I'm learning about functions of bounded variations and need help to understand the proof of this lemma: Lemma. If $f : [a,b] \rightarrow \mathbb{R}$ is of bounded variation, then f is also bounded and satisfies $\left\lVert f \right\rVert_{\infty} = \left|f(a) \right| \leq V_{a}^{b} f$. ...

I have a problem I think it is a little hard or at least has some points need to be considered to solve it. I know that if the transformation function is with the form of fog I should calculate the region with g then claulate the new region with f. But whatever I did, didn't help. The question is...

So the travelling salesman problem is a problem wherein a salesman has to travel through all cities in a way that the total travelling distance is minimal. You can rewrite this as an integer linear problem: Label the cities with the numbers $0, ...,n$ and define: $x_{ij} = \begin{cases} 1 &

Prove using the definition and the Cauchy criterium that $x 􏰀→ x3$ is Riemann integrable on $[a,b]$ and that 􏰁 $\int_a^{b}x^2dx= a^3 - b^3 /3$ Thank you.

I should solve this using the polar coordinates, but I just do not know how to start...

Can someone help me to prove this equivalence : let $f:E \rightarrow F$ we have $1)$ for all $y\in F$ and for all open $U$ from $E$ such that $f^{-1}(\{y\})\subset U$, there exists a neighborhood $V$ of $y$ such that $f^{-1}(V)\subset U$. 2) $f$ is closed I strated by $1) \Rightarrow 2)$ l...

Please is the following assertion true? Let $\tau\subset\mathbb{R}_+=[0,\infty)$ be a measurable set (the standard Lebesgue measure), $\mu(\mathbb{R}_+\backslash \tau)=0$. Then there exists an increasing sequence $\{x_i\}_{i\in\mathbb{N}}\subset \tau,\quad x_i\to\infty$ such that for any $k,i,\...

I'm a bit confused with this determinant. We have the determinant [picture determinant][1] [1]: http://i.stack.imgur.com/veqk0.png for a matrix $\mathbb{M_n}$ with n$\geq2$. I compute $\delta 2=19$, $\delta 3=65$ Then I would like to find a relation for n$\geq 4$ which links $\delta_n, \delta...

Let p and q be distinct prime numbers and N = pq. G ={natural numbers less than pq that are relatively prime to pq}. if a and b are in G and k is a natural number, then (1) G has (q-1)(p-1) elements. (2) ab mod N is never 0. (3) ak mod N is never 0. (4) ab mod N and ak mod N are each elements...

I am looking for names/examples/references for probability distribution functions which are supported on a closed interval, say $[0,1]$, and increasing there.

What is the answer to the following puzzle: If 3 and 2 = 7 5 and 4 = 23 7 and 6 = 47 9 and 8 = 79 then what is 10 and 9 = ?

I have a positive function $f(x,y)$, where $x\in{\mathbb R}^n$ and $y\in{\mathbb R}$. I know that for $y$ fixed, $g(x)=f(x,y)$ is convex, and that for $x$ fixed, $h(y)=f(x,y)$ has positive second derivative. If this enough to show that $f(x,y)$ is convex?

Let $\mathcal{X}$ be a smooth geometrically irreducible projective curve over $\mathbb{F}_q$. Fix a closed point $\infty\in \mathcal{X}(\bar{\mathbb{F}_q})$. Let $K$ be the function field of $\mathcal{X}$ and $\mathcal{A}$ the ring of functions on $\mathcal{X}$ regular away from $\infty$. Let $\p...

I read somewhere that a line is made up of infinite points. Between any two points on that line, there are another infinite points. and between any two points BETWEEN those 2 points there are another infinite points. Is this true? If so , HOW? Also, i have read in physics that the smallest p...

$(1) ([8], [10]) \in Z_{12} \times Z_{18}$ $(2) ([3], [6], [12], [16]) \in Z_4 \times Z_{12} \times Z_{20} \times Z_{24}$

why does $A−λ_iI$ lose rank? In other words, why does rank(A) is larger than rank($A−λ_iI$)?

If I have $$ n < \cfrac{n^2(s-2)-n(s-4)}{2}$$ can I make it so that one side becomes completely known when I give a value to $s$, for example(obv. not true) $$123n^2+456n+789<987s+654$$

How can prove the following optimization model is convex for $n>3$? $\psi (x)= \min \sum_{i=1}^{n} (x_i-\frac{\sum_{j=1}^{n}x_j}{n})^2$

Since math is the most important part for programming I want to implement the gradient tool and I need to understand the math behind it . Can any body also explain the differences between linear , radial ,angle,reflected,diamond gradient? 2 []3 []4

How do I prove that any finite subgroup of SO2 must be cyclic? Also, what are all the finite subgroups of O2?

My question is: Given sets A = {a1, a2} and B = {b1}, let the function f from A to B be given by the following set of ordered pairs, f = { (a1, b1), (a2, b1) }. If f has an inverse function, call it g, and write g as a set of ordered pairs. If f does not have an inverse function, explain why it d...

I'm reading Lemma 1.3.8 of Mujica's "Complex Analysis in Banach Spaces" and I'm having some trouble understanding the proof of it. The lemma states: Let $f: E \to F$ be a mapping such that $f(a + \lambda b)$ is a polynomial in $\lambda$ for all $a$, $b \in E$. If $f$ is identically zero on a ...

Is there a direct way to find the matrices representing the symmetries for example of a tetrahedron with vertices (1 1 1) (-1 -1 1) (-1 1 -1) (1 -1 -1)

I have known that ∀x [s(0)∗ x = x] is a theorem of Peano Arithmetic. but how to Prove that this formula is not a theorem of Robinson Arithmetic?

I would like to obtain the letter $K$ from $K[1,2]$ in Maple. I tried to use op(K[1, 2]). But I get $[1,2]$. Are there some function $f$ in Maple such that $f(K[1,2]) = K$? Any help with be greatly appreciated!

I’m stuck on the following problem. There are two sources $S_A$ and $S_B$ at the ends of a channel. Both are made up of a white noise component $W_i$ plus an impulsive component $I_i$: $S_A = W_A + I_A$; $S_B = W_B + I_B$. Two sensors are co-located with the sources. The first one records the...

Solve the equation $\frac{dy}{dy}=\frac{x+2y-5}{2x+xy-4} $ I tried substituting $x=X+h$ and $y=Y+k$ but the $xy$ term is creating problem. How to solve it?

$S^1 := \{z\in \mathbb{C}: |z|=1\}$ is the set of roots of unity. We can use the fact discovered by Euler that any complex number $z$ can be represented as $z=re^{\pi i k}$, where $r\in \mathbb{R}$ is the radius of $z$ and $k$ is any real number. Any complex number with radius $1$ can thus be def...

The inverse of covariance matrix can be used to find the conditional independencies among variables. This inverse is more sensitive to changes then the correlation matrix. As a result two samples with very similar correlation matrices and standard deviations, may still point to different conditio...

Let p and q be distinct prime numbers and N =pq. Do multiplication mod N. Define G = {natural numbers less than pq that are relatively prime to pq} and s =(p-1)(q-1) . Then (1) m^s =1 for all m in G. I have no clue with first part of this problem, and don't know how to start with it.

I do understand the concept of Naive Bayesian classification, as it tries to calculate the probability of an outcome of a class given multiple evidences. It comes from the Bayes theorem and it is called naive as it tries to each of piece of evidence as independent. This approach is why this is ca...

prove E(X1+X2|Y)=E(X1|Y)+E(X2|Y) This question is given in the text book as an exercise. It looks trivial but took me hours to think of a proof. I don't know what to do after expand it using definition of expectation...

Objective:To find r and area of the circle, when two parameters are given.

I would like to see how this is done, probably using van Kampen. How to see this space?

Asumme that $(S,T) $ is topology space and $A'$ is set of limit pointso of set $A$ show that $(A')'\subset A'$ is not always true. I can't find a example to show that.

This is a question in one of the previous exams in my introductory real analysis course. The question wants us wants to use the definition of the riemann integral and cache's criterium to prove that $x^2$ integrable over $[a,b]$ and prove that $\int_a^b x^2 dx = \frac{b^3 -a^3}{3}$.

Matrix $A \in R^{3, 2015}$ is given. It is known that matrix $AA^{T}$ is invertible. Show that $rank A = 3$. How to start this? What does the info that $AA^{T}$ is invertible gives us?

Find the maxima, minima and saddle points of the surface Z = (x^2 -y^2)e^((-x^2 -y^2)/2)

Question: Let the set A be defined as A = {a, b, c, d}, and let the relations R and S on the set A be defined as R = {(d, a), (a, b), (b, c)}, and S = {(a, a), (b, d), (d, c)}. Explain why the ordered pair (a, b) is or is not an element of the composition of S and R (denoted R o S). I would ...

For spaces X=(t,2t-s,s,3s-t) Y=(2p-q,p-q,3p-q,5q) find their intersection and show that their intersection is a 1-dimensional subspace of R^4. Thanks :)

I know that e^[(log base e^2) of 16] is 4, but I can only get this far: e^[(log base e^2) of 4^2] e^2[(log base e^2) of 4] I need some way to cancel the 2's so I can get e^[log base e of 4] but I don't the identity or rule that cancels the 2's.

Here I have a rather naive question concerning integral representation of probability measures. In general I have problems with it, so here there is a super basic setting: $(X, \Sigma, \mu)$ probability space, $A, B \in \Sigma$, $\chi_B \in [0,1]^X$ indicator function of $B$ (measurable), $\i...

Compute R4; L4; M2; M4 where M2 is the midpoint rule with 2 rectangles. M4 is midpoint with four rectangles. Notice you'll have to estimate with M4. x 0 2 4 6 8 y 5 6 8 11 14

This is a solution check for my quantified representation of the following statement. "For every number a, the equation $ax^2 + 4x -2 = 0$ has at least one solution." $\forall a[$iff $a \ge 2$ there exists at least one solution such that $ax^2 + 4x -2 = 0]$ $\forall a[(a \ge 2) \leftrightarrow...

Find $\lim_{n\to \infty}{n{C^n}}$ when: $|C|<1$ I want to use the squeeze theorem so I bounded it from below with: $C^n\to 0$ But I can't find the upper bound.

I was reading the following article http://www.askamathematician.com/2015/03/q-why-does-kinetic-energy-increase-as-velocity-squared/ I don't understand the math of the explanation at the final of the article. The author explain what he is using on every step, for example if it is the chain rule. ...

Recall that ln x = We are going to estimate ln 4. Use 6 rectangles and midpoint rule.

It's a pretty open-minded exercise I found online. It says, you're advising a social network company and they're trying to model an equation for $u(t)$, being this the amount of active users in the network. They know their equation will have this following "shape". $$\frac{\partial^2 u}{\partial...

Let vertex sets V1 and V2 be defined by V1= {1, 2, 3} and V2 = {a, b, c}. Let E1 = { { 1, 2}, {2, 3} }, and let E2 = { {a, b}, {b, c} } be the edge sets corresponding to the vertex sets V1 and V2, respectively. Write a function f that is a bijection from V1 to V2. a. Write your function f as a...

I want to write code for a nth root function, so I need to be sure, that the underlying mathematical function is correct. From another post over at SO, I wrote the following definition: $ \sqrt[x]{y} = y^{\frac{1}{x}} = \left\{ {\begin{array}{rl} \exp_{2}\left(x \cdot \log_2 \left(\frac{1}{y}\ri...

Suppose we have the system $$\exists y (G(y)=f \land G_1(y)\neq g_1 \land G_2(y)\neq g_2).$$ Is this equivalent to $$\not\exists y \left (G(y)=f \land G_1(y)= g_1 \land G_2(y)= g_2) \land \exists x(G(x)=f\right )$$ ? Or to $$\not\exists y_1, \not\exists y_2 \left ((G(y_1)=f \land G_1(y_1)= g...

Why are trivial solutions "wrong"? For example, if I'm solving a PDE and the eigenvalue being zero implies that the solution to the PDE is identically $0$, why do we say that the eigenvalue cannot equal $0$?

Let $ f : \mathbb{R} → \mathbb{R} $ be $C^∞$ suppose the following: (i) $f(0) = 0,$ (ii) there is a smallest $n \in \mathbb{N} $ so that $f^{n}(0) \neq 0$ Determine stability of the equilibrium at $0$ for the differential equation $ \dot x = f(x) $ in terms of $n$ and the sign of $f^{n}(0)$....

Let $E$ be a infinite-dimensional normed vector space. How do I see that there exists an algebraic basis $(e_i)_{i \in I}$ in $E$ such that $\|e_i\| = 1$ for all $i \in I$?

I had a test today and I had to find sup and inf of this set :$$ A=\{sin(\frac{nπ}{3}+\frac{1}{n}), n \in N\} $$ I calculated the value of sin for 6 cases (n=6k, n=6k+1, n=6k+2,...) and then said that the minimum value found is the inf and the maximum value is sup. Is this approach correct? If no...

This is the exact problem: `Suppose that X, Y are random variables with Sx =2, Sy = 3. Let Z = 3X - 2Y, and assume that Sz = 6. Find the covariance, cov(X, Y).'

Is it true that funciton $$y=-x^3$$ has inflection point and function $$y=\frac{4x(x^2+3)}{(x^2-1)^3}$$ hasn't inflection point? If yes why??

$X$ is exponentially distributed $\varepsilon(\theta)$. Using the Method of Maximum likelihood find the best (marking?)of sample $n$ for parameter $\theta$ .Question its centeredness and existence. Now I think I do not know the exact english translations of these notions so I will explain using t...

ive been struggling to solve the following equation. t³*dx/dt + 3*t²*x = t I tried to use its characteristic equation and got the one root for x must be Ae^(-3t/x) but i just end up confusing myself even more. I read that you can do it by separating the variables but im not sure if this ...

The integral from1,ef(e^x+lnx)dx Also, what are these type of questions called?

Let n and k be integers. Need to find all pairs of (n,k) such that (n+1)(n+2)…(n+k)−k = x^2 , where x^2 is a perfect square.

Are there any cases in which x.y (where x and y are both integers) would produce some kind of result or be interpreted as anything other than a decimal number? This question relates to this MathML question asked on StackOverflow.

Show that if $\lim x_n =x$ with $x_n \in \mathbb{R}$ and $x \in \mathbb{R}$, then the sequence given by the averages $y_n = \frac{x_1 +x_2 + \dots + x_n}{n}$ also converges to x. Solution: Denote the limit of $(x_n)$ as $a$. Let $\epsilon >0$ Then there exists a positive integer $N_1 > 0$ such...

Let a and b be 3D vectors. Which of the following expressions make sense? A. (a•b)+a B. (axb)+a C. (axb)•a D. (axb)xa

Let $\mu\in\Bbb Z^d$. Suppose that $H=\{x\in\Bbb Z^d:\langle\mu,x\rangle=1\}$ is nonempty and fix $v\in H$. Now, let $\{a_1,\dotsc,a_{d-1}\}$ be a basis for the kernel of the map $\langle \mu,-\rangle:\Bbb Z^d\to\Bbb Z$. Form a $d\times d-1$ matrix $A$ with these vectors $$ A=\begin{bmatrix} a_1 &

Let A be a 2 x 2 matrix such that A^2 = A. Show that Ax = x for every x in R(A) and if rank(A) = 1, M = R(A), N = Ker(A-I) then A is the projection along N onto M. For the first part, I said that A^2 * x = A * x so multiplying by A^-1 on both sides we have A * x = x. This doesn't sound right b...

How do I show that for $f \in C^{\infty}(T)$ for which the Laplace's equation $u_{tt}+ u_{xx} = 0$ with initial conditions $u(x,0)=f(x)$ and $u_t(x,0) =0$ has no soultion with $u(\cdot,t) \in L^2(T)$ in any interval $|t| < \epsilon$ of any $\epsilon >0$. Do I need to find a general solution a...

Let $x_1 = 2$, and define $x_{n+1} = \frac{1}{2} (x_n + \frac{2}{x_n})$ Show that $x_n^2$ is always greater than or equal to 2, and then use this to prove that $x_n − x_{n+1} ≥ 0$. Conclude that $\lim x_n = √2$. My question: So I know how to do this problem but I don't know how to prove that ...

I understand that it is a Fresnel Integral but I don't understand how to evaluate it or find its antiderivative.

I'm having trouble with formatting (lower limit is (pi/3) and upper limit is (pi/6)): \begin{align} \int csc \theta cot \theta \ \mathrm d\theta &\end{align}

f(x) = e^-(x-1) ; x>1 I need the sum of above shifted exponential distribution from 1 to infinity.

Question: Find the cardinality of the set of all points in $R^3$ all of whose coordinates are rational, and justify the answer. Idea: Call the set of all points in $R^3$ all of whose coordinates are rational A. Also, I will make a set called B, that is the set of all coordinates (x,0,0) where x...

I have a simple proof question: Suppose $a,b \in \Bbb Z$ where $a|b$. If $a|(b-c)$, then $a|c$. I have solved it below, but is my way a valid answer? Is there a better clear way of proving this? Suppose $a|(b-c)$ is true given $a|b$. Then $\frac{b}{a} = k, k \in \Bbb Z. $ Re-arrange for b: $...

Let $f:X\rightarrow Y$, and $g:Y\rightarrow Z$ be morphisms of schemes locally of finite type. Suppose that $f$ is unramified, $g$ is flat and $g\circ f$ is étale. Does $g$ is also étale?

Determine the fourier series for the function defined by: f(x) = 2x ................. 0 <_ x <_ 2pi f(x+2pi) = f(x) I have three questions. Is A0 = 0? Is An = 0? Is Bn = (Something =/ Not 0). This has been my approach so far. I can't seem to get the right answer so I have to root out what ...

Here is given table showing correlation coefficients for given variables: Сorrelation coefficients So I need to take simple calculation: enter image description here

I have encountered the following problem and I am curious how to solve it. $\textrm{Given }a_{n+1} = a_{n}(1 - \sqrt{a_{n}}) \textrm{, where } a_{i} \in (0,1) \textrm{, } i = \overline{1,n}$ I have proved that $(a_{n})_{n\in\mathbb{N}}$ is decreasing and now I have to prove that the upper bound...

I know the answer is (D) 4 but I don't know how to solve this.

Struggling with this proof, any strategies for natural deduction? ¬(P^Q) |- ¬P v ¬Q

I have a quantity $f=\sqrt{(a+b)^2-c}$ I need to know the explicit dependence of $f$ on the quantity $(a+b)$, is there any way of removing the term $-c$ from the square root and then compensate for it in a subsequent step?

Fix $a=(a_1,...,a_n)^T,b=(b_1,...b_n)^T \in \mathbb{R}^n$. Assume WLOG $a_1\geq...\geq a_n$, $b_1 \geq ... \geq b_n$. Let $s$ be a permutation of the indices $\{ 1,...,n \}$. Intuitively, the way to minimize $\sum_{i=1}^n (a_i-b_{s(i)})^2$ would be to choose $s$ to be the identity permutation....

Why is $\int_0^L [cos(\frac{n \pi x}{L} - \frac{m \pi x}{L}) - cos(\frac{n \pi x}{L} + \frac{m \pi x}{L})]dx = L $ ? $n$ is an integer, $L$ is a constant.

Let $M$ be a f.g $R$-module for $R$ commutative. Let $S$ be a multiplicative submonoid of $R$ which is f.g as a monoid. Is it true that $S^{-1}M$ is a f.g $S^{-1}R$-module?

Suppose we consider the following function $f(z) = 1 - e^z$ We want to determine the order of the zeroes of this function here. My professor wrote that the order of zero of this function is actually 1? how did he determine that ? Can someone explain?

Anyone, please explain me what am I doing wrong? I have a data (16 districts, 1 time horizon): Sales volume (number of units) Price per unit Sales __ Price __ Revenue 81996 __ 49 __ S*P 91735 __ 49 __ S*P ....and so on I have to find an elasticity in order to define what is ...

$$ \frac{8}{(s+1)^2 + 2^2} * \frac{1}{s} = \frac{8}{5} - \frac{1}{s} + \frac{16}{10}* \frac{s+1}{(s+1)^2 + 2^2} + \frac{8}{10}* \frac{2}{(s+1)^2 + 2^2} $$

Suppose $M,N$ are submodules of some other module and that $M$ is f.g. Suppose $S^{-1}M\subset S^{-1}N$. How can I prove $M[\frac 1s]\subset N[\frac 1s]$ for some $s\in S$?

A recurrent sequence a(n+1)=arcsin(a(n)), a(1) belongs to [-1,1], is given. Is it converging?

If $p$ is a prime number, show that there is no group with $p$ elements of order $p$.

From the paper: https://drive.google.com/file/d/0B8-0WXhCnPn3Vnk1RGpseGJhV28/view In chapter "2.1 Construction of Franklin Functions" there are 2 mathematical formulas. First formula is $φ_i(x) = (x - a_i)_+ , i = 1,2,3$ ... Second formula is $φ_0(x) = 1$, $φ_1(x) = √3 * (2x-1)$, $φ_2(x) = ....

Let's consider $\mathbb{R}$ with the absolute value metric. Is every non countable subset of $\mathbb{R}$ second category in $\mathbb{R}$?

If curl F = 0, is F conservative? I know F conservative implies curl F = 0, however, I want to know if it works the other way.

Is it an explicit formula for $$ _2F_2(2,2;1;1;z) $$ thanks you in advance

Rudin exercise 26 chapter 4: "Suppose $X, Y, Z$ are metric spaces, and $Y$ is compact. Let $f$ map $X$ into $Y$, let $g$ be a continuous one-to-one [not explicitly stated in the book: $g$ need not be onto] mapping of $Y$ into $Z$, and put $h(x)=g(f(x))$ for $x∈X$. Prove that $f$ is uniformly c...

How can I show that for a function $f$ that is differentiable on $[0,1]$ st $f(0)=0$ and $f(1)=1$ there is unique $c \in \mathbb{R}$ and some point $x \in (0,1)$ so that $f'(x)=cx$?

Can someone help me with the following puzzle problems?: There is a number with 6 different digits, if we pick the last digit of that number and place before that number we got $5$ times our number; James is two times older than John was when James was as old as John is now, James is 28 years o...

So I need to evaluate the boundaries. I have found that 0

I have examined partial Collatz 3n+1 trajectories going from one odd integer to the next. These lead to an infinite number of repeated patterns where the "next" odd integer is congruent to one of only six patterns: {5, 11, 17, 1, 7 or 3} mod 18. I have formulae for these patterns in terms of ...

We have a queue of N humans. At this queue there are two friends. What is the probability that between friends will be M humans. (M + 2 < N) So, what we got: total number of combinations = N!; Lets take some example: N=10, M=3. For that case, between friends can be 3 humans, any humans, so...

http://i.stack.imgur.com/NFnZt.png So I'm pretty familiar with SOH-CAH-TOA but this question in particular looks a bit different and I'm not sure how to go about it. Thanks in advance!

What is the probability of a total greater than 9 in a given roll with two standard dice?

Let a,b,c, and d be real numbers. If d < 0 and 3a^2 < 8b, show that x^4+ax^3+bx^2+cx+d = 0 has exactly 2 roots. I know that you have to use IVT to prove that there are atleast 2 distinct roots, but I don't really know how to go about it.

Let $\mathbb{K} \in \{ \mathbb{R} , \mathbb{C} \}$ and $s= \mathbb{K}^{\omega}$ be the usual sequence set with entries on $\mathbb{K}$. I proved that $\mathbb{K}$ induces a $\mathbb{K}$-vector space structure on $s$ and that the function $\rho : s \times s \to \mathbb{R}$ given by $$\displaystyle...

I want to try and construct a proof by contradiction but am having a hard time negating this statement. The statement that I am working with is There are only a finite number of points accepted into the set and this finite sequence converges to a stationary point. So I want to prove th...

So I think the interval for this question is (n,n+1) and the function is y=lnx, based what the inequality looks like, but I don't know how to approach this question the "proper" way. Also I am not sure what the (n+1)/n does. so if someone can explain it I would be very happy. Thanks

Can someone help me with the following puzzle problem? I really don't get it: James is two times older than John was when James was as old as John is now, James is 28 years old, how old is John; Thanks in advance!

I am looking for arguments to support that this is independent. A study on the effects that listening to loud music through headphones had on teenager's hearing found that 12% of those teenagers in the sample who did listen to music in this way showed the signs of hearing problems. If 60% of the...

Given a nominal rate of 6% per annum. Change it to an effective rate per month. What I do is: $$(1+\frac {0.06}{12})^{12}=(1+i)^{12}$$ where $i$ is the effective interest rate per month. Now what if the question had said that it was the effective annual rate that was 6%, what would I do then? ...

A = Z8 * Z5 * Z2 * Z3 * Z5 B = Z25 * Z4 * Z4 * Z3