what will the summation be for the following equation: 45((n(n+1))/2)

Are those expressions $ 4\sin(x)(\sin(x)\cos(2x) + \cos(x)\sin(2x) $ and $4\sin(x)\sin(3x)$ equivalent?

In a triangle if $\tan A<0$ then values of $\tan (B)\tan(C)$ should be in the range $(0,1)$. (Says my book) I'm getting that $\tan (B)\tan(C)$ should be lesser than $1$ as $$tan (A)=\frac{\tan(B)+\tan(C)}{\tan(A)\tan(B) -1 }$$ But why should it be greater than $0$ ?

I have the following question and I have no idea how I can solve it. Let F be family of functions f in A(D) where D is aunit disc. if a)f(0)=2i and b) |f(z)|>1 for all z in D then Prove that F is a normal family in A(D). I tried to use montel's but this family of functions is not locally bound...

In my textbook it says: $\frac{4}{\sqrt{2}}$V1 + $\frac{6}{\sqrt{3}}$V2 and suddenly on the next page it says 2V1 + 2V2 is there something i'm missing here? or did they just second power the sqrt out???

I am trying to prove $$\lim_{n\to\infty} \frac{n-\lfloor \sqrt n \rfloor^2}{n} = 0$$ given $n>0$. But I'm having difficulties dealing with the floor function. I tried splitting apart the limit like so: \begin{align*} \lim_{n\to\infty} \frac{n-\lfloor \sqrt n \rfloor^2}{n} &= \frac{\lim_{n\to\...

Let $X$ be a topological space and $G_n(\mathbb{C}^m)$ be the space of vector subspaces of $\mathbb{C}^m$ of codimension $n$. Let $G_n(\mathbb{C}^\infty):=\bigcup_{m=n}^{\infty}G_n(\mathbb{C}^m)$ using the natural inclusion $$G_n(\mathbb{C}^m)\hookrightarrow G_n(\mathbb{C}^{m+1})$$ Question: Is...

If $A$ is a square $n\times n$ matrix, with $\lambda_1,\ldots,\lambda_n$ being the eigenvalues of $A$, $v_1$ being the eigenvector associated with eigenvalue $\lambda_1$, and $d$ the column vector of dimension $n$, then can anyone provide a counterexample to show the following statement is false:...

Let $f,g:[a,b]\subset \mathbf R\to \mathbf R$ be continuous functions with $f^2(x)=g^2(x)\neq 0$ for all $x\in[a,b]$. Show $f=g$ oder $f=-g$ on $[a,b]$.

Given two positive real numbers, A and B, such that A<=B, take the geometric mean, giving A', and the arithmetic mean, giving B'. Repeat ad infinitum. My intuition tells me that, since both means give values between the two original numbers, they will converge as the number of repetitions approac...

Consider $\Omega = [0,1] \times [0,1]$ with sigma algebra of borel sets on $[0,1]^2$. Let $P$ be the Lebesgue measure on $\Omega$. Let $$\xi(x, y) = x, \ \ \ \eta(x,y) = y.$$ How can I find $\mathbb{E}(\xi - \eta| \xi +\eta)$? I tried to find out what $\sigma( \xi + \eta)$ and I've found that $...

My neice asked me for help with her national 5 homework. (National 5 is a new qualification in Scotland roughly equivalent to the old standard grade, so I would have thought quite easy). I have made a crude MS paint of the problem and linked it below. http://s22.postimg.org/nyhmey2a9/nat5_maths....

For $A \in \mathbb{R}^{n \times n}$ with $A=A^{T}$ we set: $$\lambda:= \max \{ \langle x, A x \rangle: ||x||_2=1\} \\ \mu:= \min \{ \langle x,A x \rangle: ||x||_2=1\}$$ Then for $x \in S_{||\cdot||_{2}}$ we have: $$\langle x, Ax \rangle \leq |\langle x, A x \rangle| \leq ||x||_2 ||Ax||_2 \leq ...

I'm trying to calculate the variance of a martingale strategy (inital_bet = 1) with infinite bankroll. Where p = P(winning), N = number of trials to win, M = payout. Since this is a geometric variable I know N = 1/p. I deduced E[M] = 2^(1/p - 1). But I'm having trouble with var(M). I tried the...

exercise : A function $f$, continuous on the positive real axis, has the property that $$\int_{1}^{xy}f(t)dt =y\int_{1}^{x}f(t)dt +x\int_{1}^{y}f(t)dt$$ for all $x > 0$ and all $y > 0$. If $f (1) = 3$, compute $f (x)$ for each $x > 0$. My progress: I derive $f(xy)y = yf(x)+\int_{1}^{...

Let $d\le 3$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I'm considering an incompressible Newtonian fluid with uniform density $\rho_0$ and viscosity $\nu$. In this case, the stationary Navier-Stokes equations are $$\left\{\begin{matrix}(u\cdot\nabla)u&=&\displaystyle\nu\Delta u-\frac ...

This problem arose in a different context at work, but I have translated it to pizza. Suppose you have a circular pizza of radius $R$. Upon this disc, $n$ pepperoni will be distributed completely randomly. All pepperoni have the same radius $r$. A pepperoni is "free" if it does not overlap an...

I'm aware of the formal definition of algebraic and geometric multiplicities, but I can't make much sense out of the names. What I mean is, if it were me, I would name these two quantities multiplicity one and multiplicity two, for all I know. But I'm guessing that the people who named these two...

Is the infinite product $\prod_{n=2}^\infty\Big(1-\dfrac2{n(n+1)}\Big)^2$ convergent ? If so , then what is it's value ? Please help . Thanks in advance

How to solve the following optimization problem: $$ \max_{x_i} ~ f(x_i,x_j) = h(\max(x_i,x_j)) - x_i\qquad i=1,2 $$ where $h(\cdot)$ is a concave function. Thanks.

Consider three events $A,B,C$ and a probability measure $\mathbb{P}$. Do you know any rule linking $\mathbb{P}(A\geq B)$ with $\mathbb{P}(A\geq C)$, $\mathbb{P}(C\geq B)$? Moreover, is it true that $\mathbb{P}(A\geq B)\leq \mathbb{P}(A\geq C)+\mathbb{P}(C\geq B)$? For any rule that you sugges...

You have 9 boys and 6 girls in a line. In how many ways can you arrange them so that no girls stand next to each other.

The linear transformation is given as T: M33 -> M33 defined by T(A) = 1/2(A+A^T). This is also known as the symmetrization operator.

My question is basically in the title: What exactly is Green's Function and why can I use it to solve harmonic oscillator problems? In other words, how is Green's function connected to physics problems like the harmonic oscillator? My question is motivated by a bonus question on my work...

Will someone please describe the following set in words.

$(a_n) =(\frac{(-1)^{n+1}n}{4n-5})$ I dont understand why the lower limit isnt equal to - 1. Since it is lowest number in the set of infumums

I'm struggling solving the following limit problem: Limit problem At first I thought I could Multiply by: conjugate But that doesn't seem to take me anywhere closer to an answer. Some help would be appreciated.

I'm programming a chess game and I'm trying to validate the movements every player tries to make. Obviously, every piece can move differently and I've had no trouble validating their moves up until now. The first thing I do in order to validate the pieces is to check for collisions. In order to ...

I found the following functional equation: Find all functions $f : \Bbb R \rightarrow \Bbb R $ such that: $xf(x) + yf(y) = (x - y)f(x + y) $ for all $x, y \in \mathbb R $ Could you please help me? I think I proved that if $f(0) = 0$ then for each $x \in \Bbb Q$ $f(kx) = kf(x)$, but I don't kno...

Given: $f(t,u(t)) = u'(t)=t\cdot exp(u(t))$ and $u(t_0)=u_0$ We have as solution $u(t)=-ln(e^{u_0}+\frac{1}{2} t_0²-\frac{1}{2}t²)$ as here: Wolfram's solution It somehow bothers me that whatever $t_0$ is chosen, it is between the two asymptotes $\pm \sqrt{t_0²+2e^{-u_0}}$. It would mean that t...

I know that higher-order derivatives are defined inductively as $f^{(n)}=(f^{n-1})'$. However does this imply that $f^{(n)}(x) = \lim_{h\to0}\frac{f^{(n-1)}(x+h) - f^{(n-1)}(x)}{h}$ ?

I know that orthogonal projection of a vector v on a subspace W is given by proj_W(v) = (u1.v)/u1.u1 * u1 + ... + (uk.v)/uk.uk *uk where {u1,..,uk} is an orthogonal basis of W. I am suppose to find a matrix B that projects a vector on W. Now, I know that by applying the above formula with tak...

I have this ODE system:$$x'(t) = ax(t)+ by(t)$$ $$y'(t)= cx(t) + dy(t)$$ how do I get the coefficients? I know how to solve ODE with two variables, but now sure how to solve the system...

So the question is, do calculators really use Newton's method to calculate things? I mean, for some equations, is it how they get the answer?

I have a random variable $X$, where $0<X<1$; and a random variable $Y$. Assume $X$ and $Y$ are uncorrelated but not independent. If I let $Z=binomial(p=X)$. Is it true that $Z \perp Y|X$

Suppose $A \overset{f}{\to} B$ is a morphism in a category (say for ease they are abelian groups). When we consider other morphisms out of $A$ (say to a fixed object $C$), suppose we ask which maps $A \overset{g}{\to} C$ factor as $A \overset{f}{\to} B \overset{\tilde{g}}{\to} C$. Here are the...

I have an equation to fit a curve of phosphorus concentration Response = 1.927596E-03 + 4.631966E-04*concentration + 3.297547E-08*concentration^2 I usually use it to predict my absorbance by an know concentration solution. For example, using the concentration 11,473: f(x) = ax^2 + bx + c ; ...

I want to show this but I dont have idea. $\ Hom(\mathbb Z_n,\mathbb Q/\mathbb Z)\cong \mathbb Z_n $ what the homomorphism between $\mathbb Z_n \to \mathbb Q/\mathbb Z$. thanks your help

We consider the following setting: Let $Z_{1},...,Z_{n}$ be iid random variables with distribution function $H_{Z}$ and $u>0$ a constant. We set $M:= \sup_{n\in N} \sum_{k=1}^{n} Z_{k} $. I found the following step in a book: $P(Z_{1}\leq x_{1},...,Z_{n}\leq x_{n},M>u) = \int_{-\infty}^{x_{1}} ...

enter image description here How do I solve the below equation with single application of trapezoidal rule?

In Least Square optimization, A=\begin{pmatrix} 1 & t_1 & t_1^2 & \cdots & t_1^k \\ 1 & t_2 & t_2^2 & \cdots & t_2^k \\ \vdots & \vdots& \vdots & \ddots & \vdots \\ 1 & t_n & t_n^2 & \cdots & t_n^k \end{pmatrix} b=\begin{pmatrix} s_1 \\ s_2 \\ ...

Here https://www2.bc.edu/~reederma/Groups.pdf on page $112$, a table of the number of groups of order $p^2q$ is given. In the explanations, there is a typo ($\frac{q+5}{5}$ instead of $\frac{q+5}{2}$) Can someone approve the following formula for $f(p,q)$, the number of groups of order $p^2...

graph Note that the graph has been constructed so that there are no cycles of length three (triangles) or four (quadrilaterals). Note also that the gra ph has a total of fifteen edges and ten vertices, and every vertex has degree 3. Now, s uppose this graph had a Hamiltonian cycle. That wo...

How can I show this? For every prime number, p, there is at least 1 solution (x,y) such that x^2+y^2=-1modp Thanks

How to solve the differential equation $$\frac{dy}{dx} + \frac{1}{x} \tan(y)= \frac{1}{x^2} \tan(y)\sin(y)$$ Hints please.

In any n+1 integers there will be a pair which differs by a multiple of n. I have tried to create a pigeon hole with numbers a0,a1,a2,...,an but i could not get a solution.

I have problems solving \begin{equation} \frac{dx}{dt}=2xy; \quad \frac{dy}{dt}=1+x^2-y^2. \end{equation} I can solve similiar system easily when I have \begin{equation} \frac{dy}{dt}=2xy; \quad \frac{dx}{dt}=1+x^2-y^2, \end{equation} by introducing complex number $z=x+iy$, then $\frac{dz}{dt}=...

Let $M$ be a 3 dimensional manifold, $N$ a surface in $M$ and $A$ the second fundamental form on $N$, $H$ the mean curvature. $h$ is the metric induced on $N$. I need to show that \begin{equation*} \int_{N_l}|A - \frac{H}{2}h|^2 = \frac{1}{2}\int_{N_l}(\lambda_1 - \lambda_2)^2. \end{equation*} w...

Suppose $X$ and $Y$ are topological spaces which are both embeddable as a closed subspace of $\mathbb{R}^n$ for some $n\geq 0$. Let $f\colon X\to Y$ be a proper map, i.e. a continuous map such that the pre-image of every compact set in $Y$ is compact in $X$. Do there exist a continuous map $...

Can someone explain to me the difference between joint probability distribution and conditional probability distribution?

I am new to this forum. I was reading this document : http://math.kennesaw.edu/~plaval/math4490/rotgen.pdf Here the author says that from this figure : http://i.imgur.com/4KyrI3e.png We can express Vper like this : T (Vper) = COS * Vper + SIN * W I dont understand this part. Can anybody exp...

I want to ask how to model if x==y return 1 else return 0 function as constraints in the model?

Suppose we are given a random sample x1, x2,..., xn from a distribution with density: f(x;θ) = θx^(θ−1) ,0 < x < 1. Find the UMP critical region of significance level α for testing H0 : θ = θ0 against H1 : θ < θ0 C={x:L(θo)/L(θ1)<Κ} ={x: θo x^(θo-1)/θ1 x^(θ1-1) Woudl this be...

I have been asked to show the Fourier Transform of f''(t), however I am confused about what apporach to take. I know that there are certain properties that follow Fourier Transforms for derivatives and I have also seeked help from this link Fourier Transform of Derivative but I am still confused ...

Let $m,n\in\mathbb{N}$, with $m,n>1$. Suppose $K\in \mathbb{M}_{mn\times mn}(\mathbb{C})$ is positive semidefinite. We can always write $$K=\sum_{i,j=1}^m E_{i,j}\otimes K_{i,j},$$ for some collection of matrices $K_{i,j}\in \mathbb{M}_{n\times n}(\mathbb{C}) $, where $E_{i,j}\in \mathbb{M}_{m\...

The two dimensional standard vector space is not a subspace of the three dimensional standard vector space. Give three distinct subspaces of C3 that are isomorphic to C2.

I'm considering a problem of computing the degree of a map $\varphi: S^{1} \times S^{1} \rightarrow S^{2}$ defined as $$\varphi(x, y) = \frac{\gamma_{1}(x)-\gamma_{2}(y)}{|\gamma_{1}-\gamma_{2}|}$$ $\gamma_{1}(x) = (\cos{x}, \sin{x}, 0) $ given by $x^{2}+y^{2}=1$ and $\gamma_{2}(x) = (\cos{y}+1...

Evaluate this integral. $\int _{0}^{1}\sqrt [3] {2x^{3}-3x^{2}-x+1}dx$

Need some help with this problems: Is there $f \in C(\mathbb{T})$ such that $\hat{f}(k) = \dfrac{1}{|k|^{1/2}}$, if $k \neq 0$? Suppose the $f_n \in L^1(\mathbb{T})$, $n = 1,2,...$ and $\| f_n - f \|_{L^1(\mathbb{T})} \xrightarrow{n \to \infty} 0$. Prove that $\hat{f_n} \xrightarrow{u} \...

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

For the problem attached, I don't understand where the 2nd double integral came from in line 4 of finding the density of $Z$. I understand up to line 3 of solving for $F_z(Z)$

So I found a particular question on my math homework today that I thought would be fun to toy with: Mary rides her bike the same distance that Leah walks. Mary rides her bike 10km/h faster than Leah walks. If it takes Mary 1 h[our] and Leah 3 h[ours] to travel that distance, how fast does e...

We have the following matrix $$ \begin{bmatrix}1 & -2 & -1 & b_{1} \\0 & -2 & -2 & b_{1}+2 b_{2} \\0 & 0 & 0 & 3b_{1}+ \frac{7}{2}b_{2}+ b_{3} \end{bmatrix}$$ which clearly does not have any solutions when $3b_{1}+\frac{7}{2}b_{2}+b_{3} = 0$. But can we not figure out more restrictions? I...

Is there a way to proof that this always works: Pick 21 different cards; Choose one card, and remember it; Place the cards on three accumulations (every accumulation is 7 cards); Choose the one with your chosen card; Pik up the three accumulations with the one with your card on the top; Do that...

Compute $$\sum_{n\ge 1} n^2x^n$$ Here's what I did: $$\sum_{n\ge 1} n^2x^n = x\sum_{n\ge 1} n^2x^{n-1}$$ We integrate the power-series and get: $$x\sum_{n\ge 1} nx^n = x^2\sum_{n\ge 1} nx^{n-1}$$ We integrate one more time to get: $$x^2\sum_{n\ge 1}x^n$$ In the interval $(-1,1)$ we hav...

I am investigating the Wilson Cowan neuron population model, and I can follow most of it, but I'm not sure what is meant by t' in the equation for proportion of neurons in the refractory period. Heres a link to the article: https://en.wikipedia.org/wiki/Wilson%E2%80%93Cowan_model The image below ...

Can someone explain to me what do we mean by Conditional on a random variable(discrete)?

I Got the set $\sqrt[k]{k+1}$ I have found the supermum by the Inequality of arithmetic and geometric means. And the result is 2. I dont have a way for solving the infimum . I tried to solve it by move to to some eqaution and try by Binomial theorem, but i didnt success. I know that the infimu...

I have a problem about topology that is supposed I must prove without the use of any metric in the standard topology of $\Bbb R$ (metrics are not started in the book by now) just using pure topology definitions. The problem say Let $S\subset\Bbb R$ and $a\in\Bbb R$. Prove that $a\in \overlin...

Absorption Law states A∪(A∩B)=A and A∩(A∪B)=A I can't seem to picture these with venn diagrams can someone help me out ? Thanks

Let ß be an automorphism of C1 such that ß(2) = 10. Compute these three conditions. 1. ß(0) 2. ß(2i) 3. ß(8 + 2i) Additionally,Can you explain automorphism with another example?

Let z = 1 + i. Find the real and imaginary parts of i.

I wish to solve this guys in 3 dimensions $(\Delta-\lambda^2) u(\vec r)=-c$ where $\lambda$ and $c$ are real and $c$ is a constant. It is the screened Poisson equation and from the wikipedia page I see that the solution should be $ u(\vec r) = \int d^3r' \frac{e^{-\lambda |\vec r - \vec r'| }}...

Find two distinct isomorphisms between the standard nine dimensional vector space C9 and3*3 matrices M33?

The following theorem is true? f: [a,b] into R

http://i.imgur.com/wasyzxz.png?2 The red part is my main problem. I don't get why I can change just the upper border. Can somebody please explain? I tried so many ways and failed..

$$lim_{x\to 0} \frac{e^x-1}{x^2}$$ it is an expression in form of $(\frac{0}{0})$ using l'hopital: $$lim_{x\to 0} \frac{e^x}{2x}$$ The expression in form of $(\frac{1}{0})$ so one-sided limits should be checked $$lim_{x\to 0^{-}} \frac{e^x}{2x}=-\infty$$ $$lim_{x\to 0^{+}} \frac{e^x}{2x}=\in...

Thank you It would be much appreciated if I get to know the answer as soon as possible Thanks again

I have such stochastic process with which I struggle all day, finally I found 2 mistakes, however answer is still unsatisfying. $$X_t = atW_t^2 - \int_0^t(W_s^2+s)ds,$$ I need to check if it is a martingale. I simply write Ito formula for $X_t(t,W_t,S_t)$, where I denote by $S_t = \int_0^tW_s^2d...

The matrix given to me is : $$A=$$ $$1\ \ r\ \ r\\r\ \ 1\ \ r\\r\ \ r\ \ 1.$$ Find the values of $r$ for which this is positive definite. So,I naturally try to find the determinant of the matrix $$A-xI$$ where $$I$$ is the $3\times 3$ identity matrix. The determinant is $${(1-x)}^3-2r^2(1-x)...

$\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$ Let $\mathbf{x},\mathbf{y} \in \mathbb{R}^n$. I wish to show $$\norm{\mathbf{x}}-\norm{\mathbf{y}} \leq \norm{\mathbf{x}-\mathbf{y}}\text{.}$$ I have already proven $$\norm{\mathbf{x}+\mathbf{y}} \leq \norm{\mathbf{x}}+\norm{\mathbf{y}}\text{....

Suppose $X$ and $Y$ are convex compact subsets in $\mathbb R^n$. Let $\langle.,.\rangle$ be the standard inner product. Does the following equality $$\max_{y\in Y} [\langle y, z\rangle- \max_{x\in X}\langle y, x\rangle]=\max_{y\in Y} \min_{x\in X}\langle y, z-x\rangle $$ hold? I saw the term o...

I am struggling with the derivation of the Lévy-measure of a Gamma-process $X_t$ with law $p_t(x)= \frac{\lambda^{ct}}{\Gamma(ct)}x^{ct-1}e^{-\lambda x}1_{\lbrace x>0 \rbrace }$. The paper I am reading says it is shown "easily" via the characteristic function and using the Levy-Khintchine formula...

I just can't figure out how to take the derivative of this function!

The questions here arise out of investigating theoretical straight-line pressure-temperature phase diagrams for suites of minerals in chemically-similar metamorphic rocks. I've tried to phrase the questions in purely mathematical terms. All points and lines lie in a single Euclidean plane. We st...

This is an exercise in Hungerford. I've tried proving this as follows (forgetting for the moment that in $A_4$, $n_2=1$ and $n_3=4$): By Sylow III, $n_2=1$ or $3$. If $n_2=3$, the number of elements of order $2$ in $A_4$ is $n_2 (2-1)=3(2-1)=3.$ Then the number of elements of order not $3$ is $...

If the map ß from polynomials of degree at most 2 P2 to C4 dened via 0 1 1 0 ß(0)= 0 ß(x)= 0 ß(1+x^2)=0 ß(1+x+x^2)= 0 0 1 1 0 0 2 2 2 (These all matrices.But,I don't know how to ...

Prove that if $S$ is non-empty compact subset of metric space $E$ and $p_0\in E$, then $\min \{d(p_0,p) : p\in S\}$ exists. Since $S$ is compact, it will be enough to proof that function $S\ni p\rightarrow d(p_0,p)\in \mathbb{R}$ is continuous. For any $x$ in $S$ and any $\epsilon>0$ ch...

I was attempting to find a matrix for the function x*(d/dx) in the span of the set {1,x,x^2} for the the standard dot product. Could someone guide me in how to do this?

Assume that we have a continuous time Markov chain $(X_t)$ on $\{0,1\}$ and $f(t):=P(X(l)=0 \text{ for all } l\in [0,t]|X(0)=0),$ then I want to show that under the assumption that $f'(0)$ exists, we have $$f'(0)=\frac{d}{dt}|_{t=0} P_t(0,0).$$ I managed to show the trivial inequality, i.e. $$...

Suppose T is a mapping such that T( 1,1 ) = ( 2,1 ) and T(1,-1) = (2,-1) (each set of numbers is a vector) Can the mapping T be Linear? Please note that the question is not asking if T is a Linear Transformation. Is there a way to show that T is Linear by plotting vectors and show linearity?...

I have looked on Wolfram for the limit of $lim_{x \to 0} \frac{lnx}{x}$ The full answer uses one-sided limits, and the product rule stating that for $0^{-}$ is a product of $-\infty \cdot -\infty=\infty$ and for $0^{+}$ is a product of $-\infty \cdot \infty=-\infty$ does the product rule stands...

Does sample paths continuity of a real-valued square integrable stochastic process imply continuity of its covariance function?

A recent article in Healthy Life magazine claimed that the mean amount of leisure time per week for European men is 48.5 hours. The distribution of leisure time amounts is reported to be approximately Normal. You believe the figure of 48.5 is too large and decide to conduct your own test. In a ra...

can someone please help? This problem asks to find the equation of the tangent line to y= 3sin(x) at x= pi/3. I tried finding two points of the graph and finding the slope of the line by taking the derivative of 3sin(x), am I in the right direction? I think I'm doing something wrong because I ...

Write out the addition and multiplication tables for $\mathbb{Z} _2$[x]/$(x^2 + 1).$

For instance: A Field with 13 elements exist. How to prove that this statement is true or false?

I am in trouble with the following excercise: We have an infinite chessboard with squares , where all squares are at first white except one initial set M(0) with n black squares,where M(0) is an initial formation of black squares.We set new formations of black square as follows: one black square ...

Does this definition contain redundancy: "A function $f$ is said to be analytic at $x_0$ if its Taylor series about $x_0$ exists and converges to $f(x)$ for all $x$ in some interval containing $x_0$". Is anything missing in the following definition: "A function $f$ is said to be analytic at $x_0$...

Let ß be an automorphism of C1 such that ß(2) = 10. Compute these three conditions. 1. ß(0) 2. ß(2i) 3. ß(8 + 2i) C1 is a vector space.

Hi can someone please help? I need to evaluate this indefinite integral: ((lnx)^5 dx)/(x) I know I need to use substitution, so if I let u= x but I can't figure out the antiderivative for the top portion. Thank you!

What is the solution to the integral of the integral: $\int_0^T 1/t dB_t$? How do i go about solving this?

I am reading some notes on probability theory, and it says $$V(X) = EX^{(2)} - (EX)^{(2)}$$ Why is it using parenthesis around the powers?

First-time poster here. While doing some research on Waring's problem and the term $\{(3/2)^n\},$ I determined that the following recurrence relation holds for a certain sequence (here $n$ is a fixed, positive integer): $$r_{k}={n \choose k}+\frac{1}{2}r_{k-1}$$ Mathematica gave me the result $...

What is really confusing me about this problem is that the field is ℤ_11; if F = ℝ the basis could be {(-1/3,1,0,0),(-1/2,0,1,0),(-5/6,0,0,1)}. However, all three of these vectors contain values not in ℤ_11, so I'm not quite sure how to proceed.

The title says it all. I'm not sure on how to approach this problem. Any help will be much appreciated.

Are $ \mathbb{A} (k ) = k^n $ and $ \mathbb{P}^{n} (k) = \mathrm{Proj} \ k[X_0 , \dots , X_n ]$ irreducibles when k is a domain ? Thanks in advance for your help.

Suppose a 2D line passes through two points P0(10, 15) and P1(200, 20) What is the perpendicular distance from point P(-500, 48) to the line? Following this formula this is as far as I got before I got lost. dx = 10-200 dy = 15-20 dist = sqrt(-190*-190 + -5*-5) dx / 190 (dist) dy / 190 (dist...

Evaluate $\displaystyle\sum\limits_{i=1}^{12} {n\choose{k}}k^2$ The answer is 159744.

Let $d\le 3$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I'm considering an incompressible Newtonian fluid with uniform density $\rho_0$ and viscosity $\nu$. In this case, the stationary Navier-Stokes equations are $$\left\{\begin{matrix}(u\cdot\nabla)u&=&\displaystyle\nu\Delta u-\frac ...

How can we verify Dvoretzky–Kiefer–Wolfowitz inequality for a uniform Distribution?

I am trying to solve the following; First, given G is a group and H a subgroup of G, what can we say about the relation $a \cong b$ if $b^{-1}a \in H$ I can show that it is reflexive as the identity is always in subgroup. if $a \cong b$ then $b^{-1}a \in H$ and so $(b^{-1}a)^{-1}=a^{-1}b$ $\i...

We are provided with a recurrence relation as follows:- F(n,k)=F(n-k,k-1)+F(n-k,k) ; F(n,0)=0 ; F(1,1)=1 I need help to solve this one.

can someone please help? I need to evaluate this indefinite integral: (x)dx /(sqrt{x^2+ 2}) I tried using substitution for letting u = x, but I can't get past finding the antiderivative after that. Thank you!

The wikipedia-article for the P-NP problem [1] says there are three possible answers to the P-NP-problem: $P=NP$ $P\neq NP$ $P=NP$ is independent of ZFC The third possible solution seems to be very interesting. Assuming it is true, there could still exist a turing machine which solve e.g. $SA...

Is the operator $A:C[0,1]\to C[0,1]$ defined by $Ax(t)=tx(t)$ is compact? And what if we change the underlying space?

I think about the spectral theorem for compact operators on a Banach Space. And I come to a question: Can the Theorem be generalized to any Normed Space or a bigger subclass of TVS

I know of: $a = p(1+\frac rn)^{nt} , $ but how would this be modified to include a regular deposit at a given interval. Thanks for your help, Julian

How do i evaluate this integral for any integer $n>1$ $$\int \limits _0^ {\infty} ln^n\frac{|1-x|}{|1+x|}dx$$ Note: for $n=1$ just we use integral by part and the integral will be divergent .My problem what about $n >1$ Thank you for any kind of help

I have the following question given r(t) = e^(t) cos(t) i + e^(t) sin(t) j, where 0 < t < infinity reparametrize the curve by arc length and compute its curvature. I know how to reparametrise it but can't seem to use the reparametrisation to computer curvature. Help

$r^3+1 = (r+1)(r^2-r+1)$ I know we can simply multiply equations in the right-hand side then we get $r^3+1$. However, is there any way to construct right-hand side without knowing it?

Let V the space of free vectors from the geometric space. Prove that: =|AB|*|AC|*cos(<(AB,AC)) is a scalar product, where AB and AC are vectors and <(AB,AC) is the angle between AB and AC counted in the trigonometrical sense. I tried choosing a base in V,so I can get the biliniar form fr...

I'm reading Conway's complex variable book and I didn't understand this proof on page 31: I didn't understand why $\frac{1}{r}>\frac{1}{R}$ implies there is an integer $N$ such that $|a_n|^{1/n}<1/r$ for all $n\ge N$. Any help is welcome.

In "A Mathematician’s Lament", Paul Lockhart derides the "status quo" of math education, claiming that "mathematics is an art form done by human beings for pleasure" but instead is taught "devoid of creative expression of any kind". His writing is provocative, and I'm sure his accusations and sug...

I am trying to understand proposition 2.1 of chapter IV of Zak's 'Tangents and Secants of Algebraic Varieties'. Let $$ S^{0}_{X}=\{(x,y,z):(x,y)\in X^{2}-\Delta_{X}, z\in \langle x,y\rangle \}, $$ $$ S_{X}=\overline{S^{0}_{X}}\subseteq X\times X\times \mathbb{P}^{N}. $$ We denote $$ p_{1}:S_{X}\...

Let $\left( X_n \right)_{n \ge 1}$ be a sequence of iid random variables with discrete distribution $p_k=P(X=k)>0$ for $k=0,1,2,...,N$, where $N<\infty$ and $\sum_{k=0}^Np_k=1$. We define $T_1=1$ and $T_n=\min \{ k \in \mathbb{N} : k>T_{n-1} \wedge X_k \ge X_{T_{n-1}} \}$ for $n=2,3,...$ and $W_...

For a complex polynomial F(z) = z^8 + 5z^7 - z^4 + 2, how many roots does it have in the disk |z| < 1? I want to apply Rouche's Theorem. So let f(z) = 5z^7. Then |F(z) - f(z)| = |z^8 - z^4 + 2| <= |z^8| + |-z^4| + |2|. At z=1, 1 + 1 + 2 = 4 < 5 = |f(z)| which implies F(z) has 7 roots in |z|<1. D...

The question is: Let {$a_j$} be an arithmetic sequence with the following conditions: $S_16$ = 376 and $a_16$ = 46. Find $a_1$

Could I have help with part biii) please? I solved b ii) by getting an equation for $f(x,y)$ by substituting from the differential equation and getting bounds for all the terms like $\frac{1}{x}$ , $exp(\frac{1}{x})$ using the bounds on $x$ and $y$ given but obviously couldn't get a specific $...

I am supposed to evaluate F(x,y)=(4x^3y^2-2xy^3)i + (2x^4y-3x^2y^2+4y^3)j along the curve r(t)=(t+sin(tpi))i+(2t+cos(tpi)j, 0<=t<=1. I could put try to do the \int f(r,t) r'(t) as t goes from 0 to 1, but that seems really messy. Is there a simpler way?

Well I'm in organic chemistry and am trying to explain to her that it's not the same thing at all. Please help me explain it to her.

Find the vector tangent to the curve of intersection: x^2 + y^2 =8 and y^2 +z =8 at the point (2,-2,2)

Im stumped by this question on a practice ACT math test. If $\frac 1x + \frac 1y = \frac 1z$ then $z =$? The correct answer is $\frac {xy}{x + y}$ How do you arrive at this answer? I don't know how to even begin with this problem.

I wanted to ask whether my thoughts are right. After computing out the angle of π/4 and the r I got this: z_{1} = 0,5^{100}\exp(\pi*i) Did I make any faults? Could it be simplified any further?

My problem revolves around the function: $$ f(x) = \frac{\sin(\tan x) - \tan(\sin x)} {\arcsin(\arctan x) - \arctan(\arcsin x)} $$ The limit of f(x) as x -> 0 = 1. However, whilst approaching 0, there is a lot of sporadic oscillations (mainly between -0.005 and 0.005). I've been asked 'Since ...

Let: $$x= \frac{1}{\cosh^{2} (t)},$$ I want to express $\frac{d^n}{dx^n}$ in term of $\frac{d}{d t}$. Thanks in advance

I'm having some trouble solving this directional derivative problem. $$ f(x,y) = sin(2x + 5y), P(-15, 6), \mathbf{u} = {1 \over 2} (\sqrt{3}\mathbf{i-j}) $$ I know the theorem for the bivariate directional derivative: $$D_u f(x, y) = f_x(x, y)a + f_y(x, y)b $$ But that is with the unit vector...

So there's this false proof going around that I can't seem to find now that says that complicated numbers don't exist. So let me explain what it's about (I've added some technical details of my own, but the idea is the same). Definition A number $a \in \mathbb{N}$ is said to be complicated if it...

Let G be a graph. Let k be the minimal degree of G. Show that G contains a cycle of length k + 1.

So, I have y as an implicit function $e^{2x}= ln(\frac{2(y-1)}{y+1}$) Any hints to how I can get y explicitly? Any hints / help?

How may I state the KKT conditions for minimize $c^{T}x$ subject to $Ax \leq b$, $x$ unrestricted?

How do you solve $\sin\theta$+$\cos\theta$=$1.2?$On the interval{$0$,$2Pi$}. Answer in the nearest radian. I've got... cos(x)=sqrt($1-sin^2x$) sin(x)-sqrt($1-sin^2x$)=1.2 (sin(x)+1.2)=(sqrt($1-sin^2x$)^$2$ $sin^2x$+2.4sin(x)+(1.44-1)=.44 $sin^2x$+2.4sin(x)-.44=0 $2$$sin^2x$+2.4sin(x)-.44=0...

Okay so I'm doing numerical integration using the trapezoid rule. I wrote the m file as such: function y = trap(f,a,b,N) h= (b-a)/N; x=0; for i=1: N-1 x=(a+i*h);y=y+eval(f); end y=2*y; x=a; y=y+eval(f); x=b; y=y+eval(f); y= (h/2)*y; It says that the value assigned to x might be unused and ...