So I am struggling to understand the definition of a coset. If I have the following symmetric group $S3=${1$, \sigma, \sigma\tau, \sigma\tau^2, \tau, \tau^2$} where $\sigma$=($\begin{array}{ccc}1 & 2 & 3 & \\ 1 & 3 & 2 \end{array}$) and $\tau$=($\begin{array}{ccc}1 & 2 & 3 & \\ 2 & 3 & 1 \end{arr...

We usually read of our universe as a flat infinite universe (in accordance with WMAP and Planck data). Yet, it is possible to have a flat closed universe with a finite volume. One clear example of this is a 3D Torus. What are the basic parameters of this body and what is its volume as a function ...

How can I prove, that the set $$P = \{(x, y) \in \mathbb{R}^{n+m} : Ax + By \geq c, \: x \geq 0^n, \: y\geq 0^m \}, $$ where $B \in \mathbb{R}^{m \times m} \;$ is positive semidefinite matrix, $A \in \mathbb{R}^{m \times n} \;$ and $c \in \mathbb{R}^{m}$, is compact only when $P = \emptyset$ ? ...

We know that $P(\liminf_n A_n)=0.3$ and $P(\limsup_n B_n)=0$. Find $P(\liminf(A_n\cup B_n))$. My solution: We know that $\liminf_n(A_n\cup B_n)\supset \liminf_nA_n \cup \liminf_n B_n \supset \liminf_n A$. Therefore we get $$P(\liminf_nA_n)=0.3\leqslant \liminf(A_n\cup B_n).$$ How should I show t...

we try to solve a magnetics problem and it end up with $\int^l_{-l}{r/({\sqrt{L^2+r^2}})^3}dL$ I don't know how to get its integral .

Let $R=k[x^4,x^3y,xy^3,y^4]$ be a polynomial ring. We can see $R$ is not integrally closed (Since $x^2y^2 \in Q(R)$ is integral over $R$ but $x^2y^2 \notin R$ ). Therefore $k[x^4,x^3y,x^2y^2,xy^3,y^4]\subseteq\bar{R}$ (where $\bar{R}$ is integral closure). How to show that otherside?

Is there any easy way of showing this integral is recurrent? i.e. it visits every point infinitely many times?

Does the norm map in a class formation always have a finite kernel?

from stein exercise 6-5 Let A be a dxd positive definite symmetric matrix with real coefficient show that integral [e^(-pi*(x,A(x))] dx = det(A)^(-1/2) (integral is on R^d , so x is) hint is applying spectral thm that A=RDR^-1, where R is a rotation and D is diagonal matrix maybe it is not ...

Is the distribution of a random variable always a probability measure? Let $\mathbb{P}X^{-1}$ denote the distribution of the random variable $X$.

Let a, b and c be such that a + b + c = 0 and P = $a^2/(2a^2 + bc)$ + $b^2/(2b^2 + ca$) + $c^2(2c^2 + ab)$.How could I find Integral value of P.

Let P be a non-deterministic push-down automaton (NPDA) with exactly one state, q, and exactly one symbol, Z, in its stack alphabet. State q is both the starting as well as the accepting state of the PDA. The stack is initialized with one Z before the start of the operation of the PDA. Let the in...

I spent a lot of time but I can't understand why $B$ is open? Can anyone show it rigorously, please?

The circle ω touches the circle Ω internally at P. The centre O of Ω is outside ω. Let XY be a diameter of Ω which is also tangent to ω. Assume PY > PX. Let PY intersect ω at Z. If Y Z = 2PZ, what is the magnitude of angle PYX in degrees?

We are given an integer $n$ s.t. $n^2+n+1 \equiv 0 \mod m$ Where $m>3$ is a prime. Find$|n|_{m}$ and show that $m \equiv 1 \mod 3$

How to show that (if K is a field) then K [x] is a unique factorization domain? And why does it follow from Gauss's lemma that (since $Q$ is a field) Z[x] is also a UFD.

For f ∶ N → R+ other class of f, O′(f) ∶= {g ∶ N0 → R+ ∥ ∃c > 0, ∀n ∈ N0 ∶ g(n) ≤ c ⋅ f(n) + c} Proof that: When limn n→∞ f(n) > 0, then g ∈ O(f) ⇔ g ∈ O′(f).

I was reading about error analysis in numerical methods, particularly about calculating the sum of arrays. I understand how naively summing all elements can lead to accumulation of error especially when the number of elements is very large and all elements are of small order. In the alternative ...

In the question here Simplifying the Kahler form, user290605 asked a question about how is that when we take the differential of Kahler form:$$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge d\bar{z}^{\bar{j}},$$ we get$$\partial_ig_{j\bar{k}}=\partial_jg_{i\bar{k}} \hspace{1cm} \text{an...

Which can be the best book for elementary Number Theory of High School students or, to get started with Number Theory which book should I read. Please suggest.

neumann's formula : $$Q_{n}(z)= \frac{1}{2}\int \frac{P_{n}(\mu )}{\mu -z}d\mu $$ prove that: $$P_{n+1}(z)*Q_{n}(z)=\frac{1}{2}\int \frac{P_{n+1}(\mu )P_{n}(\mu )}{z-\mu }d\mu +\frac{1}{n+1}$$ solution : $$z^{n+1}Q_{n}(z)-\frac{1}{2}\int_{-1}^{1}\frac{\mu ^{n+1}P_{n}(\mu )}{z-\mu }d\mu =\frac{...

I'm looking at the ODE: $\frac{dY}{dX} - \frac{ X^2 + 2 Y^2 - 1 }{ ( Y - 2 X )X } = 0$ I'm looking for an $implicit$ solution to the above. Meaning, I want to find a relation $F(X,Y)=0$, where $\frac{\partial}{\partial x} (F(x,y)) = \frac{dY}{dX} - \frac{ X^2 + 2 Y^2 - 1 }{ ( Y - 2 X )X }$. $\...

I am reading Elements of Integration by Bartle and I came across this. "If f is a bounded function defined on an interval [a,b] and if f is not too discontinuous .......In particular the lower Reimann integral of f may be defined to be the supremum of the integrals of all step functions $\gamma$...

How can I determine if the function is one-to-one .. I know that any odd function is 1-1 and any even function is NOT 1-1 but what about functions thats are nither like X^3+5 and X^3+x^2+3 How can I determine whether it is a one-to-one ?

Supposing we have a space $S$ in $2n$-dimensions. You have two sets of $n$ number of indepenent constraints $C_1$ and $C_2$ but that are related to each other (that is if you arrange them as a $2n\times 2n$ matrix rank is $n$). Consider space corresponding to $C_1$ be $S_1$ and consider space c...

Price of lemon juice bottle is 4 , price of orange juice bottle is 6 A buyer bought 20 bottles and the total cost is 96 How many lemon bottles and orange bottle did the buyer get ? I know the answer but i don't know the steps to get to it ... Please answer the question with steps Thanks

If product of matrices $A$ and $B$ is defined, then $rank(AB)=rank(BA)$. Is this always true, or just in some special cases?

Let: $$ B:= \{ (x,y)|-1\leq y\leq 1 \} $$ Is it correct to say that $B$ is closed, because: $$ B^C := \{ (x,y) | |y| >1 \} $$ is open? Thanks !

Do the following converge: $\sum_{n=1}^\infty {\frac{n+2}{n^3-2n^2+1}}$ For this one I think the answer is no I just can't prove it. I split it up into partial fractions and got: $\frac{3n+1}{n^2-n-1}-\frac{3}{n-1}$ but after that I'm stumped :( The second part: $\sum_{n=1}^\infty {\frac{1}{n...

Some while ago I encountered a theorem which goes like this: Let $V$ be a $n$-dimensional vector space, and $N:V\to V$ a (linear) nilpotent operator of index $n$. Suppose we're given a cyclic base $E=\{e,Ne,...,N^{n-1}e\}$, where $e\in V$. Then all nontrivial invariant subspaces of $N$ are given...

I posted my question on math.stackexchage . But many replied that you cannot post a homework . So can please tell me other sites on which I can post my maths homework for free and the reply is fast .

We have a sequence of n numbered parking spaces which are arranged in a line. Type A vehicles require one parking space and Type B vehicles require two parking spaces. Let H(n) denote the number of ways in which the n spaces can be filled. a) If $ n \ge 3 $, explain why if the first car is Type ...

I'm following a sample solution from my lecturer and part of it is to solve $x \equiv 9 (mod 12) $ and $x \equiv 3 (mod7) $ simultaneously. He writes it as $9 + 12k \equiv 3 (mod 7)$ which I understand He then writes: This can be rewritten as $5k \equiv 1 (mod 7)$ I can see how to solve it fr...

I found that exercise. However I can not find the answer. I know I can do it with L'hopital. However is there a way to solve it without using L'hopital nor Taylor series? If so show me please. I will appreciate it a lot. If there's no way, which solution seems easier? L'hopital?Taylor?

Given that $G$ is a finite abelian group, and for every prime $p$ that divides the order of $G$, there is a unique subgroup of order $p$. How can I prove that $G$ is cyclic?

It is a pretty basic function but I am still having problems. Expand the following function $U(x,t) = x$ into a Fourier Series on $(-1,1)$

I was resolving a question about partition, quotient set and equivalence relation, but I don't sure if my proof is correct. Anyone can help me? I'm grateful right now for all help. Follows the question and my attempt, respectively. Question: Let be $A$ an set non-empty and $P(A)$ power set of $...

I would like to improve my proof of the following result: If $H$ is a finite, elementary abelian $p$-group, then $\Phi(H) = 1$. Here, $\Phi(H)$ is the Frattini subgroup, defined as the intersection of all maximal subgroups of $H$. An elementary abelian $p$-group is an abelian group with the...

Let p and q be atomic propositions and a and b be atomic actions. S={u1,u2,u3,u4} R(a)={(u1,u1),(u2,u1),(u4,u1)} R(b)={(u2,u3),(u2,u4),(u3,u3),(u4,u2),(u4,u3)} π(p)={u1,u2} π(q)={u1,u4} Let M be the dynamic structure specified above. There are 4 states given by the elements of S. a and b a...

Suppose we have a Lattice $L$ in $R^n$ generated by a basis $L(b_1,...,b_d)$, where $b_i$ is a column vector with $n$ elements, assume $d<n$, could we reduce the basis $L(b_1,...,b_d)$ to new bases $L(b^*_1,...,b^*_d)$ using $LLL$ algorithm? Thanks in advance

We know that $\mathbb{P}(\liminf_n A_n)=0.3$ and $\mathbb{P}(\limsup_n B_n)=0$. Find $\mathbb{P}(\liminf_n(A_n\cup B_n))$. My solution: We know that $\liminf_n(A_n\cup B_n)\supset \liminf_nA_n \cup \liminf_n B_n \supset P(\liminf_n A_n)$. Therefore $$\mathbb{P}(\liminf_nA_n)=0.3\leqslant \liminf...

So if A is diagonalizable, there exists $PAP^{-1}=D$ and also $PA^mP^{-1}=I$. To prove $A^2=I$, we need $D^2=I$ but how does it work?

Is any open set a ball? So instead of open sets we can talk about balls or disks? Is that correct? Because ihave seen definitions with open sets Like the definition of a surface.So is it sufficient instead of talking about generally open sets simplify things thinking it as balls?

Building random graph with probability to connect two vertex $p = \frac{1}{2n}$, and not connect $q = 1 - \frac{1}{2n}$. Find chromatic number a.a.s.(asymptotically almost shure), when $n$ tends to infinity. I've tried to use that $\chi(G) \geq \frac{n}{\alpha(G)}$ to get lower bound, but I fai...

I would like to create statistical test that detects over-representation / enrichment of structures such as pairs, triplets, quadruplets, etc. in a given group of obejcts compared to another group. The tricky part is to take into account the substructures. For example, let's say I'm interested ...

If the principal part converges for, say, |z|>1, and the analytic part (the positive powers in $z-z_0$) converges for |z|<2, then does the Laurent series, as a whole, converge in the annulus 1<|z|<2? So, this would be like taking the overlapping region of convergence of the principal part and th...

A mathematical problem is computable if there is an algorithm that decides this problem, right? Can you give an example of such a problem?

When an area of trapezoid ABCD reaches maximum , what is length of sum two diagonal ?

I have been searching for complex analysis problems recently. In https://www.math.stonybrook.edu/~olga/mat542/hw8_542.pdf I found the problem to determine the bianalytic endomorphisms of $\mathbb{C}-\{0,1\}$. Could someone give me some hints?

In Braid Groups of Kassel, Turaev, it mentions that $\mathcal{B}_n$ is a residually finite group. The definition that they give as a residually finite group is a group $G$ such that for each $g\in G-\{e_G\}$ ($e_G$ the identity of $G$), there exists an homomorphism $f$ to a group $H$ such that $f...

Note: The hope here is for a definition that does not rely on "Jargon", and efficiently conveys the nature of "Calculus". Question: What is a practical method to Define and Illustrate "Calculus" that could be provided, while either avoiding or clarifying "Mathematical Jargon" seen in common de...

let G be a group and H a subgrouop I want to know the difference between the order and the cardinal of a subgroup

Here is an puzzle question in this website. I hope someone will be interested in and solve. http://www.gizmodo.com.au/2015/12/can-you-solve-the-uk-intelligence-agencys-christmas-puzzle/

please i really need help on this express lim(n->∞) {1/n+1 + 1/n+2 + ... +1/n+n } as reimann integral. I was able to transform it to him(n->∞)£{1/n+r} I don't know how to go about it

I need a vector field $\vec{F}:\mathbb{R}^3\to\mathbb{R}^3$ such that $$\mathrm{curl}\ \vec{F}(x,y,z) \cdot \left(\frac{-x}{\sqrt{x^2+1}},\ 0,\ \frac{1}{\sqrt{x^2+1}}\right) = 1.$$ (This equality, of course, holds if $\mathrm{curl}\ F(x,y,z) = \left(\frac{-x}{\sqrt{x^2+1}},\ 0,\ \frac{1}{\sqrt{x...

Is it true that the center of a right Noetherian ring (with identity) is always Noetheria. ?

If I have $({u,v,w})$ is a linear independent set, then for which values of $\alpha,\beta$ is {${u,\alpha u +v,\beta v +w}$} a linear independent set? I claim that for all $\alpha,\beta$ in $\mathbb{R} $ this holds. Is this true?

Prove that $$ \lim_{n\to\infty} 2^{\frac{1}{\sqrt{n}}} =1 $$ I tryed to solve this using the sandwich theorem and the fact that $\lim 2^{\frac{1}{{n}}} =1 $. We did not learn continuity of functions so I am not alowed to use this. Any suggestions?

What are some slick ways to prove, with respect to $x\in[0,\infty)$, $S_1(x)$ is increasing, and given $b\in[\frac12,1]$ $S_2(x)$, is decreasing as defined below? I have proved it by taking several rounds of derivatives for various functions in the intermediate steps. That is not very elegant. \...

what is the solution of this equation. Please help me. ∬[(x-y)u]dxdy = ? where, (x-y)u = 0

I am trying to learn how to prove that the preimage of an open set is open in general topology. Here is an example that I am not really satisfied with Proposition 3.9 (Book: Essential Topology, Crossley): If $S$ has the discrete topology and $T$ is any topological space, then any function $f:...

I am studying for an exam, and this is a question from a previous homework that I got wrong. Im not really sure how to start, and any help would be appreciated! Let $R=\mathbb{Z}_7 [x]$ and $I$ be the ideal generated by the polynomial $x^3+2x+1$ a)Let $f$ and $g$ be the elements of $R/I$ repres...

I was trying to solve the question of AeT. on the (local) Lyapunov stability of the origin (non-hyperbolic equilibrium) for the dynamical system $$\dot{x}=-4y+x^2\\\dot{y}=4x+y^2$$ The streamplot below indicates that this actually is true. Performing the change of variables to polar coordinat...

Let $X,Y$ be two independent random variables such that $P(X=1)=P(X=-1)=1/2$ and $Y$ is symmetric. Show that $X|Y|=_d Y$.

If $f$ is an isometry of the plane and $L$ is a line, prove that $f(L)$ is a line. I know that isometries preserves distance, so that is easy enough. I also know that two distinct point make up a line. Since we know that they share the same distance, I only have to prove that the image is a lin...

Soon it's the year 2016. Time to ponder how we can arrange the digits in 2016 to form a valid equation. Use any symbols you like (please explain the less obvious ones). Keep digits in the same order (should this be relaxed?). Examples: $$\lfloor e^2\rfloor + 0 - 1! = 6$$ $$\lfloor\sqrt{\sqrt{20...

Q. Of all the graduate students in a university, 70% are women and 30% are men. Suppose that 20% and 25% of the female and male population respectively smoke cigarettes. What is the probability that a randomly selected graduate student is: (a) A woman who smokes. (b) A man who smokes. In my boo...

I'm using Royden's Real Analysis for a class (4th edition), and there's one line that I'm not sure if it should say what it does. At the top of pg. 110, when they're proving the Vitali covering lemma, on the line labeled (4), it says that $E \setminus \bigcup_{k = 1}^{\infty} I_{k} \subseteq \big...

He want to prove that $A$ is open. He take any $x_0\in A$ and showed that exists $\varepsilon:=R-|x_0|$ such that for any $x\in N_{\varepsilon}(x_0)$ we have $f(x)=0$. $A$ is the set of all limit point of $E$ in $S$. How he conclude that $N_{\varepsilon}(x_0)\subset A$? Can anyone explain thi...

I have: (x-y) uxy - ux + uy = 0 ----(1) i have to get the solution: u(x,y) = (x-y)^-1 [X(x) - Y(y)] to solve this at first let, v = (x-y)u u = v/(x-y) then i can derive the derivatives of u: ux = -...

Let $\mu,\alpha_n:\mathbb R^+\to \mathbb R$ continuous function with $\mu$ bounded function. Let $N^{(n)}$ the trajectory of a Poisson process with intensity $(\alpha_n \mu)(t)$. Let $0=T_0^{(n)}<T_1^{(n)}<..$ jumps of $N^{(n)}$. Let $M_n(t)=\sum_{i=1}^{N_t^{(n)}} \frac {1} {\alpha_n (T_i^{(n)})...

I found this proof sketch in Rosenlicht's book. I get the overall idea, but i don't get why we can write $f(z)$ in the way shown in c). I would be grateful if someone explained this proof to me. Thank you.

This problem 49 from chapter 9 of Royde & Fitzpatrick's Real Analysis 4th edition. Let X be a compact Hausdorff space. show that the Jordan decomposition Theorem for signed Borel measures on B(X) [the Borel sigma-algebra] follows from the Riesz[-Kakutani] Representation Theorem for the dual of ...

I have two equations: x = √16 x^2 = 16 In first case I think there will be two value of x = +-4. Because -4 * -4 = +4 * +4 = 16 In the second case I am confused. It can also have two values i.e +- 4. But some where I read that in one of the equations return only one value and it will be +4...

Let p be an atomic proposition, let a be an atomic program, and let $π = (K, M)$ be a Kripke frame with $K = \{u, v, w\}$ $Mπ(p) = \{u, v\}$ $Mπ(a) = \{(u, v), (u, w), (v, w), (w, v)\}.$ The answer is: In this structure, $u⊨$$¬p∧$<$a$>$p$, but $v⊨[a] ¬p$ and $w⊨[a] p$. Moreover, every stat...

I recently learned about the Enigma Machine in my cryptography class, but I am a bit confused as to the number of permutations of the wheel settings. According to every article I've read on the matter, the number of different ways the wheels could be set up equals the number of permutations that ...

How can I evaluate $$\int_L{\frac{z^2}{x^2+y^2}}$$, where: L: $$x=\cos{t};$$ $$y=a*\sin{t};$$ $$z=a*t$$ $$0\leq t \leq 2\pi$$ So far I tried it this way: $$I=\int_L{\frac{z^2}{x^2+y^2}}ds=\int_0^{2\pi}{(\frac{a^2t^2}{\cos^2{t}+a^2*\sin^2{t}}\sqrt{(\frac{d(\cos{t})}{dt})^2 + (\frac{d(a*sin{t}...

Let $ U \subset X $ be an open neighbourhoud of $x$, let $i : U \to \mathbb{A}^n$ be a closed immersion and let $U$ be defined by the ideal $I \subset k[ X_1 , \dots , X_n ]$. There is no loss of generality un supposing $i(x) = (0, \dots , 0) \in \mathbb{A}^n$. Given $f \in k[X_1 , \dots , X_n ]$...

Let $f_n(x)$ be positive measurable functions such that $$\sum_{n=1}^\infty \int f_n \lt \infty$$ show that $f_n \to 0$ almost everywhere Attempt: let $K = \sum_{n=1}^\infty \int f_n$ and $S_m = \sum_{n=1}^m \int f_n$ so there is $\forall \epsilon \gt 0 $ a $L$ such that $\forall m \gt L $ we ...

I was wondering if I was on the right track with this problem given that I have not had much guidance with them. $u_{t}=u_{xx}+2u$ $u_{x}(0,t)=u_{x}(1,t)=0$ $u(x,0)=cos(\pi*x)+cos(3*\pi*x)$ So far I have: $u(x,t)=X(x)w(t)$ $X(x)w'(t)=X''(x)w(t)+2X(x)w(t)$ $\frac{w'(t)}{w(t)}=\frac{X''(x)}{...

I'm confused with the generalised version of Le Cam's Third lemma presented in Theorem 6.6 of van der Vaart asynptotics Statistics here: https://books.google.co.uk/books?id=Ocg2AAAAQBAJ&pg=PT188&lpg=PT188&dq=van+der+vaart+theorem+6.6&source=bl&ots=Rm4Rw_KJ8H&sig=iXbKRK_HhSX1qQhKjhlGOcLC7Nc&hl=it&...

Does this relation are transitive? I know that the relation is reflexive and symmetric but I can not find transitive.

I have a basic uniqueness proof to help me work on form: It should be obvious by simple inspection that the statement is true for t=9 and only for t=9. So my proof was this: Let $t=9$ then $9-9=0$, and $0 \cdot s=0$ For the uniqueness, assume that $w,x \in \mathbb{R}$ satisfy the property. Then...

I'm trying to solve Exercise 20 of Chapter 5 of Fourier Analysis by Stein. The problem is as follows: Suppose $f$ is of moderate decrease and that its Fourier transform $\hat{f}$ is supported in $I=[-1/2,1/2]$. (a) Prove the following reconstruction formula holds: $f(x)=\sum_{n=-\infty}^{\infty}...

sinx+1=cosx, xE[-pi,pi] How do you solve by squaring both sides? the solution is x={-pi/2,0} and I do not understand how by subbing -pi back into both sides of the equations makes them uequal, and the same for positive pi. Which equation are you subbing pi into to check, the original? Thanks

I looked for a simple proof Theorema the Poincaré-Miranda based on the intermediate value theorem. Not succeeding, I prepared a proof that was simpler than I thought. The fact that I have not found proof so simple Poincaré-Miranda Theorem in textbooks or articles (but proofs that are based on not...

there are four inputs for the funtion f. the output is 1 if and only if two or three of the four inputs are "1". a. create a truth table for the function f b. minimize it using K-Maps c. implement the minimized function using AND and OR gates. It was a review question and i missed the sess...

The statement I want to negate is: there exists a $\delta > 0$ s.t. $f(x) > 0$ $\forall x \in A$ where $|x-x_0| < \delta$ and $x \ne x_0$ The negation I think is correct is: for all $\delta > 0$ there exist an $x \in A$ s.t. $f(x) \le 0$ where $|x-x_0| < \delta$ and $x \ne x_0$. But a friend of...

Let $I$ be the incenter of triangle $ABC$. Let $D,E,F$ be the intersections between the incenter and sides $BC,CA,AB$ respectively. Let $M$ be the midpoint of $EF$ and let $Q$ be the second intersection between $AD$ and the incircle. Show that $MIDQ$ is a cyclic quadrilateral. I have tried some...

The modified Bessel differential equation can be obtained by replacing $x$ with $ix$ ($i$ is the imaginary unit) in the Bessel differential equation. If the general solution of the latter is $$f(x) = A J_{n}(x) + BN_n(x)$$ While the general solution of the modified Bessel differential equation ...

A function f on X is said to be $\gamma$ measurable if for every real number $\alpha$ the set $\{x \in \gamma |f(x) > \alpha)$ Then I have that the following statements are equivalent for a function f on X to $\mathbb{R}$. (a) For every $\alpha \in \mathbb{R}$ the set $A_\alpha= \{x \in \gamma...

the ^ symbol represent negation. I have proved so far... (^B -> ^A) -> (^B -> A), ^B |- "bot" <=> ^(B v ^A) v (B v ^A), ^B |- "bot" and then I have no clue for next step til getting the answer please help! thank you!

An insurance company insures E drivers under age 24 and D drivers over 24 years old. Of these drivers, e under 24 and d over 24 had an accident in a 1 year period. A driver insured by this company is chosen at random. Let A be the event that this driver is under 24 and B be the event that this dr...

If ff is an isometry of the plane and LL is a line, prove that f(L)f(L) is a line. I know that isometries preserves distance, so that is easy enough. I also know that two distinct point make up a line. Since we know that they share the same distance, I only have to prove that the image is a line...

The proof below was taken from Hungerford. I can see how $R = \cap_{Y \in G} \, Y$ defines the desired $\varphi: \mathbb N \rightarrow S$, but I don't see why he had to go through so many mechanics to do so. I can define $\varphi: \mathbb N \rightarrow S$ as follows: \begin{align} \varphi(0) &...

Let $x_0<x_1< ... < x_n$ be the roots of an n+1 degree orthonormal polynomial $\phi_{n+1}$ with respect to the inner product: $$\langle g,h \rangle = \int_a^bw(x)g(x)h(x)dx$$ and $$p_n= \sum_{j = 0}^nf(x_j)L_{n,j} \in \mathbb{P_n}$$ be the lagrange interpolating polynomial for the given data....

I set the two polynomials equal to each other and after multiplying everything by -1 I got x^2+x-2=0 However, the graph of the polynomial is concave up. Why?

I have to proove the following: Let $A$ be a mtrix of size $m, n$ with rank $n$. Prove that the minimum for $f(x) = ||Ax -b||^2$ is given by the normal equation. I have absolutely no idea - would be great, if someone could help :)

Find the volume of the region when B is the region bounded by the cylinder $$x^2 + 3z^2 = 9$$ and the planes $$y = 0$$ and $$x + y = 3$$. $$ \int_{-3}^{3} \int_{0}^{3-x}\int_{-sqrt{(9-x^2)/3}}^{sqrt{(9-x^2)/3}} \;dz \;dy\;dx $$ Is my integral correct?

Find an x such that Q(x) = Q(2^0.5, 3^0.5, 5^0.5). For my abstract algebra class. Don't really know where to start, or how to finish for that matter.

How to prove that $f: l_1 \to \mathbb{R}$ s.t. $$f((x_n)_{n \in N})= \sum_{n=1}^{\infty} (\frac{1}{n} x_{n}^{2}+ x_{n}^{3})$$ is a function of class $C^{\infty}$ and $f'(0)=0$ and $f''(0)(h,k)>0$ for $h,k \in l_1 \setminus \{0 \}$ but $f$ has no extremum at $0$ Let $x=((x_n)_{n \in N}) $ and $h=...

Find $\sum\limits_{n=1}^{\infty}\frac{n^2}{2^n}$ using the function $f(x)=\frac{1+x}{(1-x)^3}$ Power series representation of $f(x)$ is $\sum\limits_{n=1}^{\infty}n^2x^{n-1}$. Question: Why is $\sum\limits_{n=1}^{\infty}\frac{n^2}{2^n}=\frac{1}{2}f\left(\frac{1}{2}\right)$? Doesn't it has to be...

Is it possible to define exponential and currying in the category Relation? If not, what is the reason that we cannot?

This is the second part of my last question sinx+1=cosx, XE[-pi,pi] The question asks to solve using a half angle identity and in the solution it states to subtract 2pi from 3pi/2 resulting in one of the solutions being -pi/2 along with another solution of zero. I am confused how you would know...

If f: X→R, where X is some metric space, be continuous. Prove that if f(a)=0 for every element a ∈ A ⊂ X, then f(b)=0 for every b ∈ the closure of A. Thanks.

I would like to how I can compute this expectation and get the answer that is given. All terms W indicate a Wiener process. E_t[W_s^3]=E_t[(W_t+(W_s-W_t))^3]=W_t^3+3W_t(s-t)

The year 2016 is coming up, so it makes me wonder: how do you prove that there is no simple group of order 2016? (without invoking the 10,000 page theorem on the classification of simple groups)

I have the following data list in a file, say mydata.txt 0 -2.900125720 -253.200 1 -5.512974510 -253.800 9 -398.4569435 -253.16 10 -748.4988836 -253.19 I read the file in as readdata(mydata.txt, float, 3) and this generates a data list, say mydatalist := [[0., -2.900125720,...

I am trying to interpret the first question statement of the Appendix in Hatcher's topology A covering space of a $CW$ complex is also a $CW$ complex, with cells projecting homeomorphically onto cells. Consider a $CW$ complex with, for example, $2$ $k$-cells in each dimension $k$, for $k \le ...

enter image description here Any help and comments will be greatly appreciated.

Find all functions f(z) which are analytic in the region |z|<1 (or equal to 1) and are such that f(0)=3 and |f(z)|<3 (or equal to 3) for all z such that |z|<1...... How???

I have to prove the following things: a) Assume that f is a convex function on S. Prove that the set ${x element S; f(x) <a$ is a convex set. b) Prove that the sum of two convex functions is a convex function c) Prove that an affine function is convex Proving things is completely new for me :)

For an exam I have to prove that if a constrained problem has no duality gap for some $(l,g)$ and $x$, then $x$ is a global minimum point for the constrained problem --> Do you think, an example is sufficient?

A single investment has two distinct outcomes with known probabilities. For example, this investment either returns 5% (with a .35 probability or returns 12% (with a .65 probability).

Consider the following procedure. Given an integer $n \geq 2$, obtain the canonical prime factorization of $n$, i.e. $\prod_{i=1}^k p_i^{e_i}$. Take the distinct factors $p_i$ and list them in ascending order. Concatenate them into a new integer. That is, 2 and 5 becomes 25, 3 and 7 becomes 37, ...

Is it possible to take the Fourier transform of "cross-sectional plane"; in other words, is $F\{\frac{\partial^{2}{u}}{\partial{x}\partial{y}}\}$ possible where $F\{\cdot\}$ is the Fourier transform? I know how to solve $\left[F\{\frac{\partial^{2}{u}}{\partial{x}^{2}}\} + F\{\frac{\partial^{2}{u...

f(x,y)=4xy x,y[0,1] Find E(x-y) and v(x).

For $x \in \mathfrak R^d$,why is $\int_\limits{\{x; |x|\geq 1\}} \frac{1}{|x|^d} dx = \infty$ in Lebesgue integral? It's hinted to apply Tonelli Theorem (Fubini Theorem) and use the fact that $\frac 1 x$ is not integrable over $[1,\infty)$, but I don't know how.

I'm trying to figure out a graph with a range of (-∞, 0]. I can't seem to figure it out. Keep in mind that I tried telling him right away -(x)^2, but he said that he wanted one that wasn't a [0, ∞) (E.G. absolute value, even exponentials, even roots, etc.) just negated. Any ideas?

Let $ \mathcal B$ be a Banach Space. Fix $z \in \mathcal B$. Consider the set $$A :=\{y-z : y \notin \operatorname{span} \{z\}, y \in \mathcal B\}.$$ Is it true that $\alpha z \notin \overline{A}$ for any $\alpha \in \mathbb{C}$? I'm looking for a counterexample or a hint about how to prove i...

I want help with the proof of this theorem Consider the metric spaces $(\mathbb{X}_j,d_j), j=1,\dotsc,n$. Let $\mathbb{X}=\mathbb{X_1}\times\cdots\times\mathbb{X_n}$ The map $E:\mathbb{X}\times\mathbb{X} \to \mathbb{R}$ with $$E(x,y)= \left[ \sum_{j=1}^n\big( d_j(x_j,y_j ) \big)^2 \...

Let $a$ and $b$ be to real arbitrary real numbers, show that the relative error that you made by computing $a^2b$ with floating point arithmethic is bound to $5\epsilon + O(\epsilon^2)$, with $\epsilon$ the machine error. As $a$ and $b$ are real numbers I can write them as: $a = a(1 + \d...

Let $x, y, n, p$ be positive integers such that $x^2 + ny^2 = p$, where $p$ is a prime such that $p \neq n$. Show that the Legendre symbol $(-n/p) = 1$.

Let $E/F$ and $E/F'$ and $E/F\cap F'$ be separable and normal field extensions. I am trying to show that $Gal(E/F\cap F') \simeq Gal(E/F).Gal(E/F')$

I'm trying to understand a part of the proof of the following Theorem Let $T$ be such that for every vertices $u,v$ there exists a unique walk joining them, then $T$ has no cycles and $T+e$ has exactly 1 cycle. Proof. If $T$ had a cycle, say $<v_1,e_1,v_2,e_2,...,v_1>$ then there wo...

Is there some probability distribution that can be implemented/defined/etc. without irrational numbers such that it returns 1 an irrational proportion P of the time and 0 the rest of the time, for any irrational probability P? If not, for what irrational P can this be done? I am specifically tryi...