How do I see that the tensor product operation for vector bundles over a fixed base space is commutative, associative, and has an identity element?

I have a question about the process to find the mass of a sphere with a varying radial density in respect to the radius. It's something really simple, but I would like someone to explain it me. Say that the density varies with: $\rho(r)=a-br$ So when you go from $m=\rho V$ Why is it that it c...

while working on a signal processing problem i've reached to the following: So my aproach was: Am I doing something wrong? Is it valid Y(f)=[X(f) x H(f)]*W(f)=X(f) x [H(f)*W(f)] If you could help me move further simplyfing Y(f) Thank you in advance

Find distinct left cosets of SL(2,R) in GL(2,R) I am guessing that it should be for a in R, a*SL(2,R) Is this true?

I've been trying out some of the early problems in the probabilistic method, since I hear it can prove many problems in discrete mathematics. Let $A_1,..., A_n$ be events such that the independent of all events except for at most $m$ of them. Taking $0 < \epsilon < 1,$ and $Pr[A_i] \leq \frac{\ep...

Suppose that the continuous function $f:[a,b]\rightarrow\mathbb{R}$ has $\int_{a}^{b}f(x)=0$. Prove that there is some points $x_0$ in the interval $[a,b]$ at which $f(x_0)=0$. Proof: Since $\int_{a}^{b}f(x)=0$, there exists a partition sequence $\{P_n\}$ such that $$\sup\{L(f,P_n)\}=\inf\{U(...

I'm trying to prove the following lemma: We have $m$ is a real number greater than 1, and an open interval $(a, b) \subseteq [0, m)$ with $a < b$. Then if $n\text{ mod } m \in (a, b)$ for some integer $n$, then there is another integer $j > n$ so that $j\text{ mod }m$ is also in $(a, b)$. This ...

Let $z \in \mathcal B$ be non-zero. Let $T \in \mathcal B^*$ be defined as the mapping $$ \operatorname{span}\{z\} \ni \alpha z \mapsto \alpha \in \mathbb{C}$$ and $T(y-z) = 0$ for $y \notin \operatorname{span}\{z\}.$ Is it then true that $$\mathcal B = \operatorname{ker} T \oplus \operatorname...

...How can I calculate the work required to move the Earth 1 cm farther away from the sun, given that the gravitational constant is 6.7 * 10^-11 m^3/kg sec^2?

$a1. ∃x [p(x) ∧ ∀y [p(y) → q(x, y)]]$ given $a2. ¬ ∃x [p(x) ∧ q(x, x)]$ given

I believe this is a Liouville's Theorem problem, but I am unsure as to how to solve it. |f(z)|< |z^2 + 3z +1| for all z, and that f(1) = 2. Evaluate f'(2), and explain your answer.

Find the maximum profit corresponding to a $\ demand \ function \ $ of $ \ p=36-4x$ and a total $cost\ function \ = 2x^2+6$ Hello. Can you kindly help me solve this problem? Thank you in advance

I have 12 objects of 3 different types: 5 type A, 4 type B, and 3 type C. These objects must be arranged into 6 pairs such that no type is paired with another of the same type. The only arrangement I can come up with is: Three A/B, Two A/C, One B/C There might not be another way; however, I'm ...

Suppose that $f\in L^1 (\mathbb{R})$ and that for any $n\in \mathbb{N}$ there is $C_n > 0$ such that its Fourier transform satisfies $$ |\hat{f}(\xi )| \le C_n(1+|\xi |^2)^{-n}. $$ Let $$g(x) = \frac 1{2\pi}\int_{\Bbb R}\widehat f(\xi)e^{i\xi x}d\xi$$ and suppose we want to show that $g$ is smo...

So I'm refreshing on integration and venture across a property of definite integrals I've been taking for granted for quite some time: \begin{equation}\int_{a}^{b} f(x) \mathrm{d}x =-\int_{b}^{a} f(x) \mathrm{d}x \end{equation} Stewart's Calculus claims this property is owed to $\Delta x= \dfra...

If I have the numbers 1, 2, and 3 and there are 4 different places (Example: 1111 or 1211 or 2312), how many different combinations can you have for the numbers 1, 2, and 3?

Spent WAY too long trying to figure this out and I just don't know what I am doing wrong. "A rectangular region is to be fenced using 5100 feet of fencing. If the rectangular region is to be separated into 4 regions by running 3 lines of fence parallel to two opposite sides, determine the dimens...

Problem: A few questions: what parallelogram is the solution talking about? Also how is its height $\dfrac{1}{2}$ and base $\dfrac{\sqrt{2}}{2(1+\sqrt{2}}$? Here is a picture of just the squares:

I've been going about doing one of the key results in Group Theory: Given a group $G$, for all subgroups $H \leq G,$ $o(H)$ is a multiple of $o(G).$ I have a feeling a good start to this would be using the properties of $\mod H$ to start this, because it seems to me that $\mod H$ partitions $G$ i...

I'm having trouble with the following complex integral: g(w) = ∫ z exp(z^3)/(z-w)^3 dz where C is a positively oriented ellipse x^2/9 + y^2/4 = 1 Find g(2+3i) in the form a +bi I know z(t) = 3cos(t)+2isin(t) However, when I substitute this in (along with the z '(t)) the integral becomes a mess...

I am not sure how to approach finding the limits of integration. Would I need to change coordinate systems?

The problem assumes that it is true, but I'm not quite sure how to go about proving it. Where do I proceed?

I'm brushing up on algebra, reading over some questions I found online. I was wondering, if $p(x)\in \mathbb F[x]$, for some field $\mathbb F$, then what is $\langle p(x)\rangle$? Does it mean the elements like $p(x)+p(x)$, or elements like $p(x)p(x)$? Or something else? Thanks!

Why is taking the average of both the x and y values of the endpoints of a segment result in the coordinates of the midpoint?

I am trying to prove the following statement: Let $X, Y , Z$ be random variables, then $P(X \vert Y) = \sum_{z} P(X,z \vert Y)$. I have a sketch of the proof but I do not know if it is correct: $$\sum_{z} P(X,z \vert Y) = \sum_{z} \frac{P(X,Y,z)}{P(Y)} =$$ $$\frac{\sum_{z}P(X,Y,z)}{P(Y)}...

Recently I've been experimenting with polynomial interpolation to create a simple model of a battery discharge curve. My mathematics knowledge is a bit limited and this is part of my quest to find interesting things that my professors never mentioned. I've found something very interesting while ...

I was reading about the classical problem to prove homeomorphism between subspaces of $\Bbb R^2$ a rectangle of the kind $R=\{(x,y):|x|\le a \land |y|\le b\}$ and some disc $D=\{(x,y): x^2 + y^2 \le r\}$. It is clear that we can make bijections between the two spaces but the hard part is take so...

I have been told that in a fraction if we only have little ohs then we can't conclude anything taking the limit. As an example say we have $$ \lim_{x \rightarrow 0}\frac{o(x^3)}{o(x^2)}$$ then we can't conclude anything but this confuses me slightly because it seems to me we should be able to sa...

Let $(X, \mu)& be a $\sigma $-finite measure space, $f $ be a $L^2 (X)$ function, $K (x,y)$ a $L^2 (X\times X)$ function. Define $Tf (x)=\int_X K (x,y)f (y)dy$. I need to prove that $||Tf||_2 \leq ||K||_2||f||_2$. I can bound the left side with $\int_X |f (y)|^2 \int_X |K (x,y)|^2 dx dy$. But I...

The problem states: How many non-empty collection of balls can be formed from x-red balls, y-red-balls, and z-green balls. I don't want the answer, but rather the explanation of what is "non-empty collection". To me it sounds something like the elements in the power-set of colored balls minus t...

Suppose that I have the following situation: everything takes place on complex plane, I have some probabilistic measure $\mu$ and suppose that I have a family of function $(u_w)_w$ parametrised by $w$ where $w$ runs throught some subset of the complex plane (it may be whole $\mathbb{C}$). Suppose...

What is the value of $142,857 \times 7^2$? Obviously you could solve this with a calculator and be done. But is there a more clever way to calculate this?

I recently asked this question. In that question I presented a hand-waving proof as part of the question. There was some confusion as to the validity of my hand-waving proof. So I wanted to make it more precise. The difficulty that I had is that it's been 25 years since I've written formal logic ...

Is it something kind of like 30cos(30) and 60cos(30) ?

So, i'm really strugling with this one, when studying the triangle inequality, the inequality $−|x|≤x≤|x|$ pops really often, however I just can't get why this is true. I've seen only two different approaches, one that it seems to me too trivial: Since: $|x|≤a⇔−a≤x≤a$ We can write: $|x|≤|x...

A produce stand sells roasted peanuts for $2.99 per pound. The function C(p)=2.99p relates the total cost of the peanuts to the number of pounds purchased p

can anyone please give me some clarity on this question that if $||\frac{f_n}{x-i}||_{2}$ goes to 0 as n goes to infinity. how can i justify that $||f_n||_{2}$ goes to 0 also? I have been trying to use the Holders inequality but I am getting the whole thing messed up

I'm trying to create the Rubik's Revenge (4x4x4 cube) group in GAP . Take the following net of the 4x4x4 cube with each sticker labelled with a number. The front, left, upper, right, down, and back faces labelled with their respective initials. U [64][65][6...

What method should I use to find the Taylor series of $f(x)=\frac{x+2}{2-3x}$ with center 2?

Solution of $\frac{dy}{dx}=\frac{1}{y}-\sqrt{4x-x^2}$? I dont know how to get the solution of the equation.Thanks for any hint or detail answer.

$y=f(x)$ How to solute $-\sqrt{4y-y^2}y'=y''-1$ ?

The question is to show if $G$ has order 77 then $G$ has a subgroup of order 7. Without using Sylow Theorems. Attempt sketch: Let $x \in G$. By Lagrange's theorem the order of $x$ is either $1, 7, 11$. Suppose $x \neq e$ Then $x$ has order $7$ or $11$. Now suppose $|x| = 7$. Then $x$ is the ge...

I have a question concerning an eigenvalue problem that I've been working on for a while now. Essentially, I have a system of coupled, linear, non-constant coefficient, second order differential equations of the following form: $\frac{\partial^2 y_1}{\partial x^2} + a_1(x) \frac{\partial y_1}{\p...

Let X1, X2,....Xn be a random sample from the exponential distribution with the pdf f(x│θ)=(1/θ) e^((-x)/θ) where x≥0 find E(T_i ) (MEAN), i=1,2,….,n, where T_i= X_i /(X_1+X_2+⋯+X_n ) would you please help me to solve this problem

The question is Let $T: \mathbb{R}^3\rightarrow \mathbb{R}^3$ be defined by $T(v) = Av$ wheere $A$ is given by $A = \frac 13 \begin{bmatrix} 2 & -1 & 2 \\ 2 & 2 & -1 \\ -1 & 2 & 2 \\ \end{bmatrix}$ $T$ is a bijection if it is both injective and surject...

I need help finding an explicit formula from this recursive formula: a_n = a_n-1 -3/5(n-1) a_1 = 1/2

Let $p$ and $q$ be primes such that $p=4q+1$. Then $2$ is a primitive root modulo $p$. Proof. Note that $q\not=2$ since $4\cdot2+1=9$ is not prime. $\mathrm{ord}_p(2)\vert p-1=4q$, so $\mathrm{ord}_p(2)=1,\;2,\;4,\;q,\;2q,\;\mathrm{or}\;4q$. Clearly $\mathrm{ord}_p(2) \not= 1$, and $\mathrm{o...

Let , $y=x\sin x+x^2$ be a solution of the $7$-th order ODE $$a_7y^{(7)}+a_6y^{(6)}+\cdots +a_0y=0.$$Then which of the following are correct ? (A) $a_7+a_3=a_5$. (B) $\displaystyle \sum_{i=0}^7 a_i=4$. (C) $\displaystyle \sum_{i=0}^7 a_i=3$. First I differentiate $y$ ,$7$-times and then put t...

Let $p > r$. Suppose I have a $p \times r$ matrix $V$ of rank $r$ and a $p \times p$ positive semi-definite matrix $X$ of rank $q$. What does this imply about the rank of $$ V^T X V?$$ Is this matrix positive definite if $r < q$?

A normal matrix over $\mathbb C$ with all eigenvalues real is hermitian (using diagonalization) .But a normal matrix with real eigenvalues is symmetric, is this statement true? I think, all normal matrices are not diagonalizable i.e rotation matrix .please someone explain over $\mathbb R$,are t...

Let G=Z4+U(4), H=<([2]4,[3]4)>, K=<([2]4,[1]4)>. Show that G/H is not isomorphic to G/K. [2] means 2 mod 4.

If 2n people are sitting at equally spaced intervals around a circle. How many ways can they form n pairs if no two persons seated directly opposite each other can form a pair? What I did was draw some examples for low values of 'n' and also tried to see it as a graph on 2n vertices where there...

Hey i was wondering if anyone can help me with the following optimization problem: enter image description here I know the answer to (a) is 100 = 4s + 2(pi)(radius^2) and the answer to (b) is A=s^2 + pi(radius)^2 but what si the answer to (c) and (d)? for (c) do I provide the domain for the are...

Using FLT for exponent 3, I need to show that if n positive integer is divisible by 3, then there are no x,y,z positive integers such that x^n+y^n=z^. This is what I did: if 3/n then n=3.k, for some k positive integer. Then if (x^3, y^3,z^3) is a solution for the exponent k (x^3)^k+(y^3)^k=(z...

In S5, we have aba^-1=b^2, b=(12345), find a. I have tried different ways to substitute/rearrange, but none of them worked.

Problem: Find a basis of the vector space $\{x \in \mathbb F^n \mid x_1 + \cdots + x_n = 0\}$. Solution: The vector space in question is exactly $$\left \{ \begin{pmatrix} -x_2 - \cdots - x_n \\ x_2 \\ \vdots \\ x_n \end{pmatrix} \; : \; x_2, \ldots, x_n \in \mathbb F \right \}.$$ Setting one ...

Given that the utility function is $u(s_1,s_2)=s_1-a(s_1-s_2)^2$, where $a>0$. How can I show that this preference is non-monotone?

Can someone recommend a book which gives an informal but axiomatic presentation of elementary logic? More precisely, I envision a text with the following features: Takes grammatical sentences (which may be true, false, undecidable, contradictory, etc.) and logical symbols (connectives, quanti...

I was wondering how exactly to find the domain and range of a rational function. I have a test next week and I wish to know a simple and straightforward method to find the domain and range. You can use the function (x+3)(x+4)/(x-2)(x-6) And also (x+3)(x+4)(x+8)/(x-3)(x-2)

I know this may be routine and simple, but I'm not great at proofs and was wondering if someone could look at my solution and offer corrections or tips? If A and B are sets, prove that $ A \cap \emptyset = \emptyset. $ My proof: Let $x \in A \cap \emptyset $. Then x $\in A$ and $x \in \empty...

Let x,y,z be integers with x > y. Then xz > yz. Prove or disprove. If the statement is false, modify the hypothesis so that the edited statement is true.

In GAP > gap> f:=FreeGroup(2);; > gap> Rank(f); Error, no method found! For debugging hints type ?Recovery from NoMethodFound Error, no 2nd choice method found for `RankPGroup' on 1 arguments called from <function "HANDLE_METHOD_NOT_FOUND">( <arguments> ) called from read-eval loop at line 11 o...

Evaluate $\sum\limits_{k=0}^n\sum\limits_{l=0}^n\left(-1\right)^{k+l}\binom{n+k-l}{n}\binom{k+l}{n}\binom{n}{k}\binom{n}{l}$

Let $W_t$ be one-dimensional Brownian motion, to calculate $\int_0^tW_sdW_s$ by the definition of stochastic integration, one way is use $W^{(n)}_t=W_{[nt]/n}$ to approximate $\int_0^tW_sdW_s$. My question is the definition of stochastic integration require the simple function to be bounded, but...

http://imgur.com/VdGv6JH I'm a bit confused on how to solve this. This is my work so far: http://imgur.com/oCdydU7 But the last step I have done is apparently wrong. Any help would be appreciated!

Can a discrete random variables takes the values $+ \infty$ and $- \infty$ ? Can someone explain to me this with an example?

Suppose a pair of positive integers x, y are called square mates if their sum x+y is a perfect square. (Note, square mates is a made up definition.) Then, for all x in the natural numbers, there exists y in the natural numbers such that x < y ^ x,y are square mates. Prove or disprove. If this sta...

Consider the logistic map $x_{n+1}=rx_n(1-x_n)$, whose bifurcation diagram is shown below for $2.4 < r < 4.0$: I need to find a particular value of $r$ so that "attracting $2^k$ periodic points (a result after $k$ instances of period doubling) accumulate". This was one part of my lab assignment,...

What is the difference between a function and a sequence? I have googled it but I have not found any solid answer anywhere.

Let's say you have a simple parametric equation where x= sin(3t) and y=cos(7t). This is a pretty simple parametric equation that generates a relatively complicated Lissajous figure. You can check it out here pretty easily: www.wolframalpha.com/input/?i=parametric+plot (If I presented the enti...

Rudin in his PMA defines: $$E(x)=\sum \limits_{n=0}^{\infty}\dfrac{z^n}{n!} \quad (z\in \mathbb{C}) \qquad (1)$$ He write that "these limits follows directly from (1)" 1) $E(x)\to +\infty$ as $x\to +\infty$. 2) $\dfrac{E(h)-1}{h}\to 1$ as $h\to 0$. I can't understand how the directly follows f...

Let $\mathcal{A}$ be a nonempty finite set of positive integers, with $\forall$ r $\in$ $\mathcal{A}$, $\forall$ s $\in$ $\mathcal{A}$ : r|s or s|r. (i). Prove $\exists$t $\in$ $\mathcal{A}$: t|a, $\forall$ a $\in$ $\mathcal{A}$ (ii). Prove t is unique (iii). Suppose $\mathcal{A}$ $\subseteq$ ...

If I distribute the limit into the product I end up with $e^{-n} \cdot e^{n} = 1$ Where does the 1/2 show up?

Let $G$ be a topological group acting continuously on a topological space $X$. Let $A$ be a $G$ - invariant subspace of $X$. Then is it true that $\bar A$ is also $G$ - invariant? (where $\bar A$ is the closure of $A$ in $X$) My attempt - Let $g\in G$ and $x\in \bar A$. To prove that $g\cdot x ...

Why is "x^2 - 2x > 0" the same as x<0 or x>2 and "x^2 - 2x < 0" same as 0

Sequence $\{a(n)\}$ of real numbers is such that $\forall\space\lambda\in(1,2)$ sequence $a(\lfloor{\lambda}^n\rfloor)$ has a finite limit. Does it follow that $\{a(n)\}$ is convergent?

I am presented with the following problem: Let $T: R^3 \rightarrow R^3$ be linear and define it as such: $T(x) = Ax$ for some $3x3$ matrix $A$. Let $m_1, m_2, m_3 \in R^3$ be linearly independent, call this basis "$m$". Find $[T]^m _m$. In general terms how can I find the matrix of transforma...

In page-13 of Graph minors. X. Obstructions to tree-decomposition, $\gamma(G)$ introduced as maximum size of an edge. What is the definition of size of an edge? I think it may be number of edges that incident on terminals of an edge.

Is there any way to derive Maclaurin's Series for $sec(x)$ with the help of Maclaurin's Series for $tan(x)$? Maclaurin's Series for $tan()$ is: $tan(x)=x+\frac{x^3}{3}+\frac{2x^5}{15}+....$

The Tower of Hanoi is a famous math puzzle consisting of a board with three dowels and a collection of $n$ donuts of different sizes (radii). The donuts are on the first dowel arranged in size order with largest on the bottom and smallest on top. The object of the game is to move all of the donut...

How come: 3^n/2^(2n-1) < 0.5 * 10^(-5) will be equal to n = 45 ? Thanks in advance!

Let $d_1$, $d_2$, $d_3$ $\in$ {0,1,2, $\dots$ 9}. What is the fraction representation of 0.$d_1$$\overline{d_2 d_3}$ ? Prove this in general.

Let A and B be two $6x6$ nilpotent matrices. How would you prove they are similar?

How do I find the PDF for $Y$ = e$^X$ when $X$ is $N$(μ,σ$^2$) I have seen the problem where $X$ is $N$(0,1), but I am curious on how to find it given just these parameters?

I recently had an algebra exam and one of its questions was: Let $R$ be a ring with a $1$ and let $x\in R$ be nilpotent, i.e., there exists a positive integer $n$ such that $x^n=0$. Then $1+x$ is a unit. This is what I did: Observe that $$\begin{align*}\left(1+x\right)\left(1-x\right)&...

In Quantville the weather follows a peculiar pattern described in the table below: Tomorrow's Weather Sunny Cloudy Windy Rainy Today's Weather Sunny 40% 40% 20% 0% Cloudy 20% 20% 30% 30% ...

I've encountered the following questions: Suppose $X_n \rightarrow X$ in distribution, and $a<b$ Prove that $$ P( a \le X\le b) \ge \limsup_{n\to \infty}P(a\le X_n\le b) $$ I know how to find the limsup of a r.v, but i'm confused on the concept of limsup of a probability. My idea of approac...

I dont have poker game knowledge. Any suggestion for this? The probability of drawing a flush is .001980439. The probability of drawing a straight flush is 0.00001544.

when determining the intervals in which the graph of a function increase or decrease, Some boks include the ends while others do not. which one is true or which one is more prefreable?

Suppose $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are sets with $\mathcal{A} \cap \mathcal{B} \cap \mathcal{C} = \emptyset$. Then $| \mathcal{A} \cup \mathcal{B} \cup \mathcal{C} |$ = $|\mathcal{A}|$ + $|\mathcal{B}|$ + $|\mathcal{C}|$ Prove or disprove. If false, modify statement to make i...

Suppose I have a simple program that implements an algorithm (say depth-first search), written in a simple imperative programming language with the standard for loops, recursions, conditional statements and so on. It takes in a well-specified input and has a well-specified output. Suppose I want...

I have a problem about Inverse Laplace Transform, I would be appreciated to get your help for solving this problem (It took me about several hours to think but didn't come up with any solution), please find the inverse laplace transform of : Problem Here where "a" is an constant.

Let $y(x)$ be continuous solution of initial value problem $\frac{dy}{dx}+2y=f(x),y(0)=0$. Where $f(x) = \left\{ \begin{array}{rcl} 1 & \mbox{for} & 0 \leq x\leq 1 \\ 0 & \mbox{for} & x>1 \end{array}\right.$ Then FInd $y(\frac{3}{2})$ Answer is $\frac{sinh(1)}{e^2}$ I have tried the follo...

What is number of conjugacy classes in the permutation group $S_{6}$ I only knows that $o(S_6)=6!$

This question has been bugging me for a while. Suppose that $H$ is a subgroup of $G$ such that $Ha\not=Hb$ implies that $aH\not =bH$. I need to show that $gHg^{-1}\subset H$. Now, take $x\in gHg^{-1}$. Suppose it is not in $H$. Then $x=ghg^{-1}$ for some $h$. As $x\in H$, $Hx\not=H$ whih implies...

$$X=\frac{\left(\frac{c}{r^2}+\frac{1-c}{(1+r)^{T+1}}\right)}{\left(\frac{c}{r}+\frac{1-c}{(1+r)^T}-1\right)}$$

Let $p$ be an odd prime. Show that $x^4 = -1$ (mod $p$) $ \Leftrightarrow p = 1$ (mod $8$)

Let $x\in C_G(H)$. This means $xh = hx$ for $h \in H$. Then $xH = Hx$ (This is that part I'm not so sure about). Hence, $x \in N_G(H)$, so that we have $C_G(H) \le N_G(H)$.

Determine whether the first and second order weak derivatives of the following function $f: \Omega \to \Bbb R $ exist, and if exist what they are.\begin{align} \end{align} $a. \Omega_1 =(0,2\pi), f_1(x):=|\cos x|. $\begin{align} \end{align} $b$.Determine whether the function is contained in the ...

Are the folowing equivalent? Hilbert's tenth problem and the theorem of Matiyasevich, Davis, Putnam and Robinson that the diophantine sets are equial to the enumerable sets?

Do you have an example of a compact operator for which the image of the closed unit ball is not compact?

Can anyone please explain what a Weyl group is and what a Weyl group of $V_{4}$ in $S_{4}$ is? I do not understand this exercise.

Let $\Omega=(-1,1) *(-1,1). f,g: \Omega \to \Bbb R$ are defined by $$ f(x,y) = \begin{cases} 1, & \text{$x\le 0$} \\ 0, & \text{elsewhere} \end{cases} $$ $$g(x,y)=||x-y||$$ Determine whether $\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial g}{\partial x}$ and $\frac{...

I'd like help with 2 + 3Logx. With detailed workings on the question. Thanks.

Good Morning! This is Rajeev, How are you? I hope you will be fine. I am gain sending you a numeric based series puzzle for a hope of solution. If possible, please help to solve. Importance High!!!!!!!!!!!!!!!!! Dear Sir / Madam,   I am putting a signs series in front of you which based on rea...

$$\frac{8}{3}=\frac{\log{x}}{\log{y}}-\frac{\log{y}}{\log{x}}$$ When I graphed this implicit function on desmos (https://www.desmos.com/) it appeared as if there were two solutions as $x\to{0}$ from the positive direction: $y\to\infty$ and $y\to{0}$. However, neither of these solutions make much...

Simulate the following situation. Attach the code as part of your submission. At a post office, customers enter a single line waiting to be served by any one of two clerks. Every minute there is a 60% chance that a new customer arrives. If there is no one in line and a server is free, the cus...

Is x•y=60 same as y=60-x If not then y is equal to what ( in x•y=60 )? I'm confused .. I'm not sure what should y be because it multiplied ... ------------/----------/-----------/---------- This is just a copy , cannot meet .... Is x•y=60 same as y=60-x If not then y is equal to what ( in ...

I have looked at the proof Proving that the Fourier Basis is complete for C(R/$2*\pi$ , C) with $L^2$ norm but am having trouble understanding the argumentation about the Hilberspace. I think the proof is quite nice and would like to understand it, and also need to refresh my Functional analysis....

what are the difference between: A → (B → C), (A → B) → C, A∧B → C, A → B ∧C, (A → B)∧(B → C)? Why we can not write A → B → C And the same questions about: A ↔(B ↔ C), (A ↔ B) ↔ C, A∧B ↔ C, A ↔ B ∧C, (A ↔B)∧(B ↔ C)? Why we can not write A ↔ B ↔ C? Can someone explain me this ?

I read from a book that for linear regression problem with intercept: $$Y = X\beta + e, e\sim N(0,\sigma^2 I)$$ The diagonal entries of the hat matrix $$H = I - X(X'X)^{-1}X'$$ can be expressed by $$h_{ii} = \frac{1}{n}+(x_i-\bar{x_i})'(\mathcal{X}'\mathcal{X})^{-1}(x_i-\bar{x_i})$$ where ...

Suppose G is a finite group in which every element has order 2 or 1, and that G contains some element different from the identity element. How can I explain that the order of G must be even?

The space of a rectangle is 60 , If you add 3 to the length of the rectangle and 1 to the width then the new space of the rectangle is 90. What is the length and width of the original rectangle ? I tried doing this X is the length Y is the width X*Y=60 (X+3)(Y+1)=90... But it didn't work f...

How will you derive the quadratic function if you only have the zeros which are 2+ √7 and 2-√7?

I have three data: coordinates (x,y) angle (degree) distance (meters) How should I calculate new (x,y) ? for example with these data : begining coord. (100,100) , angle - 30 , walked 2meters. What is the result? Sorry for formulation of this topic and explanation of my problem, but i really ...

Most classification algorithms are developed to improve the training speed. However, is there any classifier or algorithm focusing on the decision making speed? I can achieve enough training data，and endure the long training time.

Consider the equation $$f(x)=f(\frac{1}{x})$$ Where we want $f$ to be real-meromorphic. Are all solutions $f$ of the form $$f(x) = g({x}{1+x^2})$$ Where $g$ is a continu function ?

How do we calculate time that we need for specific distance if we have non-constant acceleration. I have some formulas for other variable but don't have for time: Speed at time: $$ v(t) = v_0+at+\frac{j(t^2)}{3} $$ Distance at time: $$ x(t)=x_0+v_0t+\frac{a(t^2)}{2}+\frac{j(t^3)}{6} $$ Acceler...

I have been trying to figure out an expression for evolution of a hysteresis shown in the picture below. It is given that both A and d are functions of 'phi'. The shape of the geometry is a perfectly symmetric parallelogram. There is an expression used to define evolution of A with respect to a...

I got a bit bored with High School maths so I picked up a copy of Calculus by Spivak. I am really enjoying the book and have found that the proofs and theorems aren't as hard as others have made them out to be. However, I am struggling a bit with the algebra. The education I have received thus fa...

Why is: $\sin(\arccos(y)) = \sqrt{1-y^2}$ ? I thought maybe transforming sin to cos: $\cos(\frac{\pi}{2} - \arccos(y))$ but it doesn't get me anywhere. Hint?

Let $a<b$ be real numbers. Prove that $[a,b]$ is compact. Below I present my solution. I thought it's good enough, but my TA said it's incorrect. I don't see where there is a problem. Could you help me find it? My proof: Let $\{U_i\}_{i\in I}$ be an open cover of $[a,b]$. Let $$S=\{x\in [a...

A column vector $a$ is said to an element of R^(4) when $a$ is [1, 2, 3, 4]' (Transpose of this row vector). Therefore my question lies in the fact that I do not understand how row vectors can be represented with this notation. I also understand that Matrices can be represented by for e.g. R^...

I'm currently programming a small script and have a statistical problem: I have 24 sequences, consisting of 4 different characters and a length of 8 characters. It is necessary to group the sequences to 8 sequences per group. To make it more clear, here are two example groups: Example 1: CAAGTC...

Given a $2$-digit sequence (which can start with $0$), let us define a nudge as the operation of either increasing or decreasing one of the digits by $1$, or changing a $0$ to a $9$ or a $9$ to a $0$. For example, we can nudge $19$ to get $09$, $29$, $18$, or $10$. Suppose $S$ is a set of $2$-di...

Let $A$ be a basic finite dimensional algebra over an algebraically closed field $K$, and let $M$ be a finite dimensional right A-module. Show that $l(M)=dim_{K}M$, where $l(M)$ is the length of the composition series of $M$.

I was watching a lesson in youtube on the wave equation and couldn't figure out how the person completed one of the steps where he integrated a second derivative.

I'm not able to understand the folowing example from Dirk van Dalen's logic and structure. What is confusing me is the relation defined on the structure and the predicate $P $. We have a set $A=\{0,1 \} $ and a relation defined on $A $ as the set with two elements $\{\langle 0.1 \rangle , \lan...

Let T be a 4*4 real matrix such that $T^4=0$.Let ki =dim Ker $T^i$ for i=1,2,3,4.Then which of the following is NOT a possibility for the sequence k1,k2,k3 and k4 1)k1 = 3,k2 = 4,k3 = 4 and k4 = 4 2)k1 = 1,k2 = 3,k3 = 4 and k4 = 4 3)k1 = 2,k2 = 4,k3 = 4 and k4 = 4 4)k1 = 2,k2 = 3,k3 = 4 and k...

I have: $F(\omega|x) \sim N(1, (1+x)^2)$ $G(\sigma|y) \sim N(1, (1+y)^2)$ How can I calculate and plot the joint pdf of $\omega$ and $\sigma$ for $x,y\in [0,1]$?

I want to prove an inequality such that $$ \int_{B}|f(y)|dy\leq |B|^{1-\frac{1}{p}}\|f\|_{L^p(B)}, $$ where $B\subset\mathbb{R}^n$ is a ball, $p>1$ and $\|f\|_{L^p(B)}=(\int_{B}|f(y)|^pdy)^{\frac{1}{p}}$. This inequality is a direct consequence of Hölder's inequality. But i want to prove this in...

( (1+i)^-n ) / i = -293953 / ( ( (1+i)^-5 ) - 1) i = 0.03 1.03^-n / 0.03 = -293953 / ((1.03^-5)-1) I end up with this: -n = ln(1.03)/ln( (-293953-0.03) / ((1.03^-5)-1) ) n= -375.9133691 However, I know the answer should be 15. What am I doing wrong?

Let $C$ be a category equipped with a Grothendieck pretopology. Let $U\in C$ be an object. A sieve on $U$ is called covering if it contains a covering family $U_*=\{U_i\to U\}$ from the pretopology. Clearly, there may be many covering sieves on $U$ containing the same covering family $U_*$ and t...

Let $X$,$Y$ be uniformly distributed continuous random variables on [0,1]. How is the random variable $X/Y$ distributed?

Attempt: Assume $G$ only has elements of order 11 and that there exists $x \in G$ such that $o(x) = 11$. Then $x^{o(G)} = e$. $x^{33} = (x^{11})^3 = e \implies x^{11}$ has order $3$. Contradiction.

I would appreciate if somebody could help me with the following problem Q:How to proof mathematical induction? $${(n+1)}^{n+1}>{(n+2)}^n$$

$lim _{x\to \infty} (3x^2-x^3)^{\frac{1}{3}}+x$ can I look at $lim _{x\to \infty} (3^{\frac{1}{3}}x^{\frac{2}{3}}-x+x)$?

I stumbled across a problem where I had to prove that a given function is one-to-one and onto. Curiously, I found that in order to prove that the function was onto I had first to show that it was one-to-one. Is there an instance of a proof that a given function is onto that needs the function to ...

For which $n_o \in \mathbb{N}$ is it possible to show with induction that $2\log_2(n) \leq n$ applies? for all $n \in \mathbb{N} $ with $ n \geq n_0$? How to proceed to such questions? Hope somebody can help

If I write $ \Omega \setminus \Gamma \cup \Theta$ this means $ \Omega \setminus \left( \Gamma \cup \Theta \right) $ or $ \left( \Omega \setminus \Gamma \right) \cup \Theta$?

Given N and R we know that N, R are coprime numbers. So.. How can I proove that N - K = R: R will be coprime.

I really need the solution of the exercise 11.4 (page 174) in M. Aschbacher, Finite group theory. Here is the text of the exercise. Let $G$ be a finite group, $p$ a prime, $\Omega$ a $G$-invariant collection of $p$-subgroups of $G$, $P$ a $p$-subgroup of $G$, and $\Delta\subseteq P\cap\Omega...

Let $f:U \subset \mathbb{R}^2 \to \mathbb{R}$ be a continuous function such that $(x^2+y^4)f(x,y)+f(x,y)^3=1$ for any $(x,y) \in U$. Prove that $f \in \mathcal{C}^{\infty}$.

Let $f$ be continuous function on $[a,b]\times [c,d]$. Prove that Prove that $$\int_a^b \left( \int_c^d f(x,y)dy\right) dx=\int_c^d \left( \int_a^b f(x,y)dx\right) dy$$ First of all, note that $ \int_c^d f(x,y)dy$ is continuous in $x$ and $ \int_a^b f(x,y)dx$ is continuous in $y$ so bo...

The lemma says that if m and n are natural numbers and m not divisible by n and $\zeta_m^j \equiv \zeta_m^k \pmod n$, then $\zeta_m^j = \zeta_m^k$ $\zeta_m$ is here a primitive mth root of unity of the complex numbers so $e^{\frac{2 * \pi * i}{m}}$. Now the proof is the following: We multiply t...

We know: $\gcd(a,\phi(n)=1$ and $a,n,x>0$. Show that $\gcd(x,n)=1$ and $x^a\equiv 1\pmod{p} \Rightarrow x \equiv 1\pmod{p}$ My Attempt: Using Euler's Theorem I know that: $x^{\phi(n)}\equiv 1\pmod{p}$ where $\gcd(x,n)=1$, is this enough to prove that $\gcd(x,n)=1$? Deducing Since $x^a\equiv ...

I tried using factorization of a^n-b^n for odd n in an attempt to work through to a situation where the factors are such that they cannot have n as a factor. But I reached nowhere. Here's how I proceeded - [math]a^n-b^n=(a-b)\left(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\dots+a^2b^{n-3}+ab^{n-2}+b^{n-1}\rig...

Let G be a non-cyclic group and H be an non-trivial normal subgroup. Can we state that G/H is non-cyclic?

I am stuck at one task, it's to proof the following equation using addition theorems. From a draft I understood that the equation is correct or true. But that's it. What's a good way to explain it mathematically? $$sin (x)+ sin(y) = 2 sin (\frac{x+y}{2}) cos(\frac{x-y}{2})$$ As usual, any help ...

I need to find how the sequence converges by showing how it increases and is bounded above.

I'm stuck on this exercise from Tao's Analysis 1 textbook: show that if $q, r \in \mathbb{R}, x \in \mathbb{R}^+$ then $(x^q)^r=x^{qr}$. DEF. (Exponentiation to a real exponent): Let $x>0$ be real, and let $\alpha$ be a real number. We define the quantity $x^\alpha$ by the formula $x^\alpha=\li...

Please help answer how the sequence is bounded above and increasing to show that it is convergent