$, proof that $A^2=I$.

How do I prove that for $V$ a complex vector space and $A\in\text{End}_{\mathbb{C}}(V)$ such that $A^m=I$ that $A$ is diagonalizable?

I have this problem, I am not 100% sure how to figure it out. This is the problem!

I saw that there are "two" definition of derivative: $$lim_{x\to a} \frac{f(x)-f(a)}{x-a}$$ and $$lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ for a derivative in a specific point which definition should I use? if the two are equivalent how can I prove that $xsin(\frac{1}{x})$ is not differentiable u...

I don't know what this symbol mean in terms of high school geometry. I've search in google for it but I was unable to find anything about it(in terms of geometry).

I have answered part (i) and (ii). Is there any way to do part (iii) using the first two parts?

From Artin's second edition of Algebra. Let G be the dihedral group $D_4$$ of symmetries of a square. Is the action of G on the vertices a faithful action? on the diagonals? My solution: Since every permutation of a square (except the identity permutation) will alter at least two of the vertic...

If $\mu$ is $\sigma$-finite and $f_n\rightarrow f$ a.e., there exists $E_1,E_2,\ldots\subset X$ such that $\mu\left(\left(\bigcup_{1}^{\infty}E_j\right)^{c}\right)$ and $f_n\rightarrow f$ uniformly on each $E_j$. Proof: Since $\mu$ is $\sigma$-finite, then there exists an $$X = \bigcup_{1}^{\inf...

My range is .20 to $4.80 My median is 2.10 My mean is 2.10 Help me Thanks

I am stuck on this one, I know I have to use Ax3 which is (∀x)(A→B)→(∀x)A→(∀x)B, and convert the existential quantifier to universal, but I have problem making it to become just as Ax3

I'm going through some routine exercises in studying smooth manifolds. This one is 12.27 (d) from Lee, Introduction to Smooth Manifolds If $F:M\to N$ is smooth, and $B$ is a covariant $k$-tensor field on $N$, show that the pullback $F^*B$ is continuous, and smooth when $B$ is smooth. I thin...

I have done part (i) but have absolutely no idea how to do the rest. Could you provide hints. Thanks

Let X be Hausdorff space and X/~ be "~ Quotient Spaces", Given that X/~ is T1, Does it satisfy for X/~ to be Hausdorff?

Here is the background of formula (the text is not important, my questions is in the formula): enter image description here Why 1/p(H) * ▽_Θ p(H) = ▽_Θ log p(H) ???

Is there a proof or counterproof of the following statement? An integer $i\in$ $Z^+$ exists such that $a*b=i$ and $c*d=i$ where $a,b,c,d\in$ $Z^+$ and $a\neq b\neq c\neq d\neq 1$ .

Let $f(x)=\sum \limits_{n=1}^{\infty}\dfrac{\{nx\}}{n^2}$ where $\{\}$ is fractional part. Find all discontinuities of function $f(x)$. I think that $f(x)$ is discontinuous at every rational point. Can anyone show how to prove strictly that $f$ for example is discontinuous at $0$ or $1/2$?

I have the function f(x,y)=exp(-x^2-y^2) I have no idea how to find the center of the circle and the radius. I know the formula is (x-h)^2 + (y-k)^2 = r^2 but I can't reformat my equation in this form.

My Maple input limit(sin(1/n)*n,n=infinity); says 1. I don't understand why $$ \lim_{n \to \infty} \sin\left(\frac{1}{n}\right) \cdot n = 1 $$ I know that $\lim_{n \to \infty} 1/n = 0$, so it kind of says "0 * infinity = 1". Have I overlooked some rewriting of $\sin(1/n) n$?

There exists some proof for the theorem. Some of them use Transfinite Recursion. Some of them use same argument with the following. Proof:(copied from proofwiki) "Let $S$ be a set. Let $\mathcal{P}(S)$ be the power set of $S$. By the Axiom of Choice, there is a choice function $c$ defined on $...

Can lines intersect more than once in the Projective plane?

I know trilinear coordinates of schiffler point but I dont know how to get it. Do you have any source about it?

Let $f, f_1, f_2, \dots, f_n, \dots$ be continuous applications $f_i: M \rightarrow N$, $ f: M \rightarrow N$. Then, the following affirmations are equivalent: $(1)$ If $x_n \rightarrow x$ in M, then $\lim_{n \rightarrow +\infty} f_n(x_n) = f(x)$ $(2)$ $f_n \rightarrow f$ uniformly in each $K \s...

Is the sum of area of any regular polygon made in the two sides of a right angular triangle equal to the area of the same polygon made at the hypotenuse?? if so how to prove it?the area was equal to the semi circles with diameters of the hypotenuse and the other two sides, does it apply with any ...

Show that the force of a spring ${\bf{{F}}}(x) = -k(x-l){\bf\vec{i}}$ has a potential $V(x)$, such that ${\bf{F}} = -\nabla V$.

Here is my issue. I can do modular arithmetic. I wrote a short essay on the topic. My problem comes when trying to understand equivalence classes. So in the set of positive integers, and using mod 17. Then the equivalence class of the set of positive integers mod 17, are the integers that are d...

Let $f:(0,\infty)\to\mathbb{R}$ be differentiable.if $f'(x)\to l $ as $x\to\infty,$then show that $\frac{f(x)}{x}\to l $ as $x\to\infty$ . i have no idea where to start.any hint please

Let $U$ be an open set and $f:U\to\mathbb{C}$ be a holomorphic function with real part $u(x,y)$ and imaginary part $v(x,y)$. Is it possible that $u(x,y)^2=1+v(x,y)^3$ for $x+iy\in U$ We have Cauchy-Riemann equations. But I don't know how to use them. Can anyone give any hint? thnx for your help.

Given $$\nabla\cdot(\textbf{r}\times\nabla f)~=~(\nabla\times\textbf{r})\cdot\nabla f~-~\textbf{r}\cdot(\nabla\times\nabla f)$$ I would split the equation into 2 part: $(1)~~~~~~~~~(\nabla\times\textbf{r})\cdot\nabla f = [\epsilon_{ijk}\frac{\partial x_k}{\partial x_j}]\partial_if=0\cdot\parti...

Find the coefficient of x^3 in the expansion ((1+x)^3).(2+x^2)^10 I did the first part: which is expanding the second equation at x^3: (10C3).2^7.(x^2)^3 15,360(x^2)^3 but i cant figure out what to do from here..

The question has a restriction that only the method of emliminating ∃ can be used.

$\sum_{n=2}^\infty \dfrac{1}{ln(n!)}$ How do I start with this series? I can use any method to solve this problem. When I try using Ratio Test I get stuck with: $\lim\limits_{n \to \infty} \dfrac{ln(n!)}{ln((n+1)!)}= \infty$ I also tried using Comparison Test where $b_n=\frac{1}{ln(n)}$ but ...

i have a hard time on factorizing elements from Z[i] in special -19+43i. I know that the primes in Z[i] are: 1+i. p from N, p=4k+3 , k integer ( p=3(mod4) ). a+bi from Z[i], p=N(a+bi)=a^2 + b^2 and p=4k+1, k integer ( p=1(mod4) ). I wonder if there is an algorithm that tells you how to fact...

Prove there exists infinitely many real numbers $a$ such that $a(a - 3\{a\})$ is an integer. {a} = fractional part of a number. Here is my proof. Are there any fallacies that I’m missing? Alternate proofs are also welcome :) Let $S$ be the set of reals obeying this property. Let $\lambda$ b...

It is well known that in commutative rings, maximal ideals are prime. Can we give an ring $R$ such that every prime ideal of $R$ be maximal with $|\operatorname{Max}(R)|=\infty?$

I am trying to find the gcd of $$f(x)=x^{4}+5x+1$$ and $$g(x)=x^{2}-1$$ in $\mathbb{Z[x]}/7\mathbb{Z}$ To do such, I tried using Euclidian Algorithm, and first I divided f by g to get $x^{2}+1$ and a residue of $5x+2$ Then I divided $g$ by $5x+2$ to get $x^{2}-1$ and 0 residue So if I didn't...

Here's my problem: Let G be a subgraph of $K_{20,20}$. If G has a perfect matching, prove that G has at most 190 edges that belong to no perfect matching. See here for a more generalized version of this question, i.e. there're at most $n \choose2$ such edges in any $K_{n,n}$. However, I don't und...

Let $A_{3\times 3}$ be a matrix with the characteristic polynomial $f_A(x)=(x-1)(x-2)(x-3)$, what is $f_{A^{-1}}(x)$? The eigenvalues of $A$ are $1,2,3$ and since $Av=\lambda v$ then $A^{-1}v=\lambda^{-1} v$ so the eigenvalues of $A^{-1}$ are $1,\frac 1 2, \frac 1 3$ and therefore $f_{A^{-1...

I have recently come across the substitution $u = \tan \frac{1}{2}x$. It is said that the substitution should be used on rational functions of sin and cos. I'm wondering what exactly this means. For instance can there be sin's and cos' both in the numerator and denominator, do they have to be lin...

Use polar coordinates, taking x=rcosθ and y=sinθ to evaluate ∬ dx dy where R is the interior of the circle x^2 +y^2 =1

Find $g^{-1}(x)$ in terms of $f^{-1}(x)$ if $g(x)=1+f(x)$ I find it hard to operate inverse functions. So can anyone show me a detailed process?

We are given the sides of a polygon. We need to determine if the given polygon is convex or concave . How can this be done? What is the propery applied to determine this?

Your task is to design a rectangular industrial warehouse consisting of three separate spaces of equal size. The wall materials cost 72 dollars per linear foot and your company has allocated 34560 dollars for those walls I am supposed to use the whole budget and maximize the total area. I need...

Let $0<y<1$ be arbitrary. What is the weak solution of the differential equation $-u''+u=\delta_y$ where $u(0)=u(1)=0$ then? The weak form of the equation above is given by $\int_0^1{u'(t)v'(t)+u(t)v(t)dt}=v(y)$ for all $v\in H_0^1((0, 1))$.

Spanning tree I've been trying to figure out this question, but no matter how many times I try to answer the question, the end result will always be a circuit. Can anyone help me?

derive $x^{ln(x)-1}$ so first I have simplify the expression: $x^{ln(x)-1}=e^{ln(x^{ln(x)-1})}=e^{ln(x)\cdot (ln(x)-1)}=e^{ln^2(x)-ln(x)}$ than $(e^{ln^2(x)-ln(x)})'=e^{ln^2(x)-ln(x)}\cdot (ln^2(x)-ln(x))'=e^{ln^2(x)-ln(x)}\cdot (ln^2(x)-ln(x))'=e^{ln^2(x)-ln(x)}\cdot (\frac{2}{x}-\frac{1}...

How to get numeric value of integration: eq:=int(a+b*x^(4/5)+c*exp((1+x)^(1/7)),x=1..2) I tried to do: assume(a,real,b,real,c,real); evalf(eq,5); But it doesn't work (with output same as input). Thanks in advance,

I am trying to find all equilibria and determine witch are asymptotically stable. enter image description here So can someone please help me?

I am preparing for test so I am posting some of my many questions, my apologizes if it is considered to many. I am wondering, Could we have a transitive action of the group $S_{4}$ acting on the set $\{1,2,3,4,5\}$? What I know is that for there to be a transitive action we would need that for...

Given f(x) =(x²-2)/(4x²+mx+4) determine the interval of m which makes f continuous for all real numbers.

It is true that the set of convergence sequences is a Noetherian ring?

Induction question My solution for question I couldn't understand how to determine which condition to apply l+1 or m+1 in second induction step

I am just starting a new topic in group theory and I want to understand what is a group extension. I have the following exercise (probably simple): Check that $Z_{mn}$ is an extension of $Z_{m}$ by $Z_{n}$. I want to know what is the method of solving these kind of problems.

The exam question goes as follows: A test is done to check for a predicted atomic spectral line by counting the number of photons emitted from a sample in a narrow frequency range. The hypothesis under test is that no such spectral line exists. Due to background light, the detector will also ...

Q: an urn contains 2 red and 3 black balls. Players 1 and 2 withdraw balls from the urn consecutively without replacement until the second red ball is selected. player 1 draws first, then player 2, and so on. Find the probability that player 1 selects the second red ball My approach: if i let P_...

For any a ∈ R, let f_a : R → R be the shift function defined by f_a (x) = (x−a). Show that f is continuous if and only if, whenever a sequence of real numbers {a_n} converges to zero, f_an converges pointwise to f. [I'm stuck on this problem, therefore a full solution is appreciated]

Suppose $(X,M,\mu)$ is a measure space, and that $f$ is a real-valued function on $X\times \mathbb{R}$. Show that if the sequence $f_x$ are continuous functions for every $x\in X$ and the sections $f^{y}$ are measurable functions for every $y\in\mathbb{R}$ then $f$ is measurable with respect to t...

I'm currently taking an abstract algebra course that I am finding challenging although very interesting. I am reviewing the sections pertaining to Rings, Integral Domains, and Fields, and am a little confused as to how the computational exercises at the end of each section relate to the concepts...

Let $\Omega \subseteq \Bbb R^N$ be open, $\omega = \sum_{j=1}^n \omega_j\,{\rm d}x_j$ be a $1$-form in $\Omega$ such that $\sum_{j=1}^N|\omega_j(x)|\neq 0$ for all $x \in \Omega$, and $\theta = \sum_{i=1}^n \theta_i\,{\rm d}x_i$ be another $1$-form in $\Omega$ with $\theta \wedge \omega = 0$. ...

I need some help on proving a theorem about topologically equivalent metrics. It seems that somehow I am missing a final but vital step on the proof, so I am going to write the theorem and represent my attempt to prove it. Theorem: Consider a non empty set $\mathbb{X}$ and two metrics $...

In the triangle above : $$XY \parallel BC$$ $$XO \parallel AC$$ Now, Is it true? $$\frac{OX}{YC} = \frac{OY}{BY}$$

I'm having real trouble finding MLE of B0 and B1 for a regression estimator with a random x In other words, using E(Y|X)=B0 + B1'x and covariance sigma^2, find the MLE. I know how to plug it into the multivariate normal and stuff, but I'm having trouble solving for it.

Why is this true? It seems like the inverse of an image should be in G and retain all the properties of a group.

Find the maximum $c\in[1,+\infty)$ such that for every sequence $\{x_k\}$ satisfying $\space0\leqslant x_{n+m}\leqslant c(x_n+x_m),\space\space\forall m, n\in\mathbb{N}\space\space$ it follows that $\space\exists\lim\limits_{n\to\infty}\frac{x_n}{n^c}<\infty$.

derive $10^xlog_{10}(x)$ $10^x\cdot ln(10)\cdot log_{10}(x)+\frac{1}{x\cdot ln(10)}$ Is it right?

n is a prime and then (n-1)!+1 is divisible by n. How to prove that? Hint: [Zn{0},*n] is a cyclic group of order (n-1)

It is given that order of exactly eight elements of group G is 3. We have to find the number of subgroups of order 3?

The time it takes to service a car is an exponential random variable with rate 1. If A.J.'s car and M.J.'s car are both brought in at time 0, with work starting on M.J.'s car only when A.J.'s car has been completely serviced, what is the probability that M.J.'s car is ready before time 2? The a...

You invested $968 710 in a treasury bill with the face value of $1 000 000 with 91 days left till maturity. After 60 days you have the option to sell it for $989 250. Which option is more profitable? My solution: r1=(1000000-968710)/((91/360)*968710)=12,78% r2=(989250-968710)/((60/360)*968710...

I'm learning about dual problems and was trying to get to an understanding of how to take the dual of a problem that has a sum in it. For example if we try to optimize the sum of all values while keeping them under a given total max sum i=1...n(sixi) where sum i=1...n(pixi)<=p x>0 how wou...

Write the following subgroups as a direct sum or direct product of $p$-groups: $$\text{a)}\ Z^{*}_{15},\ \ \text{b)}\ Z_{12} \times Z_{12} \times Z_{60}$$ where $Z^*{n}$ is a standard additive group modulo $n$ and $Z_{n}$ is multiplicative group modulo $n$.

The following is a question from a past exam that I am studying: Two firms A and B make color and black and white television sets. Firm A can make either 200 color sets a week or 200 black and white sets. Firm B can make either 400 color or 200 black and white and 200 color, or 400 black and wh...

In a homework sheet with true or false questions I have found the folowing statement $\displaystyle \lim_{h \to 0}\left[ f(x+h)-f(x) \right]=0$ True OR False At the first shight of it True seems the right answer but then this came to my mind If $f$ is not continues at $x$ then $$\lim_{x ...

I'm trying to find an open affine susbset whiche is not principal. the idea is to consider the ring $\displaystyle\frac{l[u,x,y,z]}{yz+uy^{2}+xz^{2}}$ (where $l$ a field) and show that $V:=D(y)\cup D(z)$ is non principal. Hartshorne has a criterion for affineness in the page 81 (ex 2.17). I hav...

What are the main differences between these two areas? Does geometric topology in general, use more analytic techniques? Which one would most consider harder?

what would the answer of this this be in the form of NAND only https://gyazo.com/3b3c87dbfe86203d032f8cf37fe0c44b

I need to model human hairs in Matlab by solving ODEs, but I'm struggling to understand what to do. Here is all of the provided information: I've been asked to write a function returning the $(x,y,z)$ coordinates of the hairs given $L$, $R$, $f_x$ and a list of values for $\theta(L)$ and $\phi...

I have big problem with the the limit of the following function: \lim_{(x,y)\to (0,0)} \left(\frac{x^2 y}{x^4 + y^2}\right) I have tried to estimate lower and upper bound for Three-Series Theorem and to to convert it to polar coordinates and it didn't work well :)

A closed rectangular container with a square base is to have a volume of 2000 cubic centimeters. It costs twice as much per square centimeter for the top and bottom as it does for the sides. Find the dimensions of the container of least cost. So the formula for the volume is: 20...

How to show that the extension of group $Z_{2}$ by $SO(n)$: $$Id \rightarrow SO(n) \rightarrow O(n) \xrightarrow{det} Z_2 \rightarrow 1$$ is a direct product for odd $n$ and indirect product for even $n$.

I find this tutorial for chi square distribution and excel solver and i dont now if i can use it for solving Newton's law of cooling ?

The gross weekly sales at a certain restaurant are a normal random variable with mean $2200 and standard deviation $230. What is the probability that a. the total gross sales over the next 2 weeks exceeds $5000 b. weekly sales exceed $2200 in at least 2 of the next 3 weeks? Don't just give me ...

What is the least common multiple of 5! and 6!

I have a equation of motion for a forced pendulum show below ((d^2*theta)/(dt^2)) = -(g/L)sin(theta) + C*cos(theta)sin(Dt) L=10cm, C=2(s^-2) and D=5(s^-1) I am trying to make this equation dimensionless by setting the follow equations (omega)^2 = g/L , beta=D/omega , gamma=c/(omega)^2...

is there any one can helps me solve this problem of integral $$ \int \frac {x^2-y^2}{(x^2+y^2)^2} dy $$ ................................................................................................. ............................................................................................

I would be very grateful if somebody could verify my proof. Assume that $f$ is continuous at $p$. We can always choose $\delta_1>0, \delta_2>0$ such that $d'(f(x),f(p))\le\frac{1}{2n}$ whenever $d(x,p)\le\delta_1$ and $d'(f(y),f(p))\le\frac{1}{2n}$ whenever $d(y,p)\le\delta_2$. Set $\delta=\...

I have calculted $dw = D_1f_1 + D_2f_2 + D_3f_3 $, but I am not sure what I can conclude if the sum is equal to $0$. Anyhelp please.

Let p be a prime where $p$ $\neq$ 2. Prove $U(p) = U(2p)$ and $U(p^2)=\mathbb{Z}_p $ X $\mathbb{Z}_{p-1}$ I know $U(n)$ is the set of all numbers relatively prime to n. I know this statement is true, it is quite obvious, but I am having trouble proving. Can you help?

Let $0 < a < b$ and $T\colon L^\infty((0,1)\times (a,b)) \to L^\infty((0,1)\times (a,b))$ be the operator defined by $$Tf(x,y) = \begin{cases}f(x+\frac yb,y), &0<x<1-\frac yb,\\\frac 12f(x+\frac yb-1,y),& 1-\frac yb<x<1,\end{cases}$$ where $x \in (0,1)$ and $y \in (a,b)$. Is it true that $$...

Show by transfinite induction that there exists a Bernstein set $B$ such that $B + B$ = R, where $B + B$ = {$b_0 + b_1$ | $b_0$, $b_1 \in B$}. I believe I need to enumerate the family of Perfect sets, P such that ($B + B$) $\bigcap$ P $\ne$ $\emptyset$ and P $\not\subset$ $B + B$, as well as $B ...

I'n not really sure how I'm supposed to prove this. All I know is that I need to use Ramsey's Theorem.

I've been working on some knot invaririants and specialy the Jones Polynomial. I was able to prove that $ V_{K_1 \# K_2} = V_{K_1} V_{K_2} $ for two knots $ K_1 $ and $ K_2 $ . So I found my self asking if the same raltion holds in the case of links. All books states that indeed $ V_{L_1 \# L_2} ...

A die is rolled five times. What is the probability of obtaining at most three 4’s?

Suppose f: [a,b]$\rightarrow \Re $ is continuous. Let M=max{|f(x) : x$\in$[a,b]} Show that $$\lim_{n\rightarrow \infty}(\int_a^b |f(x)|^n)^{\frac{1}{n}}=M$$ My attempt: Suppose for contradiction that $\exists$ c$\in$[a,b] such that $$f(c)\gt \lim_{n\rightarrow \infty}(\int_a^b |f(x)|^n)^{\frac{...

What are the matrices $A \in \mathscr{M}_n(\mathbb C)$ for which the similarity class is closed? What about the same question if we replace $\mathbb C$ by $\mathbb R$?

Show that $L^p([a,b])\subset L^r([a,b])$ for any $1\leq r<p\leq\infty$. I thought this, or something similar, was a theorem somewhere, but now I can't find anything on it. If anyone knows of a link or something, please put it in a comment below. Now, a real question: how do I prove this? It s...

I have a complex equation and need some help separating it into two first order differential equations. My equation is below (((d^2)*theta)/dt^2) = -sin(theta) + (gamma)*cos(theta)sin(betax) How do I separate this into two first order differential equations? I have no idea on the theory of t...

We have a linear operator $T: V \to V$. Suppose $W_j$ is the nullspace of $(T - \lambda_j I)^{r_j}$ for some $0 < r_j$, where $\lambda_j$ is a root of the minimal polynomial of $T$ of multiplicity $r_j$. I would like to show that $T(W_j) \subset W_j$. Here is my attempt: We want to show that $\...

In a large population of patients, 20% have early stage cancer, 10% have advanced stage cancer, and the other 70% do not have cancer. Six patients from this population are randomly selected. Calculate the expected number of selected patients with advanced stage cancer, given that at least one of...

We define ∼: Z × Z as x ∼ y ⇔ |x − y| is divisible by 7. What is [1] ∼ and Z /∼ ? Is (13, 4) ∼ × ∼ (6, 9) where : X × X as x ∼ y ⇔ |x − y| is divisible by 5? Please explain.

Am I right in thinking that since H is a cyclic subgroup, then h=g^a is a generator of H? Therefore it must contain all positive powers of h.

I am getting very confused with signs of Damped Motion. The question goes like this: A light elastic string, of natural length $2a$ and modulus of elasticity $mg$, has a particle P of mass $m$ attached to its mod-point. One end of the string is attached to a fixed point A and the other end is a...

I have the question: Given that f(r) is a twice differentiable scalar function of r = |r|, r = (x, y, z), and f′ = df/dr and f′′ = d2f/dr2 are its first and second derivatives, show that ∇ · ∇f(r) = f′′ + (2/r)*f′ I have been given the result that ∇f(r) =f′ * rhat (where rhat is the unit positio...

Is there any transformation that could turn this into a convergent series for s>0

Is $2 = 123$ modulo $6$? How do I attack this question? Do I just answer: What is $123$ MOD $6$? The answer is 3. So: Is $2 = 123$ modulo $6$? Answer is NO!

Prove that $f(x)= \sqrt{4+x^2}$ is continuous at $x_0$ using the $\epsilon -\delta$ definition of continuity. Textbook proof: $|\sqrt{4+x^2}-\sqrt{4+x^2}|=\frac{|4+x^2-(4+{x_0}^2)|}{\sqrt{4+x^2}+\sqrt{4+{x_0}^2}}=\frac{|x^2-{x_0}^2|}{\sqrt{4+x^2}+\sqrt{4+{x_0}^2}}=\underbrace{\frac{|x+x_0|}{\sq...

What is the anti-derivative of x^(x-1)?

If I localize at a minimal prime ideal $p$ in a noetherian ring $A$, is $A_p$ finitely-generated as an $A$-algebra?

I was reading a paper having the following wave equation $ u_{tt} - \Delta u = 0 $ then multiplied by $ u_ {t} $ and is $\frac{d}{dt} \frac{1}{2} ||u_ {t} ||^2+||\Delta u ||^ 2 = 0 $ My problem is with the term or $ ||\Delta u ||^ 2$, I can not reach him, I tried using the definition of Lapl...

I cant find the answer from the textbook, can anyone help me with this... I don't know if this could be the answer: $$p= \frac {a•b}{a•a} a$$

Find the upper and lower sums for $f\left(x\right)=\begin{cases} 1 & 1\leq x<3\\ 5 & x=3\\ 3 & 3<x\leq4 \end{cases} $ Attempt $I_{1} = [1,3-a] \ with \ M_{1} = \sup_{I_{1}}f = 1 \ and \ m_{1} = \sup_{I_{1}}f = 1$ $I_{2} = [3-a,3+a] \ with\ M_{2} = \sup_{I_{2}}f = 5 \ and \ m_{2} = \sup_{I_{2}}f...

The following is a question from a past exam that I am studying: For a 3-person game of perfect information. Let S denote the set {1,2,3}. First player A chooses i ϵ S. Then player B, knowing i chooses j ϵ S, j != i. Finally player C, knowing i and j, chooses k E S, k != j or i. The payoff gi...

Why the last term of cosx Tailor series is the o(x^2n+1) but not o(x^2n) ?

In this question, the most upvoted explanation of the identity in my title is this reply. I don't have the reputation to comment on the existing thread, so I'm asking here, because I am having a hard time following the explanation. I can prove this identity via Venn decomposition as in the checke...

It is a two part question. Let $K/F$ be a Galois extension with $[K : F] = n$. If $p$ is a prime divisor of $n$, prove there is an intermediate field $L$ with $[K : L] = p$. Prove or disprove that there is an intermediate field $M$ with $[M : F] = p$. Could anyone help me for some hints please...

I have this question in a homework sheet: So far, I have done parts $i), ii) $ and $iii)$ but I am really struggling with part $iv)$. Any help would be much appreiated as I have made no solid progress as of yet. Thank you

In a proof I'm reading, the author remarks in the first line of the proof that a set of finite outer measure is contained in an open set of finite measure. I have spent some time thinking about this and it is not obvious to me that this is the case. Can someone please explain why this should be o...

I got π=-4/3+4/5-4/7+4/9... but i can see that using this it will take me a very long time to reach the decimal expansion im looking for. i thought about setting (-1)^n (1/2n-1) less than .0000000001 but im not sure if that is a proper method.

$$P(AlB)=P(A)P(B\mid A)/[P(A)P(B \mid A)+P(A^c)P(B \mid A^c)] $$ $$P(A \mid B) = [P(B \mid A)P(A)]/P(B)$$ 1) Are these two equations so called bayes formula?? 2) how do you distinguish whether to use the first one or the second one? 3) If i assume A and B are independent, and let P(A)=0.2, P...

Let $\pi:X\to\mathbb{P}^2$ be a blow-up of $\mathbb{P}^2$ in 9 points (in general position for example, but it doesn't matter). Then the canonical divisor of $X$ is equal $K_X=-3\pi^*H+\sum\limits_{k=1}^9E_k$, where $E_k$, $k=1,...,9$ are exceptional divisors. It is known and not hard to compute ...

A cookie shop sells 5 different kinds of cookies. How many different ways are there to choose 16 cookies if... (a) you have no restrictions? (b) you pick at least two of each? (c) you pick at least 4 oatmeal cookies, at least 3 sugar cookies and at most 5 chocolate chip cookies? This is what...

I am kind of new in Math and I need to choose three basis functions g0, g1 and g2 for the least squares method (normal equations). Can anyone please first tell me what type of this equation is - f(x) = c0 + c1 sin x + c2 cos x (Polynomial/linear...I know neither is correct)(in which c0, c1 and c2...

What is the radical of this ideal, $I = (X+Z, (Y-1)^{2}, Z^{2}-1+Y)$? I found that his radical is $(X,Y-1,Z)$, correct? Thank you!

Can someone please help me with the following textbook problem? Find the closest point and the distance from b=(1, 1, 2, -2)T to the subspace spanned by (1, 2, -1, 0)T, (0, 1, -2, -1)T and (1, 0, 3, 2)T. I believe that I am supposed to use Gram matrix, but any help will be great. Thanks.

integral of e^x/x^2 I can't for the life of me figure it out. Can anyone help?. Here's a link if you need to see it :P. http://apcentral.collegeboard.com/apc/members/repository/sg_calculus_bc_99.pdf

Compute ∅ (40), 𝜎(124), 𝑑(124) and check the equality in Σ∅(𝑑) = 40. Here's what I've done so far: Not really sure about the summation equality. ∅ (40) = ∅ (5) x ∅ (8) => 4x4 = 16 𝜎(124) = 1+2+4+1+62+124 = 224 𝑑(124) = 1,2,4,31,62,124 = 6 Σ∅(𝑑) = 40 => ??????????