I have been learning calculus from a tutor and I have been trying to solve a problem that he gave me. The problem is to find the maximum area of a right triangle with a constant perimeter [;P;]. To start solving this problem I wrote down the different equations for the area and perimeter of a rig...

Suppose that $U$ is closed. and $V=\Bbb R^n \setminus U$. Now, is V closed or neither open nor closed? Thanks in advance!

I have the following theoreom: Let us consider the matrix $A\in\mathbb{R}^{n,n}$. $\lambda$ is an eigenvalue of $A$ iff $(\lambda I - A)$ is not invertible, or, equivalently, $\det (\lambda I - A) = 0$. I managed to prove the part $\Rightarrow$, but not the other way around, i.e. given that...

I got a markov kernel $\sigma_x(A)$ and arrival times of a Poisson process $\Gamma_i$. I need to calculate the sum \begin{align} \sum_{i=1}^\infty \sigma_{\Gamma_i} (A). \end{align} I need to get \begin{align} \int_0^\infty\sigma_r(A)dr, \end{align} but i have no idea why both of them should b...

Show that line in an edge of a polytope is boundary point of the polytope

Let $f:D \to R$ with $x_0$ as an accumulation point of $D$. Prove that $f$ has a limit at $x_0$ if for each $\epsilon >0$ there is a neighborhood $Q$ of $x_0$ such that, for an $x,y \in Q \cap D$, $x \ne x_0$ and $y \ne x_0$, we have $\lvert f(x)-f(y) \rvert < \epsilon$. Pf:Let $x_0$ be an accu...

Help me, please, to solve this task: Prove that the determinant of skew-symmetric matrix of even order have a full square as a polynomial in matrix elements

$l_\infty$ is the space of bounded sequences and $c_0$ is the space of sequences converge to 0, is a closed subspace of $l_infty$. I am trying to prove that for any $x \in l_\infty$, $d(x,c_0) = \limsup_{n\to \infty} |x_n|$.

How do I show that the series $\sum \frac{n^n}{n!} x^n$ diverges at $x =e^{-1}$? I understand that $a(n) = (\frac{n}{n+1})^n$ is a strictly decreasing function and therefore $(\frac{n}{n+1})^n > \frac{1}{e}$. Hope it helps.

There's a step in a proof in Munkres' Topology that doesn't make sense to me. Theorem: Let A be a connected subspace in X. If $A \subset B \subset \bar{A}$, then B is also connected. Proof: Let A be connected, and $A \subset B \subset \bar{A}$. Suppose that $B = C \cup D $ is a separation of B....

Let $X$ be a Polish space. Let $B$ be a Borel subset of $X \times X$ with the following property: $$\forall x \in X \ \left|\left\{y : (x,y) \in B\right\}\right| \le\aleph_0.$$ Show that there exists a sequence of Borel sets $B_0, B_1, \dots, B_n, \ldots \subset X \times X$ such that $B = \bigcu...

There seems to be huge amount of discussion about converting "first order logic to CNF". But don't see much about "first order logic to DNF" conversion. What is the reason?

The joint probability density function of X and Y is given by f(x,y)=(15/25000)(xy+y^2) if y^2−25≤x≤−y^2+25 and f(x,y)=0 otherwise. Whenever I integrate for Y disregarding x I don't get a probability distribution of X that then integrates to 1. I believe that x has to be between -25 and 25 with...

Here is the integral question I know it might not be so hard, but I just cannot think of a way to solve it.

How would I determine if the following series is absolutely convergent, conditionally convergent or divergent? $\sum\limits_{i=1}^n$$\sqrt[n]{2}+1$

Let $f$ be a positive function defined on an interval $[a, \infty)$, such that $\lim _{ x\rightarrow \infty }{ { f(x) } =0 } $ ?. Prove that $\lim _{ x\rightarrow \infty }{ \sqrt { f(x) } =0 } $ ? I can't seem to be able to come up with any way to approach this proof. A hint to put me in the r...

Was just looking at this question, just would like to make sure the proof is sound. is the following statement true? $∀x,y ∈ R:∃z ∈ R$ such that $xz=y$ so immediately I saw because $x$ ,$y$ ,and $z$ are real numbers I knew that the statement was true. I took it further and demonstrated a proof...

2 people standing in front of each other in a rail with distance of 2 meters in which player A stand on point -1 and player B stand on point 1. They have only one gun with one bullet. Player A can fire on points -1 or -0.5 or 0 (it is intuitive that he could do it if be alive in points -0.5, 0)...

Let $\alpha>0$ and define the truncated random variables $X_n^{a}=X_n$ if $X_n<a$, and $X_n^{a}=a$ if $X_n\geq a$.How to prove $E(X_n^{a})\rightarrow E(X_n)$ as $a\rightarrow\infty$ ?

Find the set of real numbers $u$ such that $u/(u−1) ≤ 2$. I don't really know how to get started with this. Thanks

this is a chess like question concerning how many games are required to tell who is the stronger player. The application is in fine tuning chess engines. It is typical that during this process a chess engine will play some games with a modified version of itself. After the tests the most succes...

enter image description here HELP

Hi guys I was reading about this topic Maximum Likelihood degree. I am not very familiar with statistics and the maximum likelihood estimate, but I want to learn what is the Maximum Likelihood degree and I have read some literature http://arxiv.org/pdf/math/0406533.pdf I think I can even follow...

$X$ is a compact K$\ddot{a}$hler manifold or smooth projective variety. is there an example that a primitive class of $H^*(X, \mathbb{C})$ is wedge product of other two primitive classes?

I am studying probability and finding hard to understand the following equation. $$\pi_i=Pr[P_i(j) \leq \min\{P_s(j);s \neq i\}]=\int_0^\infty \underset{s\neq i}{\prod}[1-G_s(p)]dG_i(p)=\frac{T_i(c_id_{i})^{-\theta}}{\sum T_i(c_id_{i})^{-\theta}},$$ where $G_i(p)=Pr[P_i \leq p]$ and $G_s=1-e^{-T...

I have a unit raduis sphere with $n$ point on it. What is the best representation of these points in a 4 dimensional space for preserving the original distances? Your help is appreciated.

I apologize if the error is easily spotted. But, here it is: If we are computing $\int^1_0{\frac{x^2-y^2}{(x^2+y^2)^2}}dy$ and use the substitution $y\rightarrow x \cdot tan(\theta)$ to get $$\int_0^{\pi/4}\frac{x^2-x^2tan^2(\theta)}{(x^2+x^2tan^2(\theta))^2}sec^2(\theta)d\theta=\frac{1}{x^2}\i...

the reviews of the book on the amazon say this one is detailed and good. i once took analysis course and want to study materials again. so would this book be good for my purpose to review analysis on my own? or just baby rudin suffices? any advice will be appreciated.

Evaluate to a minimal expression: b* a* ∩ a* b* To me, the only elements to both sets are the empty string, strings containing only a, and strings containing only b, so isn't the answer just a* ∪ b* ? This was a midterm problem and the prof marked the answer incorrect. Please help. Thanks.

I am having problems with a). I figured out that the angle is 87.48 degrees. And the length of an arc C (the displacement) was found to be 3.35899 rad

Inner product is defined to be x1*conjugate of x2 + y1*conjugate of y2 +z1*conjugate of z2 Is the inner product of (1,0,0)(1,2,0), 1*1 or 1*-1? Also I have the complex vector space with span ((1,1,0,0),(1,1,1+i,1+i),(1 + i,1+i,i,i),(1,1,1,1)). I know that they are LI since the only combination ...

I am given the results of 30 students for a Data Management test: $78, 81, 55, 60, 65, 86, 44, 90, 59, 81, 77, 71, 65, 39, 80, 72, 70, 64, 68, 78, 88, 73, 61, 70, 75, 96, 51, 73, 65, 67.$ The question I'm confused with goes as following: a) Determine the range for this data. That's easy - I su...

What does it mean to take an integral: $\int y dx + xdy$ I would guess that this means that this integral over a region multiplies the infinitely small increments of $x$ ($\Delta x$) by $y$ and the infinitely small increments of $y$ ($\Delta y$) by x, but I still can't really picture what this i...

I am trying to make sense of what the following ring really is. I have that $$\mathbb{Q}(\epsilon)=\mathbb{Q}[x]/x^2$$ where $\epsilon$ denotes the coset of $x$ in the quotient ring. So I know elements of $\mathbb{Q}(\epsilon)$ can be written as $a+b\epsilon$ where $a$ and $b$ $\in \mathbb{Q}$ ...

I have recently come across two formula's that I am unfamiliar with and would like to know if they are both aspects of the same thing: $$\color{purple}{\cfrac{1}{f^{\prime}(a)}=f(a)(f^{-1})^{\prime}}\tag{1}$$ $$\rho_x (x)=\rho_\alpha(\alpha)\left|\frac{\mathrm{d}x}{\mathrm{d}\alpha}\right|^{-1}...

e^xi=x/(1-i)+y/(2+3i) Find all solutions for x and y `

(My question is similar to this one at a high level, but I am looking for something more rigorous.) I have started into Michael Spivak's "Calculus" textbook. Problem 3 (v) on page 14 asks for a proof that "$\frac{a}{b} \big/\frac{c}{d} = \frac{ad}{bc}$, if b, c, d $\neq 0$". The only proof tha...

Define $f:(0,1)->R$ by $f(x)=\frac{\sqrt{9-x}-3}{x} $ I know that $f(x)=\frac{\sqrt{9-x}-3}{x}*\frac{\sqrt{9-x}+3}{\sqrt{9-x}+3}=\frac{9-x-9}{\sqrt{9-x}+3}=-\frac{1}{\sqrt{9-x}+3} $ Since $f$ is only defined in $(0,1)$, $-\frac{1}{5}<-\frac{1}{\sqrt{9-x}+3}<-\frac{1}{6} $ So can I just pick so...

Suppose I derive $\lim_{n\rightarrow\infty}{a_n}\geq\lim_{n\rightarrow\infty}b_{n}=\infty$. Can I conclude $\lim_{n\rightarrow\infty}a_n=\infty$?

I checked the differentiation of $(x)(x+y)$ using an online derivative tool which gives the result: $\frac{d}{dx}\left(\left(x\right)\left(x+y\right)\right) = x+y+x\left(\frac{d}{dx}\left(y\right)+1\right)$ But using a different tool I found that derivate is: $\frac{d}{dx}\left(\left(x\right)\...

I know that the r^2 value for the data is 0.9832. Is there a way to use that value to find the percent variation in Y is explained by X? Or do I need to use the data given to me?

So I have the vectors (1,0,0),(1,2,0),(1,2,3). $v_1$=(1,0,0) which also has norm 1 $v_2$=(1,2,0)-$\frac{<(1,2,0)(1,0,0)>}{1}$*(1,0,0) = (0,2,0) $v_3$=(1,2,3)-$\frac{<(1,2,3)(1,0,0)>}{1}$*(1,0,0)- $\frac{<1,2,3)(0,2,0)>}{2}$*(0,2,0) =(0,-2,3) Now the answer is (1,0,0),(0,1,0),(0,0,1), I hav...

This is what I came up with R^n={(x,z) ∈ R^n |∃y,(x,y) ∈ R^n ∧ (y,z) ∈ R^n} It's just that the R^n bugs me a lot...Can someome explain? I have read the book and look through the web but I don't really get it.

I am thinking of the following theorem: All varieties are projective over algebraically closed fields. Let $f: X \to Y$ be an unramified finite map to a nonsingular variety $Y$. Then $f$ is locally described as the projection to $Y$ of a subvariety $X \subset Y \times A^1$, where $X$ is defined...

What is an example of an open set $A_1$ in the reals which contains the interval (1,2) but so that $A_1$ is not itself an interval. Find another $A_1$ except this time $A_1$ is a closed set containing (1,2) but is not itself an interval. I think that the open set $A_1$ would be something like...

Have to show that first scherk surface is minimal : We have f(x,y)=(x,y,ln(cosx/cosy)) I am trying to show that using H=1/2(k+k')=0 (this is the definition given in the course) k and k' the principal curvatures but I don't get 0. Calculating the second fundamental form : fx=(1,0,-tanx) fxx=(...

Find all the triples (a,b,c) of positive integer such that $(1+1/a)(1+1/b)(1+1/b)$ Where a,b,c can be distinct

What is the best way to go about proving the statement in the title?

I'm studying optical fibers and trying to analyze how the core radius fluctuations along the fiber length affect the performance of optical fibers. How can I generate random noise dr which satisfies the following equations? enter image description here

If 1>1-ab>0, which of the following must be true? (1) a/b >0 (2) a/b <1 (3) ab <1 Option 1 and 3 are correct, but is option 2 necessarily correct?

Is there any continuous surjective map from $[0,1) $ to $\mathbb R$? If it exits then what is the way to construct such a map?

I am not from Maths field, but I need your help to convert the 3^n form to 2^n. I need to change the base from 3 to 2. The resultant expression can be of any form.

Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be defined by $T(v)=\begin{pmatrix} 2 & -1 & 1 \\ -3 & 4 & -5 \\ -3 & 3 & -4 \end{pmatrix}v.$ Determine the minimal polynomial of $T$ with respect to $\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.$ $T(v)= \begin{pmatrix} 2 & -1 & 1 \\ -3 & 4 & -5 \\ -3 & 3 & -...

I have a question about Novikov's condition. Let $L$ be a local martingale such that either $\exp \left(\frac{1}{2}L \right)$ is a submartingale or $E[\exp\left(\frac{1}{2} \langle L,L \rangle_{t} \right)]<\infty$ for every $t>0$. Then $\mathcal{E}(L)_{t}:=\exp \left\{L_{t}-\frac{1}{2} \langle ...

Click on the link for the graph and its spanning tree

I'm currently studying for the GRE and a specific type of question has me stumped. I have a two part question, but first here is one problem and my work: "If x is the remainder when a multiple of 4 is divided by 6, and y is the remainder when a multiple of 2 is divided by 3, what is the greatest...

It is wellknown that $n=ab\in\mathbb{Z}, a,b\in\mathbb{Z}, a,b\gt1 \implies a\le \sqrt n \lor b\le\sqrt n$. Let $z=a+ib$ be a nonprime Gaussian integer, such that $z=(c+id)(e+if)$. Do we have that either $|c+id|\le\sqrt{|a+ib|}$ or $|e+if|\le\sqrt{|a+ib|}$? I am confused here because Gaussian ...

Let $d1$ and $d2$ be two metrics on non empty set $X$ is $d$ = $/min{d1, d2}/$ is agai metric on $X$. M looking for a counter example for minimum of two metrics is not a metric ...help me out!!

Okay, so everyone knows the joke that has been going on. "What's 9 + 10? 21". Yeah, so I was wonderning an you get 21 with 9 and 10 at all, this means any mathematical aspect can be used.

Let $\mathbb R=\mathbb Z_5[i]$ denote all polynomials $a_0+a_1i+a_2i^2+...+a_ki^k$ for any nonnegative k. What I got so far is that the polynomial is actually $a_0+a_1i-a_2-a_3i+a_4+a_5i+...$ which is $a_{4k}-a_{4k+2}+(a_{4k+1}-a_{4k+3})i$, which is in the form of $a+bi$ where $a,b\in \mathbb Z_...

Let $\overline X$ be a binormal random variable with distribution $N_{\overline X}(\overline m, \Sigma)$ where, $\overline X = \left( \begin{array}{c} x \\ y \end{array} \right)$, $\overline m = \left( \begin{array}{c} m_1 \\ m_2 \end{array} \right)$ $ \Sigma = \left( \begin{array}{cc...

Let $L=L_1∩L_2$, where $L_1$ and $L_2$ are languages as defined below: $L_1=\{a^mb^mca^nb^m∣m,n≥0\}$ $L_2=\{a^ib^jc^k∣i,j,k≥0\}$ Then $L$ is Not recursive Regular Context free but not regular Recursively enumerable but not context free. My attempt : $L_1$ is CSL(context sensitive langua...

Let $f:\mathbb R\to \mathbb R$ be such that $\int _{-\infty}^{\infty} |f(x)|dx<\infty $. Define $F:\mathbb R\to \mathbb R$ by $F(x)=\int _{-\infty}^x f(t) dt$ Does it follow that : $1.f$ is continuous ? $2.F$ is uniformly continuous ? $3.|f|<M$ for some $M>0$? I think $1$ is false since we ...

The question is : A football match may either won, drawn or lost by the host country's team. So there are three ways of forecasting the results of any one match, one correct and two incorrect. Find the probability of forecasting at least three correct results for four matches. I calculated it as...

4x^3+21x^2-18x=0 Need a step by step because I'm clearly messing up along the way of solving this ! Thanks!

How t prove that $SL(n,\mathbb C)/B\cong SU(n)/T$, where $B$ is the subgroup of upper triangular matrices and $T$ is the subgroup of the diagonal $n×n$-matrices?

Good day, please a dude how to prove this: $$Suppose\ f\ analytic\ and\ {f}'(z)\neq 0\ in\ a\ region\ \Omega.\ If\ z_{0}\ \in \Omega\ and\ f(z_{0})\neq 0, show\ that\\ for\ all\ \varepsilon > 0,\ there\ are\ z_{1},z_{2}\ such\ that\left | z-z_{0} \right |< \varepsilon,\left | z-z_{2} \right |< ...

Let $C\subset\mathbb{C}$ be an analytic arc and $p\in C$. For a harmonic map $f$ defined on a neighborhood of $p$, I want to find a formula for the slope of $f$ in $p$. Any help need. Thanks.

$\newcommand{\b}[1]{\mathbf{#1}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\N}{\mathbb{N}}$ Question I solved this exercise in Munkres.(20.4) But I don't know if I did it righ t or not. I really appreciate it if anyone can take a look at it and give me some pointers. Problem Consider the pro...

So I was working on showing that $$\text{Th}(\Bbb{R}, 0,1,+, \le) = \text{Th}(\Bbb{Q}, 0,1,+, \le) $$ My Initial ideas for working on this problem was to systematically start by showing: $$\text{Th}(\Bbb{R}, 0,1,+) = \text{Th}(\Bbb{Q}, 0,1,+) $$ And $$\text{Th}(\Bbb{R}, 0,1,\le) = \text{...

Show that (e^(-x))×(x^n) is bounded on [0,infinity) for all positive integral values of n. Using this show that integration of above function exists from 0 to infinity???

Let $n>1$ be a integer and let $\lfloor\dfrac{n-2^{k}}{2^{2k+1}}\rfloor=0$. Show that there exists a constant $t$ such that $$\sum_{j=0}^{k}\left(\lfloor \dfrac{n-2^{j}}{2^{2j+1}}\rfloor+1\right)-\frac{4n}{9}\le t\log_{10} n .$$

For context, what he means by a "base" is a basis for a topology on $X$, and the oscillation of $f$ over $B$ is $\omega(f,B)=\sup_{x_1,x_2\in B}|f(x_1)-f(x_2)|$, i.e. the diameter of the image of $B$ under $f$. Now, in this theorem: I do not see any reason why $\varepsilon/3$ was used to obtai...

(M-1)x^2+2(m+3)x+(m-7)=0 Please find a relation between x' and x" independent from m then deduce the double roots of this equation . Please solve!!

Is $2^i - 2293$ always composite for i=1,2,3,...? I have known: if $2^i - 2293$ is prime, i must have the form $i = 24 k+1$ In[2]:= Table[FactorInteger[2^i - 2293], {i, 1, 241, 24}] Out[2]= {{{-1, 1}, {29, 1}, {79, 1}}, {{173, 1}, {193943, 1}}, {{6737807, 1}, {83550917, 1}}, {{399550573, ...

How can i solve this integral: $$\int\dfrac{4(f'+x f^{2})}{1-2x^{2}f} . e^{\int{\frac{4x(x f'+f)}{1-2x^{2}f}}dx} dx$$ where $f$ is a differentiable function of $x$. I would be happy to get some hints.

Assume $f(x),g(x)$ is continuous on $[a,b]$. show that there exists $\xi \in [a,b]$, such that $$g(\xi)\int_a^\xi f(x)\text{d}x=f(\xi)\int_\xi^b g(x)\text{d}x$$ I tried to use intermediate value theorem to $F(t) = g(t)\int_a^t f(x)\text{d}x-f(t)\int_t^b g(x)\text{d}x$. but I failed to find two...

There are six cards with numbers written on them: 1, 2, 3, 4, 5, 6 . Two of them are drawn at at time and put together to form fractions, e.g. 4/5 How many proper fractions can be formed? What is correct equation to solve this type of questions? thanks

Is it possible to optimize this objective function as such or transform it into an convex formulation? The unknown continuous variables $x_i \in [-1, 1]$ are nodes in a graph and for each edge $(i,j)\in E$, the goal is to maximise the difference between each nodes $x_i$ and $x_j$. I formulated ...

In order to make it clear, I ask three questions: Does $ |2^m - 3^n|<10^6 $ have any integers solution for m>20 ? $ \lim inf |2^m - 3^n|$ is infinite ? $ \lim inf |2^m - 3^n|/m$ is finite ?

Let $p$ be an odd prime. Why is $\gcd(p^2 + 1,p^3 + 1) = 2$ ?

Let $a_1, a_2, a_3...$ be the sequence of all positive integers relatively prime to 75, where $a_1<a_2<a_3...$ with $a_1=1, a_2=2, a_3=4, a_4=7$. Find the value of $a_{2008}$. What I have done: If ${a_n}$ is relatively prime to 75, then it is relatively prime to 3 and 5. Consider the following ...

Reviewing for an exam, and the book says False. But I can't seem think of a way to prove this wrong...

as per the nptel lecture 23 on discrete mathematics https://www.youtube.com/watch?v=uOE7rSKTDbo&list=PL0862D1A947252D20&index=23 , the professor proves that every partition induces equivalence. but is it necessary that the elements in the partition blocks are necessarily reflexive symmetric and t...

Show that for any measurable function $f$ in a measure space, we have: $$ ||f||_p \leq \max\{||f||_r ,||f||_s \} $$ whenever $0<r<p<s$. Now by splitting the integrals into parts where $f>1$ and $f\leq 1$, we could easily show that $$ ||f||_p \leq ||f||_r+||f||_s \ $$ But I have no idea how to sho...

Let $B$ be a subset of $A$ such that no element in $B$ is twice the other. Find the maximum number of elements possible in $B$ if $A=\{1,2,...,n\}$.

I ask how do i show if there is a commone tangent between the graph of the two function $f(x)=x²$ and $g(x)= lnx $? Thank you for any kind of help

Let $X_1$ and $X_2$ have a bivariate normal distribution with parameters $\mu_1 = \mu_2 = 0$ and $\sigma_1 = \sigma_2 = 1$ and $\rho = 1/2$ Find the probability that all the roots of $X_1x^2+ 2X_2x + X_1 = 0$ are real. Hint: First establish that $X_1 + X_2$ and $X_1 - X_2$ are bivariate normally...

If I haven's misunderstood the meanings of Sharp-P and FP, I think: FP is the class of functions computable in polynomial time. #P is the class of functions that count the number of polysized certificates which return yes after we verifies them in polynomial time for instances of an NP p...

I'm in a pre-calc class, and we're looking at logarithms and exponential functions. One of the exercises I'm struggling with is: 5e^2x = 6 + 29e^x I would ususally multiply each side by a log value to get those exponents by themselfs, but that 6 is throwing me off. How do I get rid of it? Or do...

Hari(H), Gita(G), Irfan(I) and Saira(S) are siblings (i.e., brothers and sisters). All were born on 1st January. The age difference between any two successive siblings (that is born one after another) is less than three years. Given the following facts: Hari's age + Gita's age > Irfan's age + S...

Is this formula $\lim_{\eta \rightarrow \infty }(\sum_{i=1}^n x_i^\eta )^{\frac{1}{\eta}} = max(x_i)$ right ? where $\{x_i\}$ are real bounded numbers

Use combinatorics to count how many ways can 25 identical pens be distributed to four students so that each student gets at least three but no more than seven pens. What I have done so far is look like it make sense but it doesn't work out I was thinking about view it as bit string of 0 and 1....

Give a CFG for the language $\{x \in \{a,b\}^* | x \not= ww \text{ for some w} \in \{a,b\}^*$ In my attempt to do this I understand odd length strings are automatically in the language, but don't know how to handle even length strings. What would the CFG look like? I know it starts with S-> E|U ...

I'm really stuck with this one and I'm thankful for any help. Consider the following operations on the set of integers: $\hspace{8em} a\star b := a^2 + b^2 \hspace{5em} a\diamond b := a+b+2ab$ Prove or disprove the following statements: a.) The relation is associative b.) The relation is com...

Every bijection between 2 affine spaces with the same dimension sending 3 collinear points into 3 collinear ones (in an 1-dimensional affine subspace) must be an affine mapping?

I have this question for a sample quiz, I don't know any other way to answer it as it's not explained in the book. The question is: "In a faculty, there are 1300 students. Of these students, 300 are taking a unit in mathematics, 500 are taking a unit in physics. There are 150 students tak...

Given a real m*n matrix A (m rows, n columns), a real positive number b. Each entry in A can be positive or negative. Find a subset of columns to maximize the number of rows whose sum of entries in those selected columns is at least b. The equivalent mathematical formulation: $$ \max_{x\in\{0,1...

Is anybody can help me by explained to me step by step how to solve this question. The function f(z) has a double pole at z=0 with residue=2 and a simple pole at z=1 also with residue=2. It is also analytic at all other finite points of the plane and is bounded as |z| -> infinity. Also f(z=2)=5 ...

I don't quite understand why this is the case? Since when differentiating |2x^2-3x| you get ((2x^2-3x)(4x-3))/|2x^2-3x|.. when its 2x^2-3x, the derivative is 4x-3 and when its -(2x^2-3x) its -(4x-3)? |4x-3| = +/- (4x-3)? I think I might have understand something wrong here but I am not sure what....

Prove that $R^{\omega}$ has infinitely many components in the box topology.

What does the following notation mean, see "(i)":

Let $c_0$ be the space of all complex sequences $(a_n)$ such that $$\lim_{n \to \infty} |a_n| =0$$ with norm $\|(a_n)\|_{c_0} = \sup_{n} |a_n|$. Is it fair to say that: Let $\{(a_n)\}_{n \in \mathbb{N}}$ be a cauchy sequence in $c_0$. Let $\epsilon >0$. Then there exists $N$ such that for $n...

I got as far as lim x approaches infinity for lny=xlntanhx. I'm not sure what to do there. I know tanhx as x approaches infinity is one but 1^infinity isn't the correct answer.

are the physical equations results independent (Schrödinger equation, statistical mechanics, etc... of the ZFC model "used"?

So, I am struggling with the last part of my expression, where I got the Expression . I made a Karnough map and it clearly shows that the last (x_2 inversion, x_3, x_4 inversion) is not needed at all, but I have no idea no how get rid of it.

By the definition of square root we can take square root of 16 as 4,-4 but in problems why we take 16^(1/2) as 4? And also how can we say,If a^m=b^n then a=b(m/n)? because when we take x^2=a we solve it as x=+-{a^(1/2)}.

Here's the question: y" + y = x^2 + 1, y(0) = 5, y(1) = 0 I manage to get a solution to be this: y = -6cot(1)sin(x) + 6cos(x) + x^2 - 1 - 2cos(1)sin(x) + 2cot(1)sin(1)sin(x) Can somebody help me check if my yp is correct? Thanks!

Let $AA_1, BB_1, CC_1$ be the altitudes of $\Delta ABC$ and let $AB \neq BC$. If $M$ is midpoint of $BC$, $H$ the orthocentre of $\Delta ABC$ and $D$ the intersection of $BC$ and $B_1C_1$, prove that $DH \perp AM$ I was trying to come up with a pure geometric proof but haven't got far. These...

show that (8^.5)*(9-(77)^.5)^.5 is same as 2(11)^.5-2(7)^.5 ^.5 means square root of the (). I have tried multiplying the radicals but that didn't work. the resulting radicals do not add up or get subtracted. I have tried taking commons also but that also doesn't work

A very simple question, but I can't seem to find anything relating to it : Is there any research, are there any results that have focused on or given insight on $\sum 1/p^p$, ${p \in \mathbb P}$ ? A very basic series, converges extremely fast, its value is around .29. What more can there be s...

I am reading the lecture notes. On page 46, why $R_{X}$ as a vector field on $\mathcal{H}$ is $L_{pXp^{-1}}X$? Why $R_{\kappa} = 0$, $R_{\alpha}=2y \frac{\partial }{\partial y}$, $R_{\nu_+}=y\frac{\partial }{\partial x}$? Thank you very much.

i want just to uhnderstand why in the hilbert space $$l^2(Z)$$ we have the spectral measure verifie $$\mu^x=1/2(\mu^x_{e_{1}}+\mu^x_{e_{2}})$$ where $$e_{1}(n)=\delta_{1,n} $$and $$e_{0}(n)=\delta_{0,n}$$ this what we use in the schrodinger operator

Hy I wonder how to find the tangent line ( at inflex point ) to curve...if I doesnt have function, only the measurement of the fuction / t-time,y - values/...so I have the plot of the function. for example: http://www.polymath-software.com/Pol...atlabGraph.jpg thank you /I want to in Matlab for...

$\partial_tf(x,t)=\frac{\partial ^2f}{\partial x^2}$ $f(x,0)=g(x)$ $(x,t)\in\Omega\times(0,T]$,$\Omega$ is a bounded open subset of $R^n$. g(x) is smooth function,then, I want know ,how to show that $f(x,t) $ is continue about $t$.And $\Delta f$ is continue about $t$. And $f(x,t)$ is smooth ab...

I am trying to find the real number $k$ so that $$ln(n) = n^k$$. I know it is a number between $1/e$ and $1/2e$ but I am not sure how I could find the number.

The value of $m$ for which straight line $3x-2y+z+3=0=4x-3y+4z+1$ is parallel to the plane $2x-y+mz-2=0$ is $(A)-2\hspace{1cm}(B)8\hspace{1cm}(C)-18\hspace{1cm}(D)11$ My Attempt:The plane $2x-y+mz-2=0$ is parallel to the $3x-2y+z+3=0$ and $4x-3y+4z+1=0$. So the lines are perpendicular to the n...

We have got set of students in lecture room. Every student is in relation with yourselft, student X and student Y are in relation, if X is on left of Y (from view of teacher). Relation R: $(X,X) \in R$ $(X,Y) \in R \wedge (Y,X) \notin R$ $(X,Y) \in R \wedge (Y,Z) \in R \wedge (X,Z) \in R $ ...

while reading on the Godement resolution regarding stalks of a sheaf and the relation to the whole of the sheaf itself, https://en.wikipedia.org/wiki/Godement_resolution I found this statement: It allows one to view global, cohomological information about the sheaf in terms of local informati...

I have been given the following information: Change in sequential positive information next quarter 0.55 Change in sequential negative information next quarter 0.45 Change in sequential positive information in prior quarter 0.55 Change in sequential negative information in prior quar...

Hello is anyone can help me with this problem: A non zero natural number N is such as N(N+2013) is a perfect square. Show that N can not be a prime number Find a N value such as N(N+2013) is a perfect square I've tried to proceed using a proof by contradiction) assuming that N(N+2013) is a ...

Develop the finite element equilibrium equations for a heat transfer problem and then develop a computer programme for its solution. Use a FOUR isoparametric finite element grid to test your programme with the domain below. given that the centre of the inner cylinder is 2.5cm from the centre of t...

I'm reading a paper ("La Formule de Verlinde" by Christoph Sorger) and at a certain point, the author switches from algebra geometric language to complex geometric language. He uses the symbol $dz^2$, where $z$ is a local complex coordinate on a Riemann surface. Can anyone explain what this means...

I am trying to solve. $y′+2xy=2x^3y^3$? Help me please-_-

Let $A=PDP^{-1}$ where $P = \begin{bmatrix}1&0&0\\0&2&1\\2&-5&-3\end{bmatrix}$ and $D=\begin{bmatrix}8&0&0\\0&0&0\\0&0&7\end{bmatrix}$. Find all solutions to $x = \begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}$ to equations $A\mathbf{x} = 8 \mathbf{x}$. I tried the brute force approach and calculated...

Let $(M,\tau)$ be a von Neumann algebra, i.e. a unital subalgebra $M=M''\subset \mathbb{B}(H)$ with a finite (faithful) trace $\tau: M\to \mathbb{C}$ (faithful means that $\tau(x^*x)=0$ implies $x=0$; moreover $\tau(q)\leq 1$ for any projection $q\in M$). Consider a projection $p\in M$. If $p$ i...

Probably a trivial question. Let's say I have the following system of equations: \begin{cases} f\left(x,y\right)=0\\ \\ y=g\left(x\right) \end{cases} and I want to study its saddle-node bifurcations (where an eigenvalue goes to zero), and its Hopf bifurcations (where two eigenvalues have the fo...

I'm reading a proof of a Khintchine inequality : Let $(r_{1}, \dots , r_{n})$ be iid random variables with $P(r_{i} = > \pm 1) = \frac{1}{2}$. Let $f = \sum\limits_{j=1}^{n}a_{j}r_{j}$, where $a_{j} \in \mathbb{R}$. Then $||f||_{2} \leq \sqrt{e}||f||_{1}$. The proof uses Holder : $...

At my multivariable calculus class we gave this definition for the limit of a function: Definition: Let $ \mathbb{R}^n \supset A $ be a open set , let $f:A \to\mathbb{R}^m $ be a function, let ${\bf x_0}$ be a point of $A$ and ${\bf P}$ a point of $\mathbb{R}^m$. To say that $f$ ...

I am confused with the difference between Probability and Likelihood, because both of them use the same symbol. I have made a hypothetical setting to specify my question: If $X_2$ is only depends on $X_1$, $X_3$ is only depends on $X_2$, that is $(X_1)\to(X_2)\to(X_3)$. (just like markovian mod...

I'm trying to draw something and I need to know what subject in maths involves space like how much room there is in something

why $$D(\overline\Omega)=C^\infty(\overline\Omega)$$ where $$D(\overline\Omega)$$ set of function with compact support in $$\Omega$$

I'm trying to combine percentiles (50th and 90th) from 2 different datasets. I don't have access to the datasets at the time of combining. Apart from the percentiles I have the number of elements on each dataset. Now, I know that any attempt at combining them will be only an approximation to the...

The equation of the plane which passes through the point of intersection of lines $\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{2}$ and $\frac{x-3}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ and at greatest distance from the point $(0,0,0)$ is $(A)4x+3y+5z=25\hspace{1cm}(B)4x+3y+5z=50\hspace{1cm}(C)3x+4y+5z=49\h...

Here's the question. ∂^2/(∂t^2 ) u(x,t)=c^2 ∂^2/(∂x^2 ) u(x,t) We are supposed to use this form of fourier transform to solve our PDE f(hat)(s) = 1/√2π ∫_(-∞)^∞▒〖f(t) e^(-ist) dt〗 Can anyone enlighten me on how to do this question? Thanks!

Let $(B_t)_t$ be a Brownian motion, then I am given a stopping time $\tau_s:=\min(\inf\{t \ge 0; B_t=a\}, \inf\{t \ge s; B_t=b\}; \inf \{t \ge 0;B_t=c\}),$ where $a<0<b<c.$ Now, I want to show that $P(B_{\tau_s}=c)$ is a continuous function in $s.$ What I think is easy is to show that it is co...

I am currently reading J.P. Escofier's "Galois Theory" and in the text he discusses the galois group of $\mathbb{Q}(\sqrt[3]{2}, j)$ which is isomorphic to $S_3$. I have become lost in his discussion of finding the field of invariants of a given subgroup of the Galois group, particularly in refer...

Is it true that if A is a proper subset of B, then the Lebesgue measure of A is strictly less than the Lebesgue measure of B?

Boundary point are differ from boundary point in a metric space. How? Explain with sutaible examples.

I was wondering if anyone could help with this ε–δ definition of a limit. I have looked it up in my calculus book and online and I just don't understand how to do it. Prove, using the ε–δ definition of a limit that lim(x,y)→(0,0);(x,y)∈Df(x, y) exists where D ⊆ R^2 is the domain of f and f(x,...

Let $n ∈ \mathbb N$. Consider an $(n×n)$-matrix A with real components and a column vector $b ∈ \mathbb R^n$. They give rise to an affine transformation $T : \mathbb R^n → \mathbb R^n$ with $T(x) = Ax+b$. Consider the Euclidean metric on $\mathbb R^n$: Let I be the identity $(n × n)$-matrix. Sup...

X is a random variable with density $$f(x)=2e^{-2x+2} , x\geq1$$ and 0 otherwise. Determine $Mx(θ)$, the moment generating function for X, and the values of θ for which $Mx(θ)$ is defined. Use $Mx(θ)$ to determine $E(X^2)$ Working: I know that $Mx(θ) = E(e^{\theta x})= \int_{0}^{\theta}e^{\th...

Number of lines which are normal as well as tangents to the curve $y^2=x^3$? What is the general method to solve such problems?I could'nt proceed much.

In a quad. ABCD with AB=CD,P and Q are mid points of diagonals AC and BD.P and Q joined and extended hits both sides AB and CD at S and T respectively.How can I prove that angle AST=angle DTS

This seems a little inconvenient, and using $$\phi(n)=\prod_{p|n} \left ( p-1 \right )$$ seems much more convenient for computational purposes. So I am guessing that there is a specific reason for expressing it as such. What is that reason?

2^(1/2) * (9-77^(1/2))^(1/2) How to simplify this so that it has no nested radicals? This question is same as that already posted but with a different point of view.

I seem to recall a result that says any set of axioms can be converted to a set of equivalent axioms that use only binary predicates and constants. Can anyone point me to that result?

First of all, I know nothing about Markov chains, and I'd like to prove the following without using the theory around them. Let $(M_{n})_{n\geq 1})$ be a random walk over $\mathbb{Z}$, starting at $M_{0} = 0$. Let $$S_{a, b} = \lbrace M_{i} \neq 0, a \leq i < b\text{ and } M_{t} = 0 \rbrace$$ a...

The Problem Suppose $f$ is twice differentiaable on [a,b], $f(a)<0$, $f(b)>0$, $f'(x)\geq \delta >0$, and $0\leq f''(x)\leq M$ for all $x\in [a,b]$. Let $\xi $ be the unique point in $(a,b)$ at which $f(\xi)=0$. Choose $x_1\in (\xi, b)$, and define $\{x_n\}$ by $x_{n+1}=x_n-\frac{f...

I need to show that $f'(x) = (f(x-2h) - 4f(x-h)+3f(x)) / 2h +0(h^2)$ with Taylor series expansion of $f(x-h)$ and $f(x-2h)$. I got the expansions but I don't get the final answer correct, so I think I am missing something.

I want to show that for all $A, B$ n by n-matrices with real components, $\|AB\| ≤ \|A\|\|B\|$. I know the proof of this inequality using two vectors. Is it the same for matrices as well? Any help and hint would be appreciated.

$Prove that$: $2*sinA/cos3A + 2*sin3A/cos9A + 2*sin9A/cos27A=tan27A-tanA$

Then, supposing that $[E:F] = p^k$, for $p$ prime. Then $f$ is solvable by radicals. I can't think in a tower of roots to solve the problem. Help, please.

A group $G$ acts on a set $A$.. With this statement we can say that : (1) There's a permutation from $A$ to $A$ as : $$ \sigma_g : A \: \: \:->\: \: A$$ $$ \sigma_g (a) =g \wedge a$$ where $g \wedge a$ says that $g$ acts on $a$. (2) The map form $G$ to $S_A$ defined by : $$ \phi(g) = \sigma_g...

The pattern is of a triangle followed by 3 stars and 4 circles and it continues to repeat. What will the 999th shape in the pattern be? Explain how you can tell.

Suppose there is any number N . N = x1^a1 * x2^a2 * x3^a3 and so on . We have to find x1,x2,x3,..... xn (all x are prime) and a1,a2,a3....an . How to find out N using minimum calculation ? e.g 16200 = 2^3 * 3^4 * 5^2 . x = 2,3,5 a = 3,4,2 .

Let $X_t=X_0+M_t+A_t$ a continuous semi-martingale. Let $g: \Bbb R \to [-1,1]$, of class $C^{\infty}$, with $g(x)= \left\{ \begin{matrix} -1, & x \le 0 \\ 1, & x \ge 1 \end{matrix} \right.$. Let $f_n : \Bbb R \to \Bbb R$ with $f_n(0)=0$ and $f_n'(x)=g(nx) \ \forall x \in \Bbb R$. Show $\lim \lim...

I have to determine the value of truth(always true, always false, sometimes true) of the following statement: f(n)= Ω(n) and g(n) = o(n^2), find the value of truth of f(g(n)) = ω(n) I don't know eactly what I'm supposed to do. Any help will be much appreciated

Find and prove the limit as n tends to infinite for the following sequence: $$\sum_{n=0}^\infty z^n$$ Where z is a set of the complex numbers and $|z|<0.6$

Why $\partial_tT=(\delta_\dot gT)(g)=(\delta_hT)(g)$? $h=\partial_tg,T$ is a $(k,l)-tensor$.

The letters in the word "PLACES" are permuted in all possible ways and arranged in the alphabetical order.Find the word at 48 position. a)AESPCL b)ALCEPS c)ALSCEP d)AESPLC MyApproach As per dictionary I started with AC--->$4$!=$24$ AE--->$4$!=$24$ So,I think the word start with AE b...

One sends a units of red flow from node 1 to node 9 and b units of blue flow from node 4 to node 6. The red flow can only go south and the blue flow east. The two flows down not mix. The load on a given link is defined as the sum of the sizes of red the blue flows and one would like to send the r...

The Duffing equation in its full form is $$\ddot{x} + \delta \dot{x} -ax + \beta x^3 = \gamma \cos(\omega t)$$ Now for specific values of the parameters several attractors exist (or not). Let's assume that $\alpha = \beta = \omega = 1$, $\delta = 0.15$, while $\gamma = 0.2445$. For these values...

I've been asked to "fill in the blanks" on the following: Elimination produces A = LU . Te eigenvalues of U are on its diagonal; they are all the __________. The eigenvalues of L are on its diagonal; they are all _________. The eigenvalues of A are not the same as ____________. I know that the ...

I can find hamming distance between $v \in U^n$ and $u \in U^k$, $k \le n$, $U$ - small alphabet(for example $B$ or $DNA = \{A,C,G,T\}$), beginning some $j$: $Hamming\ distance^j(u, v) := Hamming\ distance (u, (v_j, \ldots , v_{j + k - 1}))$, $1 \le j \le n - k + 1$. I want to find $i$ : $Hamm...

How do I get from this: $F = AB' + AC' + AD + C'D'$ to this: $F = AB' + AD + C'D'$ Not sure how the $AC'$ disappeared.

Given a ring R and the discrete valuations v,w such that (R,v) and (R,w) are discrete valuation rings, I have to show that u=v. Can anyone give me a hint on how to go about this? I have no idea where to start. Up to now i could show that valuation of units must be zero, and I suspect that I hav...

I have a problem that I cant figure out. how does the 2nd line become 3rd line. I need details explanation especially on the 1+i-1 of the 3rd line. Thanks

The question is : I know that because it is known that there is a limit i can choose an epsilon, so if i'll say epsilon = 1 then 4 < f(x) < 6. i also know that 1 - delta < x < 1 + delta yet i am not sure how that helps me. Thanks in advance !

Can someone explain the solution to this. I don't get the assumptions in his solution. "A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is 1/2. Then the length of the semi−major axis is?" Solution: "Without loss of generality, let h > 0, let (h, 0) b...

Consider sample space x = {v1,v2,v3,v4} and 3 Events E = {v1} F={v1,v2} G={v1,v2,v3} suppose their Probability are P(E) = 1/10 P(F) = 5/10 and P(G) = 7/10 Now define a fourth Event H = {v2,v4}.find P(H)

Let OABC be any planar quadrilateral. Let $G_1, G_2 and G_3$ be the centroids of OAB, OBC and OAC respectively, and let G be the centroid of the triangle $G_1G_2G_3$. Show that the points O, G and the centroid D of triangle ABC are collinear. I have no idea how to start this any help would be ap...

my answer that I have gotten is b'c'd + a' b d' however, the answer given to me was b'c'd can someone tell me whether I am correct

i know that we can identify $H$ with his $H^*$(dual) then if we have an other sub-space $V$ of$H$ then we will have $V\subset H=H^*\subset V^*$ can we identify here $V$ with $V^*$ then we will have V=H sure we can not but why ?

How to integrate $e^{-x^{2}}$ ?$ When I used geogebra I got the answer as $\frac{1}{2}\sqrt{\pi}(erf)(x)$. What is $ (erf(x))$ ? How to arrive at this answer ?

Consider the below theorem on Sub-Groups Theorem : Let $(G, \circ)$ be a group and $H$ be a non-empty finite subset of $G$. Then $(H, \circ)$ is a subgroup of $(G, \circ)$ if and only if $a \in H, b \in H \implies a \circ b \in H$ Proving Forward : If $(H, \circ)$ is a subgroup of $(G, \cir...

I've got a task, where a set A and a map g were given. The task was to calculate g(A). And I don't know what to do exactly, so I'd appreciate if s.o. could explain it with an example or give me a link, where it's explained. For example let A = { x in IR | |x-3|= 2 } and g (x) = (x+2)/x. So if I...

all about clash of clans game. I have total of 6 barracks. Two of them produces one barbarian each every 20 seconds, the another two barracks produces one archer each every 25 seconds, and the last two produces 1 minion each every 45 seconds. the capacity of army camp is 200. one archer has an ...

Let $(a_n)$ be a bounded sequence and this inequality holds: $2a_n\leq a_{n-1}+a_{n+1}$ Prove, that $b_n=a_{n+1}-a_n$ converges to zero. My Proof: $2a_n\leq a_{n-1}+a_{n+1}$ $\iff2(a_{n+1}-b_n)\leq a_n-b_{n-1}+a_{n+1}$ $\iff a_{n+1}-b_n\leq\frac{1}{2}a_n-\frac{1}{2}b_{n-1}+\frac{1}{2}a_{n+1...

I have a script of code which models a planetesimal that is accreted into a planetary atmosphere. In the code, I include the physics of frictional ablation and thermal ablation. Frictional ablation is governed by the coupled evolution of 5 equations and thermal ablation is governed by sublimattin...

I would be really grateful if someone could check what I have done here; it should be quick: Let $\Phi$ be a random variable taking values in $[0,\pi]$ with PDF $f(\phi)=\frac{1}{2}\sin\phi$. Define: $$h(\phi)=a\cos(\phi)+b$$ where $a,b$ are positive constants. I need to find the distribu...

Normally, linear regression asks for a pair of parameters m,b such that for a set of given points {x_i,y_i} the variance of y-m*x-b is minimized (this minimizes the distance in y-direction only). Instead, I would like to find a line y'=m*x'+b such that the pairs (x_i,y_i) have minimal perpendicu...

Let $G: \mathbb{R}^n \rightarrow \mathbb R$ be given by $$G(z)= \frac{1}{2}|(I-A)z+Az_0 - v_0|^2 +f(Az-Az_0 +v_0),$$ where $A$ is an $n\times n$ matrix and $f$ is a smooth function on $\mathbb{R}^n$ and $z_0$ and $v_0$ are fixed vectors in $\mathbb{R}^n$. I want to calculate the derivatives $DG...