Question: Find the angle between the curves y = sin 5x and y = cos 5x. I have used the formula tan θ = absolute((m1-m2)/1+(m1 * m2)) and found the gradient values by deriving each function. I get an angle of 27.124 degrees. But the answer in the book says : 31/6 degrees ?

I want to show that for a Banach algebra $A$ and elements $x,y \in A$, we have $$ r_A(xy) = r_A(yx), $$ where $r_A$ is a spectral radius. This is how I am trying to do that: $$ r_a(xy) = \lim_{n \rightarrow \infty} \| (xy)^n \| ^{\frac1n}=\lim_{n\rightarrow \infty} \|x(yx)^{n-1} y \|^{\frac1n}. $...

(1) Let $f$ be a continuous function on $[0,1]$. Then for every partition $P$ of $[0,1]$, the lower and upper Riemann sums of $f$ over $P$ satisfy $L(f,P)\neq U(f,P)$. (2) Let $f:[0,\infty)\to\mathbb{R}$ be a continuous function such that $\lim_{x\to\infty}f(x)=0$. Then $f$ has a maximum value o...

The title of Sloane's A001037 is: Number of degree-n irreducible polynomials over GF(2); number of n-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period n; number of binary Lyndon words of length n. The first few terms of the sequence are (for n = 1,2,...

Are the formulas that make up a disjunction called conjuncts? I am new to logic and need to know this for an assignment.

This is a discussion opener about something neat I found. I put some specific questions in there, but please chime in with any interesting insights. I have noticed some interesting properties of Mandelbot series that lead to a different way to plot the M-Set to elucidates certain details and pr...

The book "Probability, Random Processes, and Statistical Analysis" (written by Hisashi Kobayashi and Brian L. Mark and William Turin), talks about the role of entropy in characterising typical sequences (page 257). It says that in a coin tossing experiment: When we change the experiment of fa...

When I check for the proof of singular value decomposition, they all assume the following is true: The image of the unit sphere under any $m * n$ matrix is a hyper ellipse. However I could not find a decent proof for this, even though I googled for hours. Maybe I am using wrong keywords. Could ...

Consider $n \leq 2k$ and $A_1,...,A_m$ is a family of $k$-element subsets of $[n]$ such that $A_i \cup A_j \neq [n] \forall i,j \in [m].$ I want to show that $m$ is bounded above by $(1-\frac{k}{n})\binom{n}{k}$ I understand that the Erd\H{o}s-Ko-Rado theorem says that if $2k \leq n$ then every ...

Here is the question. If $\int ^\infty _1 f(x)dx$ converges and $\lim_{x\to \infty} f(x)=L$, prove that $L=0$. Any help would be appreciated.

I am trying to set up a polynomial such that it has a wiggle root at 0. I have gotten it such that there are wiggle roots at -1 and 1. My question is: 1. What part of the polynomial determines whether the root is a wiggle root or not? and 2. How do I make that change in my current polynomial. Th...

This is exercise 6.26.8 from Tom Apostol's Calculus I, I'd like to ask someone to verify my proof. I'd be also interested in alternative proofs: If $ f(x+y)=f(x)f(y) $ for all $ x $ and $ y $ and if $ f(x)=1+xg(x) $, where $ g(x) \to 1 $ as $ x \to 0 $, prove that (a) $f'(x)$ exists for every $x...

Following question seems so simple, yet I could not come up with a solution. I started to think that there might be sth wrong with the question. Could you please take a look? For a matrix $X:||X||_2<1 \iff matrix \begin{bmatrix} I&X^*\\X&I\\\end{bmatrix} $ is positive

If a,b,c are contained in Z, gcd(a,b)=1 and c|(a+b) Then prove gcd(a,c)=gcd(b,c)=1 I think this can be proven with linear combinations but I'm not sure how to go about starting the proof.

Suppose $A$ is a Banach algebra. Is it true that a) $x$ and $xy$ are invertible, then so is $yx$; b) $xy$ and $yx $ are invertible, then so are $x$ and $y$?

The squircle is given by the equation $x^4+y^4=r^4$. Apparently, its circumference or arc length $c$ is given by $$c=-\frac{\sqrt[4]{3} r G_{5,5}^{5,5}\left(1\left| \begin{array}{c} \frac{1}{3},\frac{2}{3},\frac{5}{6},1,\frac{4}{3} \\ \frac{1}{12},\frac{5}{12},\frac{7}{12},\frac{3}{4},\frac{1...

If I'm asked to calculate $319^{566} \mod(23)$, and I know $319 \equiv 20 \mod(23)$, is it mathematically to correct to then calculate $20^{566}$ instead? I feel the answer is yes, but I've an exam tomorrow and would rather a concrete notion rather than an inclination. Thank you guys!

Problem In pentagon $ABCDE$, $AB=BC=2,CD=\sqrt{2}$,and $EA= \sqrt{3}$. If $\angle{A}=90^{\circ}$, and $\angle{B} = 120^{\circ}$, what is the area of $ABCDE$? I just need some reaffirmation that there is no solution to this problem. There are infinitely many pentagons with the properties in ...

I'm starting to prove some of the major relations between graph problems and linear algebra. Take $G$ to be an undirected graph, and $A$ as its adjacency matrix. Let $\bar{d}$ be the average degree of the graph and $I$ to be the max degree of $G.$ I want to show that the largest eigenvalue $\ti...

Suppose $B$ is an $n \times n$ diagonal matrix. How to show that for any $n \times n$ matrix A, diag($BA$)=$B$ diag($A$)

Then, for each $x \in X$, there exists $B_x = B(x;r_x)$ such that $x \neq y \Rightarrow B_x\cap B_y = \emptyset$. My attempt For each $x\in X$, let $r_x = \inf\{ d(x,a): a\in X-\{x\}\}$. If we suppose $r_x = 0$, then $\forall \varepsilon>0$, $\exists$ $ a \in X-\{x\} : d(x,a) < \varepsilon$. ...

Also f is differentiable. I could conclude that x < 3 by the 3rd condition, and tried to use the Mean Value Theorem but found it hard to go on. Can anyone give some hints? Thanks!!

I know 13^5 + 13^3 = 169 but can someone please explain to me step by step how to do this? How does this happen? 13^3 x 13^2 + 13^3 (13^2 + 1) -> 13^2 = 169

There is a general achievement test that I have taken and here is a screenshot of how they says it is calculated: http://i.imgur.com/gNUxa62.png 'On a scale of 0 - 50, with a mean of 30 and a standard deviation of 7', what would the standardised score be if for example I got 35/40 for one of the...

Let $m$ be some prime greater than or equal to $5$. There exists an $n$ such that $n^2+n+1 \equiv 0 \mod m$ Find the order of $n$ modulo $m$, and show $m \equiv 1 \mod 3$

Let X and Y be sets and let f: X-->Y be a function between them. Suppose that U1, U2 are subsets of X. Show that if f is injective then f[(U1 union U2)^c]= f[U1^c] intersect f[U2^c] I'm not exactly sure how to go about this proof. My first thought was that the left hand side is De Morgans Law b...

if lim x-->a of f(x) = infinity, then we know: for all M>0, there exists D1 such that if |x-a| < D1, then f(x) > M/2 if lim x--> a of g(x) = infinity, then we know: for all M>0, there exists D2 such that if |x-a| < D2, then g(x) > M/2 So for all positive M choose D = max{D1, D2} Then if |x - a| <

If we define $$ $$\mathcal{J} = \{\text{ all intervals contained in [0,1]}\} $$ Then $$ B_0 = \{ \text{ all finite unions of elements of }\mathcal{J}\} $$ Is an algebra (so it is closed under the formation of complements (but it is not a $\sigma$ algebra) . Now consider $$ B_1 = \{\text{ all fin...

Expand the function $U(x,t) = 1$ as a eigenfunction expansion of $cos(\frac{ 2n+1}{ 2} )πx$ for $n ≥ 0$. I missed my last ODE class and am having trouble with this problem. Can anyone be of help?

$\sum_{n=0}^\infty$ $(n+1)(n+2)(\frac{i}{2})^{n-1}$ I tried to separate real number(n=0,2,4...) and complex number that is not a real number(n=1,3,5,...) But it didn't work. So I did another way; Using Cauchy's integral thm: Let $f(z)=(\frac{z}{2})^{n+2}$ Then $4f''(i)$= $(n+1)(n+2)(\frac...

Let $F:V\rightarrow{}\mathbb{R}^{+}_{0}$ be a function on domain $V=\{(x_{1},x_{2},x_{3},x_{4})|(x_{1},x_{2},x_{3},x_{4})\in(0,1)^{4}, x_{1}+x_{2}<1, x_{3}+x_{4}<1\}$. Here is what I know about $F$: (i) $F(x_{1},x_{2},x_{3},x_{4})=0$ if $x_{1}=x_{3}$ and $x_{2}=x_{4}$ (self-similarity). (ii) $x...

Ever since I started learning formal logic I've had these kind of doubts: Is analysis ever studied in a completely axiomatic/formal proofy way? What I mean is, given a set of axioms and inference rules, to prove things via formal proofs. Example: Let $f: \Bbb R\to \Bbb R: x\mapsto x$. Theor...

Define $$\phi:\mathbb R[x,y]\rightarrow \mathbb R[x]$$ by setting $$\phi(P(x,y))=P(x,4)$$ Is $\phi$ a ring homomorphism? So I got the $\phi(a+b)=\phi(a)+\phi(b)$ part down, but am having trouble proving $\phi(ab)=\phi(a)\phi(b)$. How would I prove that without expanding out each term?

Let $f:[a,b]\to\mathbb{R}$ be continuous and increasing, show that $$\int_a^bxf(x)dx\geq\frac{b+a}{2}\int_a^bf(x)dx$$ I am thinking of using integration by parts. First let $$F(x)=\int_a^xf(t)dt$$ Then $$\int_a^bxf(x)dx=bF(b)-\int_a^bF(x)dx=b\int_a^bf(x)dx-\int_a^b\int_a^xf(t)dtdx$$ So far I only...

Consider the function $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ given by $$f(r,\theta)=(rcos\theta,rsin\theta)$$ I like to show that $f$ is one one in sone neighborhood of any non zero point $(r,\theta).$ I tried as $(rcos\theta,rsin\theta)=(scos\phi,ssin\phi)$ Which gives $r=s$ and $\theta=\phi+...

let $f(z)= \frac{1}{z-1}$, Finding the real and imaginary parts $u=\frac{x-1}{(x-1)^2+y^2}$ and $v=\frac{-y}{(x-1)^2+y^2}$. I have to describe the contour lines of constant $x$ and $y$ in the W plane. I did something with python, not quite sure if I got it right, %matplotlib inline import matpl...

Using only modus ponens and substitution. Prove: q -> r -> [ [ p -> q ] -> [ p -> r ] ] using the three axioms: 1) p -> [ q -> p ] 2) s -> [ p-> q ] -> [ [s -> p] -> [ s -> q ] ] 3) p -> f -> f -> p where the symbol f is "false." I am having the hardest time trying to solve this proof, any ...

A. $X=\Bbb Z \times \Bbb Z \subset \Bbb R \times \Bbb R$ B. $X=\Bbb Q \times \Bbb R \subset \Bbb R \times \Bbb R$ C. $X=(-\pi,\pi) \cap \Bbb Q \subset \Bbb R$ D. $X=[-\pi,\pi] \cap (\Bbb R - \Bbb Q) \subset \Bbb R$ I choose option A as an answer because $\Bbb Z \times \Bbb Z$ ...

so I have a problem. Say I had a variable n, which is equal to any real number between -10 and 10. than say I wanted to plot an x-y series based off each possible value of n. for example, say x = n, and y = n + 1. is there any ways that anyone knows of to make a single plot-able equation using ...

F(x,y,z) is a vector field in space and f(x,y,z) is a scalar field in space. curl(grad(div(F))) curl(div(grad(f))) grad(div(curl(F))) grad(curl(div(F))) div(curl(grad(f))) div(grad(curl(f))) I'm trying to study for a multivariable final and I am having trouble understanding when and why these e...

$D_{(0,s)}(x,y)=(sx, sy)$. Is this an isometry of the Poincare Half Plane and if so, how can this be proved?

I am reading A. L. Oniscik's paper Decompositions of Reductive Lie Groups, and the author cited a proposition that a complex reductive Lie group $G=ZS$ is locally isomorphic to the reductive algebraic group $\widetilde G= \mathbb C^{m}\times S$. I want to have a reference about this proposition....

$A=\int _{|z+1|=2}\dfrac{z^2}{4-z^2}=?$ The function $f(z)=\dfrac{z^2}{4-z^2}$ has a pole on the points $z=-2,2$ which don't lie on the circle given by ${|z+1|=2}$ as $|-2+1|=1\neq 2$ and $|2+1|=3\neq 2$. Hence $f(z)$ is analytic on the given domain.Thus $A=0$. But answer is not matching .Ple...

A system can have three different types of defects: A1, A2, A3. We are given the following probabilities. P(A1) = .12 P(A2) = .07 P(A3) = .05 P(A1 U A2) = .13 P(A1 U A3) = .14 P(A2 U A3) = .10 P(A1 and A2 and A3) = .01 What is the probability that the system has both type 1 and type 2...

please help me with below Find the constant k so that F(x,y)={k(x+1)e-y , 00 Is a joint probability density function. Are X and Y independent ? Thanks in advance

I have a system of partial differential equations consist of 6 equations on 9 variables. $p = \frac{\partial f[x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3]}{\partial x_1};$ $q = \frac{\partial f[x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2, z_3]}{\partial x_2};$ $r = \frac{\partial f[x_1, x_2, x_3, y...

I am having trouble to understand why this given vector spans R^3. Vector={(−4, −6, 0), (2, −1, −4)} I understand that a non-zero determinant shows that the given set spans the space. However, this is not the square matrix. Therefore, I can not use determinant. I searched online and found this...

Having trouble with this problem: 2^2x+1=(1/32)^x Do I need to set the exponents equal to each other?

Hello guys i have a small question :) Why is the integral of a square wave with 50% duty cycle, centered at 0 not equal to 0? I mean a sine wave with these properties have an integral of 0. So why is the square wave's integral the area under the wave over half a period? Thanks a lot.

State the Midpoint rule for: $$\int_c^dg(t)dt$$ with $m$ subintervals Is this how you state it? $$M_n=\frac{d-c}{m}[g(t_m)]$$ or $$M_n=\frac{d-c}{m}[g(t_1)+g(t_2)...g(t_m)]$$

Can any function be written as a composition of other functions? For example, can a polynomial h(x) be written as k(g(x)) where g(x)=x^2 and k is a function in the set of polynomials?

This is the question ==> (http://puu.sh/lBsGQ/0f8de6aea6.png) So what I have done is let z = x+iy (z-2)/(z+5) --> (x+iy-2)/(x+iy+5) --> [((x-2)+i(y))/((x+5)+i(y))] * [((x+5)-i(y))/((x+5)-i(y))] (rationalizing here) --> [(x-2)(x+5)+y^(2)+i(y)(x+5)-i(y)(x-2)]/[((x+5)^(2)+y^(2))] As arg((z...

A fair coin is tossed 8 times then find the probability that resulting sequence of heads and tails looks the same when viewed from beginning or from the end? How to approach this question because making cases would be such a difficult task?

I saw the following claimed : Let's say we have the functional equation $f(R+S) = f(R) + f(S)$ where R and S are projections in a vector space, and f is a real valued function. Then its general solution is : $f(R) = c\ Tr(R)$, where $c$ is some constant and Tr is the trace. The reason given ...

This is the question --> http://puu.sh/lW9XL/b1e04691d2.png I've dealt with quadratics in this form but never with cubics. I tried searching up the cubic formula but I think it would take way too long to be practical in an exam situation..

Let f:R->R be the function degined by f(x) = Sin[x]/(|x|+Cos[x]) check differentiability at x = pi/2

The problem statement is: Suppose that the real series $∑_0^{∞} a_n$ and $∑_0^{∞} b_n$ converge absolutely. Part 1 Prove that there is a function $u(r,θ)$ which is harmonic in $1<r<2$ and continuous onto the boundary such that $$u(1,θ)=∑ancos(nθ)$$ and $$u(2,θ)=∑bncos(nθ)$$ Part 2 Is $u(r,...

Let $f(x)=\frac{\alpha x}{x+1},x\neq-1.$Then for what value of $\alpha$ is $f(f(x))=x?$ Given $f(f(x))=x$ $\Rightarrow f(x)=f^{-1}(x)$ means we need to find the point where the function and its inverse intersect and such points are found on the line $y=x$ so we need to solve $f(x)=f^{-1}(x)=x...

2/1 - 1/2 - 1/3 +2/4 -1/5 - 1/6 +2/8 - 1/9 -1/10 + 2/11...

Someone has already asked whether an exponent less than $n!$ is possible for a symmetric group $S_n$. It has been answered that it is for $n \ge 4$. I would like to know if there is a general formula to determine the exponent of $S_n$.

Is it true that $$m(A+B)\geq m(A)+m(B)$$? For any measurable sets $A,B$ where $A+B=\{x+y,x\in A,\,y\in B\}$ Thanks

Find the sum of all solutions of the equation $$[x]=m\{nx\}$$ where $m,n$ are positive integers. I don't know, it is just making a lot of cases. Give some hint. Thanks.

The sum of the series $1+\frac{1+2}{1!}+\frac{1+2+3}{3!}+....$ equals? The answer is $\frac{3e}{2}$. But I dont Know How? I have tried following: $1+\frac{1+2}{1!}+\frac{1+2+3}{3!}+....$=$\sum\limits_{n=1}^{\infty} \frac{n(n+1)}{2n!} $ I know that sequence of partial sum $S_n$=$1+\frac{1+2}{...

I am trying to makes sense of the proof to following problem: Given: $A_n = \displaystyle \frac{\sum_{k=1}^n a_k}{n}$. Can $\{A_n\}$ converge if $\{a_n\}$ diverges; $\forall n,a_n>0; \limsup{a_n}=\infty$. Proof: Lets define $a_n = \begin{cases} k & n=k^3\\ \frac{1}{k} & n \in (k^3,(k+1)^3) \...

I have the expected value of the square of the position operator given as $$\langle x^{2} \rangle= \int_{-\infty}^{\infty} x^{2}e^{\frac{amx^{2}}{h}}dx$$ I understand that the integrand can only be evaluated by the form $$ \int_{-\infty}^{\infty} x^{2} e^{-ax^{2}}dx=\frac{1}{2}\sqrt{\frac{\pi}...

Suppose there is an unobserved RV that can take two levels only (i.e., V~Bernoulli with values {0,1} and a 50%/50% probability). In my model there is a "signal" Y available that allows one to learn about V. Without focusing on the learning problem, lets just simply assume that upon the observati...

I am at moment trying to solve a system of linear equation, and I am not sure if the value I retrieve is even possible, or my program return some garbage value... The equation i am trying to solve is : J(q)dq=du J(q) = [6x7] matrix dq = (not known) du = [6x1] So since i have to solve for dq ...

I have trouble in understanding the definition of $\sigma $ - Algebra. I tried googling but couldn't make much of it.

Find the equations of the tangents of the parabola y^2 = 12x, which passes through the point (2,5). I tried to do SS1=T^2 But from that i am not getting Please can you do it for me

How many positive integers $n<10^4$ are there such that $2^n - n^2$ is divisible by 7?

Suppose $f$ and $g$ are holomorphic in the region $G$ and $\gamma \sim$ $_G$ $0$. Prove that if $f(z)=g(z)$ for all $z\in\gamma$, then $f(z)=g(z)$ for all $z$ inside $\gamma$.

Let $\sum_{k=a}^{n} \frac{1}{k^{1/2}}$ I have written a simple C++ program that computes the series for different values of n. What is the mathematical approach to finding this series.

Would it be 15 knots per hour or like 15sqrt(3) knots?? I'm not sure if I understand this please help.

There is a group of 30 line segments of different length such that a triangle can be formed by each three of them. Find the maximum number of different obtuse triangles formed by segments from that group.

please help me. Please. Find the points on the graph of ⅓ x3+x2-x-1 at which the slope is (a)-1; (b)2 I don't know where to begin.

Show that if $a>1$ then $\log a - \int_a^{a+1} \log x dx$ differs from $\frac{-1}{2a}$ by less than $\frac{1}{6a^2}$. For some $\theta$ between $a-1$ and $a$ and odd $n\in\mathbb{N}$ we have equality: $$\log a - \int_a^{a+1} \log x dx=\log a - \int_{a-1}^atdt+\int_{a-1}^a \frac{t^2}{2}dt-...

Give example of Dual space of Co and L1

I'm learning about functions of bounded variations and need help to solve this problem: Show that $\left|\left| f \right|\right|_{BV} = \left| f(a) \right| + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$. My attempt and Thoughts: First. We want to show that $\left|\left| f \right|\ri...

I got this set - $lim_{ n _{\to \infty}} \left(\left(\sqrt{9x+1}-\sqrt{3x}-\sqrt[3]{2x}\right)\left(\sqrt{2x+1}\right)\right)$ I know that $lim_{ n _{\to \infty}} = \infty$ by intution. But I dont know how to calculate that. Thanks a lot.

What are the prime ideals in $\frac{\mathbb{Q}[x]}{\langle x^n-1 \rangle}$ for $n=4$ and $n=5$? How do I tackle this?

This question (circles, power of point, cross ratios ) has been solved using Steward’s theorem. I try to solve it in another way but find myself stuck at some point. If we let $BC = x, CD = y, DP = z$, the required equality ($CD \cdot BP = BC \cdot DP$) is just $y(x + y + z) = x(z)$ or more si...

My task is to find a field $K$ where the vectors $$ \begin{pmatrix} 2 \\ 3 \\ 1 \\ \end{pmatrix} , \begin{pmatrix} 0 \\ 4 \\ 0 \\ \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 2 \\ \end{pmatrix} ...

I'd like to show that $$\sum\limits_{n = 1}^\infty {{{{x^{n + 1}}} \over {n(n + 1)}}} $$ absolutely converges for |x| < 1

I'm trying to understand whether the complement of a Matching is an Edge-cover and, in that case, whether the rule is valid for the opposite situation. By doing some examples I see that the complement of a Matching seems to be always an Edge-cover (is it correct?) but the complement of an Edge-c...

Suppose we have $f \in L(I)$ and derivative $f'$ exists almost everywhere . It is $f'$ measurable ? I have no idea how to begin to construct the proof .

I need to prove that the group $\mathbb{Z}_p\oplus\mathbb{Z}_p\oplus\mathbb{Z}_p$ is not generated by 2 elements. How?

Let $Y_t$ be a centered Poisson process, why \begin{equation} \lim_{n \to \infty} \sup_{s<t} |n^{-1}Y(ns)| = 0 \qquad a.s. \end{equation} This is a fundamental step in the proof of the law of large number for continuous time Markov Chain. I'm following the proof on the book by Ethier and Kurtz b...

Is sample mean always the BLUE of population mean?

To test the correctness of the SAGE software, how is it tested? Is there a standard set of math problems that SAGE is asked to solve as part of the build process?

Let $A=\{(x,y)\in\mathbb{R}^{2}:x+y\neq-1 \}$ Define $$f:A\rightarrow\mathbb{R}^{2}$$ by $$f(x,y)=(\frac{x}{1+x+y},\frac{y}{1+x+y}).$$ I have to prove that the the function $f$ is one one but not onto. It is clear that $f$ is continuously differentiable but Inverse map theorem gives locally inje...

Laplace transform property: $f(at)\leftrightarrow \frac{1}{a}F(\frac{s}{a})$ where $f\leftrightarrow F$. Question: Is $a>0$ necessary for this property?

The following LaTeX document renders exercise 2.5.2 from "Mathematical Logic" by Ian Chiswell, Wilfrid Hodges, Oxford University Press (2007). \documentclass[oneside,12pt]{book} \usepackage[a4paper]{geometry} \usepackage{microtype} \usepackage[T1]{fontenc} \usepackage{enumitem} \usepackage{amsm...

I would like to ask about an explicit suggestion/reference for the following type of heat processes: Roughly, assume we have a "wedge" $W$ of the following form - a domain in $\mathbb{R}^n$ with a tip at 0, having some hyperplanes (passing through 0) as sides and, say, the piece of the unit cyll...

This is probably wel known, so please let me know. Let $s(n)$ be the sum of all digits of the integer $n$ (in base 10). Consider a polynomial $f$ with integer coefficients. I want to show that for any large enough $n$, $s(f(n)) \leq s(n!)$. Any suggestion would be helpful.

Define the exponential integral $E(z)=\int_{-\infty}^z\frac{e^t}{t}dt$ (this differs from the usual definition of $Ei(z)$ only by $\pm i\pi$). Consider now the following contour integral of a function $f(z)=\frac{e^{az}}{z-b}$: the contour starts from real $-\infty$, goes to 0, then to imaginary...

Suppose , X Y independent geometric random variables with the same parameter p.we want to find the captive probability function of X given that X+ Y =n, where n >1.

I want to evaluate the Riemann integral $\int_0^1 {{x^2}dx} $ I want to find upper and lower estimates of the form: $$U \ge {1 \over {6{N^3}}}(N(N + 1)(2N + 1))$$ $$L \ge {1 \over {6{N^3}}}(N(N - 1)(2N - 1))$$ Then show they're equal and then evaluate the mentioned Riemann integral. I think the ...

I wonder how hard identity testing (similar to polynomial identity testing) can be for a free algebra. I thought that in a certain sense, the problem should always be semi-decidable, because the free algebra is defined with respect to the identities which follow from the given set of equational a...

Suppose $I, J$ are arbitrary index sets. Let $X$ be a set and $X_{i, j} \subseteq X$. I claim that $$\bigcap_{i\in I} \bigcup_{j\in J} X_{i,j} = \bigcup_{(j_i) \in J^I}\: \bigcap_{i\in I} X_{i, j_i}.$$ $$\bigcup_{i\in I} \bigcap_{j\in J} X_{i,j} = \bigcap_{(j_i) \in J^I}\: \bigcup_{i\in I} X_{i,...

Let $H$ be a subgroup of a finitely generated polycyclic group $Q$ such that intersection of $H$ with each normal subgroup of $Q$ is non trivial. I need an example to show that it is not necessarily that $H$ is normal in $Q$.

How do I show that Lagrange's polynomial is the only one (with degree < n) that takes the given values at given points? ($f(x_{1})=y_{1} \space f(x_{n})=y_{n}$)

Suppose we are given a set of n points in the euclidean plane , they are distributed arbitarily ( not in general position). what is the minimum number of lines in the plane needed to cover them all?

Let $K$ be a field complete with respect to a discrete valuation $v$. Let $K_s$ its separable closure, $w$ the unique valuation extending $v$, $(\mathcal{O}_{K_s}$, $\mathfrak{M})$ respectively the ring of integers and its maximal ideal of $K_s$. Let $p \in \mathbb{Z}$ be a prime number such th...

I'm trying to show that if we have a graph G that is a minimal block with at least 4 vertices, that one the vertices must have degree 2. We have defined a minimal block as a 2-connected graph such that that the removal of any edge e results in a subgraph G-e that is not 2 connected. I'm assumin...

How to solve y'=x^2+y^2; y(0)=1; Specify any segment in which there is a solution of this equation and to construct successive approximations y0, y1, y2 to the solution of this equation. Help me please.

How do I prove that $$lim \sqrt(x)=3$$ while x->9 using epsilon delta proff

please I need help. k1, k2, m powers. given that k2>=k1 prove that k2^m>=k1^m. I know that I need to find a one to one function f from k1^m to k2^m in order to prove that. I also know that if |A|=k1, |B|=k2, |C|=m and k2>=k1 there is a one to one function g from A to B. Is someone can guide m...

How could you define "Fuzzy Integral"? Could you recomended me any interesting book to self study about Fuzzy integrals? Thanks

Let X be a set and an outer measure on X is a function $\theta:\mathcal{P} X \to [0,\infty]$ such that (i) $\theta \phi= 0$ (ii) if $A \subseteq B \subseteq X$ then $\theta A \leq \theta B$. (iii) for every sequence $<A_n>_{n \in \mathbb{N}}$ of subsets of X, $\theta(\cup_{n \in \mathbb{N}} ...

Let $D=\{f:\mathbb{R}\rightarrow \mathbb{R}\,|\, $f$ \,\mbox{ is twice differentiable}\,\}$ be a ring and let $J=\{f\in B\,|\, f'(0)=f(0)=0\}$ be an ideal in B. Find all maximal ideals of $B$ that contain to $J$. I prove that the sets $M_1=\{f\in B\,|\, f(0)=0\}$ and $M_2=\{f\in B\,|\, f'(0)=0\}...

Let X=Y-Z, where , YZ is a non- negative random variables for which applies YZ = 0. a) Show that cov (Y,Z) <=0 b ) Show that var (X)>= var (Y) + var (Z)

$X$ and $Y$ are two sets and $f:X\to Y$.If $\left\{f(c)=y;c\subset X,y\subset Y\right\}$ and $\left\{f^{-1}(d)=x;d\subset Y,x\subset X\right\}$ then the true statement is $(A)f(f^{-1}(b))=b\hspace{1cm}(B)f^{-1}(f(a))=a\hspace{1cm}(C)f(f^{-1}(b))=b,b\subset y\hspace{1cm}(D)f^{-1}(f(a))=a,a\subset...

Let $K$ be a field of characteristic zero and $p$ a prime number such that $p^2$ divide at degree of all irreducible polynomial not linear in $K[x]$. Prove that $K$ is algebraically closed.

Let $p: \mathbb{R} \rightarrow \mathbb{R}$ be polynomial $p(t) = a_0 + a_1 t+ \cdots + a_n t^n $ $(a_n \neq 0)$. I'd like to prove the following statement by induction: $\forall K>0$ there exists $r_k$ such that $|t|\geq r_k \Rightarrow |p(t)|\geq K$. My attempt Base case: For $n=1$, we have ...

Can there be a ring such that all the elements are zero divisor? As I read about the poof of every finite commutative ring has unity, the prove is to use one of the non zero divisor namely 'a' as the permutation. So if a^0b,a^1b ... a^nb=a^0b gives a complete cycle on every non zero element b, a^...

In Combinatorics, transfer matrix is a matrix defining linear recurrence relations. I am asking specifically about the matrix $\left(\begin{array}{cccc} 2 & 1 & 0 & 0\\ 1 & 1 & 0 & 0\\ 0 & 0 & 2 & 0\\ 1 & 0 & 0 & 2 \end{array}\right)$ Which defines the following set of recurrence relations: $a...

In Z/(12) modulo 12 ring the LCM of (6,8) Does Not Exist ? HOW ?

Is it possible to evaluate these 2 calculations using a compound angle formulae and not a calculator? Sin(105°) Cos( - 30°)

How would I go about solving this? Before you ask, I don't even know where to start :(

I have a LaPlace transformed function that I'd like to transform back. It's quite a complex function however, which is why I am stuck C(x,s) = $(\frac{m}{s})$*e$^{\lambda^2*(\frac{v-\sqrt{\frac{4*D*s}{\lambda^2} - v^2}}{2*D})* x}$ I'd know the inverse LaPlace transform of a less complex but co...

I only know how to prove the reverse. Anyone give any ideas for this proof?

Suppose we have $p$ beads of $n$ different colors on a loop. $p$ is a prime number and we consider the loop to be the same if one is a rotation of the other. Then how many distinct beads are there? By using Burnside's Lemma, I have the result of $\frac{(p-1)n+n^p}{p}$, but not quite sure about my...

I tried expanding out on the left hand side using the formula for the LCM but i couldn't see where to go from there.

I have a second question today. In Harris' "Algebraic Geometry: A First course" he constructs (on page 200) an isomorphism between the tangent space of the Grassmannians and some homomorphisms: He begins with the cover of the Grassmannians defined in Section 6, i.e. for a (n-k)-plane $\Gamma \s...

I. Wikipedia: An integer $n$ is a congruent number if there are rationals $a,b,c$ such that, $$a^2+b^2 = c^2\\ \tfrac{1}{2}ab = n$$ or, alternatively, the elliptic curve, $$x^3-n^2x = y^2\tag1$$ is solvable in the rationals. II. Mathworld and OEIS: They define them (see this and this) furth...

I am currently working on this problem in Frank Morgan's Geometric Measure Theory book. $\mathbf{M}$ is $\mathbf{F}$ lower semicontinuous on $\mathscr{D}_m$. What I have done so far: $T_i \to T$ under the real flat norm: $\forall \epsilon>0, \exists N$ such that $\min \{ \mathbf{M}(A-A_i) + \...

I need to plot a graph of the following function for values time, 0≤t≤2π0t2π (radians). y=5sin(2t+π6) y5sin2tπ6 I need to describe and define the amplitude, periodic time and frequency. Can the formula be put into Excel somehow to plot the graph? Thanks trigonometry graphing-functions

In a class consisting of 18 boys and 12 girls. Professor asks a question . Each boy has a probability 1/3 to know the answer, while corresponding probability for each girl is 1/2. What is the expected number of students who can answer correctly ?

I am wondering what the [A b] here stands for, is it matrix and its multiplied result? So could anybody please give me some hints to start the problem as I only have the intuition that if Ax=b and x is not $0$, then rank A has to be equal to [A b].

I was solving a question related to functions and i come across a limit which i cannot understand.The question is If $a$ and $b$ are positive real numbers such that $a-b=2,$ then find the smallest value of the constant $L$ for which $\sqrt{x^2+ax}-\sqrt{x^2+bx}<L$ for all $x>0$ First i found ...

My understanding is that it is impossible to prove and disprove the Continuum hypothesis in ZFC. Would it be possible to prove or disprove it in some other axiomatic set theory?

How to prove that this statement is tautology using logic laws (q ˄ (p ↔ ¬ q) ) → q I got stuck here: (q ˄ (p ↔ ¬ q) ) → q -(q V (p ↔ ¬ q) ) → -q -q V -(p <-> -q) -> -q

(This post made me re-visit congruent numbers.) I. Wikipedia: An integer $n$ is a congruent number if there are rationals $a,b,c$ such that, $$a^2+b^2 = c^2\\ \tfrac{1}{2}ab = n$$ or, alternatively, the elliptic curve, $$x^3-n^2x = y^2\tag1$$ is solvable in the rationals. II. Mathworld and ...

Good day, Here is my equation $$ (x-1)^2 (x-h)^2q_1(x,\Omega)y''(x)+(x-1)(x-h)q_2(x,\Omega)y'(x)+q_3(x,\Omega)y(x)=0 $$ ( with $q_1(x,\Omega)$[polynomial function 38 degree], $q_2(x,\Omega)$ [polynomial function 39 degree], $q_3(x,\Omega)$ [polynomial function 40 degree] regular fu...

Let P(n) be the statement that a postage of n cents can be formed using just 4-cent and 7-cent stamps. Show by mathematical induction that P(n) is true for n ≥ 18. Hint: carefully determine what the base cases are. Solution: Base cases: P(18) is true as 18 = 4 + 2 ∗ 7. P(19) is tru...

Let $G$ be a Lie subgroup of $GL(n,\Bbb R)$ and $\mathfrak{g}\subseteq M(n,\Bbb R)$ its Lie algebra. Suppose that we have a smooth curve $$\gamma:\Bbb R\to G$$ with $\gamma(0)=I$. Then, it induces a curve $$\alpha:\Bbb R\to \mathfrak{g},\quad \alpha(t)=\gamma(t)^{-1}\frac{d\gamma}{dt}(t).$$ I am ...

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=0$ and $f'(x)>f(x),\forall x \in \mathbb R$ ; then is it true that $f(x)>0,\forall x>0$ ?

Let $\Omega\subset\mathbb{R}^n$ and $T$ is an operator (For example H-L maximal operator $M$, Calderon-Zygmund operator $K$ etc.) $$ (1)... \|Tf\|_{L_p(\mathbb{R}^n)}\leq C \|f\|_{L_p(\mathbb{R}^n)} $$ and $$ (2)... \|Tf\|_{L_p(\Omega)}\leq C \|f\|_{L_p(\Omega)}. $$ Can we say anything about $(1)...

Let two points $a,b\in\mathbb{S}^{2}$, where $\mathbb{S}^{2}$ is the two-dimensional simplex in $R^{3}$ with $\sum{}x_{i}=1$ for all $x\in\mathbb{S}^{2}$. $x_{1},x_{2},x_{3}$ are the coordinates of $x$. The Kullback-Leibler divergence is defined as follows: $$ D_{\mbox{KL}}(b,a)=\sum_{i=1}^{3}b...

Let $R$ be a Dedekind domain, with fraction field $K$, let $L/K$ be a separable extension generated by one element $x$, let $g$ be the minimal polynomial of $x$ over $K$. Do we have $\mathrm{Tr}_{L/K}(x^i/g'(x))\in R$?

The problem is: find $C_n$ which is the number of all ternary strings (length $n≥0$) that don't have consecutive 1's and 2's using combinatorics. Any tips or advices?

Integrate $\displaystyle \int{\dfrac{x}{1+x^4}}dx$. The best way I can think of doing this is by breaking $\dfrac{x}{1+x^4}$ into partial fractions but that would be messy.

Suppose we have an abstract $\sigma$-finite measure space $(X,\mathscr{A},\mu)$ and let $\nu$ be another measure on $(X,\mathscr{A})$ aswell, with the property that $\nu <<\mu$, i.e $\nu$ is absolutely continuous w.r.t $\mu$ . Does it necessarily imply that $\nu$ is $\sigma$-finite on $(X,\math...

Suppose $\Omega \left( \mathbf{\alpha }\right) $ is a $T\times T$ full rank matrix where $\mathbf{\alpha }$ is a $p\times 1$ vector, then what's the exact expression for $\frac{\partial \Omega ^{-1}\left( \mathbf{\alpha }% \right) }{\partial \mathbf{\alpha }^{\prime }}?$ If $\mathbf{\alpha }$ is ...

If $sin x +cos x = a$ Then how do I get value of modulus of $sin x - cos x$, value in terms of $a$

I'm using a 4th order Adams predictor-corrector method to numerically solve a regular differential equation. Now I would be interested to be able to include a noisy term to the equation -as in the Euler-Maruyama method, the classical and easy way to simulate a Brownian motion via a Wiener process...

today I have big problem. Our teacher gave us this HW with exponential equation for marks. I do not want to get bad mark so I am here. Please help me. I did a lot. And I also know how to solve basic exponential equation problems, but not these. Please help me. enter image description here

Say I have a ODE: $f'(x)=-kf(x)$ $f(0)=a$, how can I explain that if $a \not= 0$ then never can f(x)=0 for all x. (Please do not solve the equation explicitly)

Let $R$ be a commutative ring with unity having at most $5$ distinct ideals (including $\{0\}$ and $R$ itself) ; then is it true that $R$ is a principal ideal ring i.e. is every ideal of $R$ principal ?

In corollary 59.2, Munkres states: Suppose $X = U \cup V$, where U and V are open sets of X; suppose $U \cap V$ is nonempty and path connected. If U and V are simply connected, then X is simply connected. Then comes exercise one. As a note, Munkres states that the union of two simply connected ...

Solve equation $$(x+y\sqrt{2})^6+(u+v\sqrt{2})^6=5+7\sqrt{2}$$ for $x,y,u,v$ rationals. Sorry but I have not any idea.

I'm asked to sketch the subset for: {z ∈ C : |z + i| + |z + 1| = 2} I've gotten to the point where I've got the modulus form of |z + i| + |z + 1|: sqrt(x^2+(y+1)^2) + sqrt((x+1)^2+y^2) = 2 How do I change this to the equation for an ellipse ?

I m sorry . This question is so primary. T is a bounded linear operator on U and U is a Banach space. (||T||<1) then id-T is a surjective operator.

S is a finite set, and v: S → ℕ is a valuation function. We have S := {0,1,2} and the valuation function v v: S → ℕ, k → 2k+1 The task is to list all values for v's utility function u: utility function u How would I go on about doing so? We weren't introduced to the concept of valuation ...

Let $g,h \in k[x,y]$ such that $\sqrt(h,g)=(x,y)$. Is $k[x,y] \backslash (f^2+h^2)$ integrally closed?

Let $A\in \mathcal{R}^{nxn} $ be a real matrix and $x_0,x\in \mathcal{R}^{nx1} $. What is the gradient of the following function at $x$ assuming $Ax \neq x_0$: $||Ax-x_0||_1$ thanks