I am wondering if my solution is correct Question: Let $A ⊆ \mathbb{R}$ be nonempty and bounded above, let $s\in \mathbb{R}$ have the property that for all $n \in \mathbb{N}, s+1/n$ is an upper bound for A. Show that s=sup A. My solution: We observe first that, for each $x \in A$ we have $x \...

When I was teaching plastic decay formula in a class ,a deep question asked to me .someone asked "How can mathematics help to ,save the planet earth?" I made almost the same example $$m\overset{T}{\rightarrow} \frac{m}{2} \overset{T}{\rightarrow}\frac{m}{4} \overset{T}{\rightarrow}\frac{m}{8}......

We have the following problem: At the beginning of every year, a gardener classifies his soil based on its quality: it's either good, mediocre or bad. Assume that the classification of the soil has a stochastic nature which only depends on last year's classification and never improves. We hav...

A loan was taken out on 1 September 1998 and was repayable by the following scheme: The first repayment was made on 1 July 1999 and was £1000. Thereafter, repayments were made on 1 November 1999, 1 March 2000, 1 July 2000, 1 November 2000, etc until 1 March 2004, inclusive (note that the loan was...

i am working on a practice exam for my calculus 2 course and am working through the practice exam and i have come to a question i have no idea of how to start or where to go with it. here is the question: Knowing that (d^k/dx^k)(1-x)^-1=k!(1-x)^(-k-1), we can say the the maclaurin series of f(x)...

When we have a differential equation and we apply the superposition principle, does it stand that the initial equation has a solution if and only if the subproblems have solutions? Or is it just if?

Suppose we have a space curve $r(t)=\left(x(t), y(t), z(t)\right)$ and we consider the curve of all centres of curvature of $r$. First, would this be the correct formula? $$r(t) + {1 \over \kappa(t)} N(t)$$ (where $N$ is the normal vector in the TNB-frame and $\kappa$ is curvature). And is there ...

I know that the area under a curve represents the displacement of an object, but what does the area enclosed by two curves and the volume of the solid formed by a revolution represent?

A function $G$ is defined on a set $S$ with size $k$ : $G(a_1,a_2,a_3,.....,a_k)$. $G(a_1,a_2,a_3,.....,a_k) = 1$ if and only if a convex polygon can be created by taking these $k$ elements as the side lengths. Otherwise $G(a_1,a_2,a_3,.....,a_k) = 0$. You are given the identity permutation over ...

Goal: Prove that $5^{1/3}$+$7^{1/2}$ is irrational. Idea: We can prove this is irrational by supposing it is rational and finding a contradiction. So, $5^{1/3}$+$7^{1/2}$ = $p/q$ where p and q are integers that have no factors in common other than 1. The issue here is that I can not seem to find...

Forgive me if I'm suffering from a fundamental misunderstanding. I have a function where a is a real positive constant: $$ \frac{(-1 + \pi \sqrt s \cot( \pi \sqrt s ))*(-1 + \pi \sqrt{-a +s} \cot(\pi \sqrt{-a+s})}{4t(-a+s)}$$ This has an infinite number of discontinuities along the positive rea...

Suppose $f$ is differentiable on $(-\infty , \infty)$. We say that $x$ is a fixed point if $f(x) = x$. If there is a constant $A$ such that $0 <A <1$ and $|f'(x)| \leq A$ for all real $x$. Then there is a fixed point $x$ such that $x = \lim_{n\to \infty} x_n$ , where $x_1$ is any arbitrary real ...

The question is: We are supposed to find the first 4 terms in the power series. I was able to work through the question but the my answer is wrong, the correct answer is supposed to be the strange thing is that the answer I got was basically this but the coefficients are square roots of wha...

let $(X_n)$ i.i.d integrable random variables and $S_n=\sum_{i=1}^{n}X_i$ How can i compute $E[X_1 |X_2]$; $E[S_n|X_1]$; $E[S_n |S_{n-1}]$?

Given that $G$ is a non-trivial group, prove that Aut$(G\times G)$ has an element of order 2. At this time I lack good intuition for automorphism groups, so I would appreciate some hints about how the statement above be proved.

Ok so I know that math is math. But if you try to figure out analysis. Anal-lysis. Lysis means break down. So... Does math analysis... Break down Uranus? Why doesn't it break down Neptune too?

How would I find the filled Julia set for $f(z)=z^3$? I know it should be the filled unit circle, but I don't quite understand the math. This is what I have so far: Fixed points $z^3=z$ so $z=1,-1,0,\infty$ $f'(1)=3>1$--repelling $f'(-1)=3>1$--repelling $f'(0)=0<1$--super attracting $f'(\infty)=0<

Why is it the case that the box topology and product topology agree if we consider finite products?

I am working on systems of equations in Pre-Calculus, and I presented the teacher a question that I had been wondering for a while. $x^2 = 2^x$ The teacher couldn't figure it out after playing with it for quite a while. What are some ways it can be solved algebraically? Of course it can be so...

I am not sure how to compare (logx)^4 with x^3 or x^4 to get the answer (either 3 or 4).

How to calculate the integral $$ \int_\sqrt[]{y/3}^1 exp(x^3) dx $$

y''+4y'+4y = (3+x) e^(-2x) So I'm working with undetermined coefficient and figured out solution for the left side. But what is the particular solution for the right side? I tried these but they don't work as all terms cancel to zero: y = (Ax+B)xe^(-2x) y = (Ax+B)e^(-2x)

I need help solving the recurrence relation: $U_{n+1}=(U_{n})^{2} (n+2)$, with $U(1)=2$. I've tried wolfram alpha, but something really horrible came up. The methods I've tried have just failed so I need some ideas please.

How to calculate integration 0 to infinite e^x*(log x)^2.I have tried putting x=log t but can't reach the solution.

How to explain this: P(B|A)+P(B|(complement A) ) = 1 Can anyone help me? I can't see how to explain this. Thanks!

I am a beginner of PDE, and surprise that some nonlinear equation will become a linear equation after variable substitution,for example. So, I am curious that are there general theory making equation become linear. If not, why we don't do so ? I want to know the difficult of this way.

I was studying some neural networks back propagation from http://jeremykun.com/2012/12/09/neural-networks-and-backpropagation/ I did not catch how did he achieve this part. Could someone explain to me? Thanks!

I am working with double and triple integral in multi-variable calculus and have found that it is extremely useful to convert between different coordinate systems including: Spherical: Cylindrical: Polar: and Cartesian Well, I know the conversion values and how to transition variables over. M...

Problem said: Suppose people can be divided into two classes: those who are accident-prone and those who are not. The statistics show that an accident-prone person will have an accident at some time within a fide 1-year period with probability 0.35, whereas this probability for a non-a...

Let $G$ be a finite group. Then there exist fields $L$ and $K$ such that $L$ is an extension of $K$ with Galois group $G$. I think that since $G$ is finite, then $G$ must isomorphic to a subgroup of permutation group $S_{|G|}$. I think that Galois extension has connection with symmetric group $S...

Let X1 and X2 be random variables such that E(Xi)=μi and Var(Xi)=αi^2. A. Find E(X1+X2) and E(X1-X2) in terms of the μ's and α's. B. Suppose that E(X1X2)=α. Find Var(X1+X2). When does it equal Var(X1)+Var(X2)? C. Assume that X1 and X2 are independent. Find Var(3*X1-2*X2). Please explain how t...

I'm really confused by this and I'm not sure how to answer it so a detailed explanation on how to would be great, thank you!

I'm currently reviewing for finals and I tried to solve this question, but the program I'm using tells me I got it wrong. Can someone tell me where I screwed up? this is the question- evaluate the indefinite integral of int/(sin2x)(cos2x+1)^(1/2)dx I used substitution to replace (cos2x+1) with '...

I want to solve integration 0 to infinite (log^2 x)*e^-x .

I have found the solutions by a little calculation (2,3,5,7) and (2,3,4,5). But I don't know if there's any other solutions or not?

Fourier transform of function $\cos(w_m-w_0)t$ is $\pi\left [ \delta (w-(w_m-w_0)) \right + \delta (w+(w_m-w_0)) ]$. What would be FT of $\cos((w_m-w_0)t+\phi)$?

Is this true? $\sum a_n $ absolutely convergent and $\sum b_n $ convergent $\implies \sum a_nb_n$ absolutely convergent. I don't know how to proceed .Please help.

Find the surface area of the part of the surface given by $x=3y+z^2$ that lies between planes $y=0$, $y=z$, $z=0$ and $z=3$ Is this integral I have set-up correct?

Let's say I have the heat equation $\frac {\partial u}{\partial t} = k\frac {\partial^2 u}{\partial x^2}$, $0 \lt x \lt L$, $t \gt 0$, subject to the boundary conditions $\frac {\partial u}{\partial x}(0, t) = 0$, $t \gt 0$ $\frac {\partial u}{\partial x}(L, t) = 0$, $t \gt 0$, $L$ of course b...

Find the surface area of the part of the surface given by $x=3y+z^2$ that lies between planes $y=0$, $y=z$, $z=0$ and $z=3$ Is this integral I have set-up correct? $\displaystyle \int_{0}^{3} \int_{0}^{z} \sqrt{10+4z^2} dy dz$ Please look into it

Problem: If a,b,c,d are real, prove that $$a^2+b^2=2$$ $$c^2+d^2=2$$ $$ac=bd$$ Is true if and only if $$a^2+c^2=2$$ $$b^2+d^2=2$$ $$ab=cd$$ My proof is as follows: Note that each of the second set of equations nearly corresponds to the lengths of the sides of a right triangle. The eq...

The lengths of the sides of a triangle form an arithmetic progression. Prove that the radius of the inscribed circle is one third of one of the heights of the triangle.

I need some help to solve this integral $$\int \frac{dx}{ \sqrt[]{-x^{2}-3x -3} } $$

Would it be correct to prove this statement as follows? Since a generator must go to a generator under an automorphism, $(0,0)$ must always be sent to $(0,0)$ under any $\phi\in$ Aut($\mathbb{Z}_2\times \mathbb{Z}_2)$. Now, for Out($\mathbb{Z}_2\times \mathbb{Z}_2)$, the only possible permuta...

Let S describe the set $\bigcup_{n=1}^\infty A_n$ where $A_n = \{(x, y) \in \mathbb{R}^2 | y-x^{2n} \geq 0 \}$ I understand that S describes the set of points "inside of the parabola y = $x^2$, but am not sure how to prove this. Any ideas?

$$I=\int_0^1 \frac{\sin(\ln^4 (1-x))}{x}$$ What is the closed-form evaluation of this integral? I honestly do not have a single clue how to move on from here. (There is no application to the integral, but it is out of curiosity.)

These are proven elements which can enhance cognitive abilities within short period of time. You might have to understand at this point. Here comes the role of the cognitive enhancing properties. I guess it's cool that there are so many hordes out there with . Life seems to be incomplete without ...

Prove that if $v(z) = \mathrm{Im}[(\frac{1+z}{1-z})^2]$, then $v$ is harmonic and $\lim_{r \uparrow 1} v(re^{i\theta}) = 0$. Explain why this does not contradict the maximum principle.

I thought that the former I correct, but then I read this as part of certain proof: Should the sigma be omitted on taking the partial derivative?

Suppose $f_n$ are continuous real valued functions on $[a, b]$ such that $\sum_{n=1}^\infty f_n$ converges, does it follow that the sequence $a_N =\sum_{n=1}^N f_n$converges uniformly?

Is $\{nx^n(1-x)\}_{n=1}^{\infty}$ pointwise convergent on $\mathbb{R}$? If it is, what's the pointwise limit?

I have to know the substitution principle for an upcoming exam. This is the definition given in my book (sorry it's split up all weird). I understand the notation I believe. However, I am having a very difficult time understanding what this might look like. Can anyone give me a simple example t...

Find the domain of $\sec^{-1}(\frac{2+x}{1-x})^{\frac{1}{x}}$ Since the domain of definition of $\sec^{-1} x$ is $x\leq-1$ or $x\geq 1$. So $(\frac{2+x}{1-x})^{\frac{1}{x}}\leq -1$ or $(\frac{2+x}{1-x})^{\frac{1}{x}}\geq 1$ I am stuck here.Please help.

Show that if $E$ is a finite dimensional subspace of $l_p$, there exists an integer $m$ such that $$\| P_m(x) \| \geq (1 - \dfrac{1}{n}) \| x \|$$ where $x \in E$ and $P_m$ is a projection map from $l_p$ onto a subspace generated by its first $m$ basis vectors. The question comes from the proo...

I want to express the real root of $x^3-3x+7$ using radicals. My attempt is contained in the answer below.

$G$ is the direct product of its Sylow subgroups $P_i$. Then if $H \le G$, $P_i \cap H \le H$. I'm stuck now on how to proceede to find a terminating central series for $H$. Also, I know $P_i H / H \le G/H$ so I'm assuming proving the above will prove that the quotient is nilpotent too?

A reciprocal of p's non-integer decimal part is equal to $p+1$ where $p >0 $. Can you please explain this to me.

In the following BBN network, 1)what is meant by P(Martin Late|train strike,Norman Late)? Does this mean probability of martin Late given that Train Strike And Norman is Late? 2) what is meant by P(Martin Late, Norman Late|Train Strike)? Is this the probability that Martin and Norman are l...

In a group of 265 people, 200 like singing, 110 like dancing and 55 like painting. 60 like both singing and dancing, 30 like both singing and painting and 10 like all three activities. How many persons like only dancing and painting. So let $S$ be the set of singers, $D$ the set of dancers and ...

If E→M and F→M are two vector bundles over manifold M, then $C^{\infty}$(E,F) is a $C^{\infty}$(M) module. I am confused about this claim for the operation of module. To f$\in$$C^{\infty}$(E,F) and g$\in$$C^{\infty}$(M), how could g operate on f? f is from E to F while g is from M to R.

I know these relations, (a * b * c * d...)%m = ((a%m) * (b%m) * (c%m) * (d%m)......) %m and (a+b+c+d+........)%m = (a%m + b%m + c%m + d%m)%m But , how does this relation work? 1297 % 6 = ((129%6)*10 + 7)%6 129 % 6 = ((12%6)*10 + 9 ) %6 and so on ..... If I expand 129 % 6 ...

There exists a $RSA$ cryptosystem with $e=2$ , where $e$ is the encryption exponent ?(In general $e>2$)

I have been trying to understand the betweeen multiple logistic regression model and fitted response logistic function of full model. I think log(pi/(1-pi)) is multiple logistic regression model. And estimated pi = exp(Xb)/(1+exp(Xb)) is fitted response logistic function. I read the book and I ...

I want to know if $\sum_{i\neq j}$=$\sum_{i=1}^{n}\sum_{j=1}^{n}$?

$X_{1}, ... , X_{n}$ $i.i.d$ with density function $$ f(x|Θ) = exp^{-(x-Θ)}, x >= Θ$$ I could find method of moment estimate of Θ. method of moment estimate of Θ : $$\hat Θ = \bar{X}-1$$ How can I find the distribution of MME ? I found MLE, distribution of MLE, MME but I can't find distribut...

enter image description here Please explain D answer is why ?

I tried to define a procedure in Maple as follows. removeAnElementInList:=proc(i, l) local r, j; r:=[]; for j from 1 to lengthOfList(l) do ( if(j<>i) then r:=append(r, l[j]) end if ) end do; return(r); end proc; But Maple returns an error: invalid sequence (at "then"). But I d...

is the Mean theorem valid for complex analysis i mean if f is analytic in a convex region, can we alawys find a point c on the line segement between two points of this region(a,b) so that: f(a)-f(b)=f'(c)*(a-b) ??

Prove that the above inequality holds for sufficiently large n. pi(2n) - 1.5 pi(n) >= O ( log n / (log log n )^2 ) Log n denotes to natural logarithm and Pi(n) is prime counting function.

This is from my practice final exam. A variation with '8' instead of '7' can also be seen in an exam paper from MIT ocw probability course here: http://ocw.mit.edu/courses/mathematics/18-440-probability-and-random-variables-spring-2014 7 people throw their hats into a box and then randomly redi...

Given 2 primitive pythagorean triples with parameters as per: https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple First triple has parameters $(m,n)$. Second has parameters $(r,s)$. Note that $m>n,r>s$ We define $g$ and $h$ according to these equations: $$g^2 = (mn(r^2-s^2))^2 ...

The problem statement is: Show that there exists numbers $a$ and $b$ such that $$det (A + sxy^*)= a+bs$$ here $A$ is an $nxn$ matrix with real entries, and $x,y\in R^n$. I've been kind using brute force and using multi-linearity of the determinant several times, and the computations are gett...

If (S,T) be a topology space and $A$ be a subset of $S$ and $bd(A) = \overline{A} ∩ \overline{A^c} $ Show that $int (bd (A)) = ∅$ if and only if $bd (bd (A)) = bd (A)$

I don't understand why the Hausdorff dimension of a countable set in $\mathbb{R}^n$ is $0$. Can someone please give me a hint? Thank you!

I need help with this Fourier transform computation. F(w)=\int_{\infty}^{\infty} e^{-|x|+ix}e^{-iwx}$

Can someone explain the proof of the result 7.131 on page 450 of Eberhard Zeidler's 1st volume on Quantum Field Theory?

$G$ is a finite group and $H_1$,$H_2$ are two disjoint subgroups of order $2$. $H$ is the subgroup of smallest order that contains both $H_1$ and $H_2.$ What is the cardinality of $H$ $?$ $A.$ always $2$. $B.$ always $4.$ $C.$ always $8.$ $D.$ none of the above. No...

I'm wondering how to read this sum: $w_k = \sum_{p+q=k} u_p v_q$

I cant see which version and how they use Baires theorem to get that atleast on $MB_{n}$ is dense in some open set. Any version of Baires theorem needs open or closed sets. I can get neither on the $MB_{n}$'s.

Suppose $C[0,1]$ is the ring of all real valued continuous functions and if $M $ is a maximal ideal of $C[0,1], i want show that the following set is nonempty. $$\{a \in [0.1] \| \forall f \in M , f(a)=0) $$

Are there some function $f$ in Maple such that $f(K[1,2]^(1/3)) = 1/3$? Any help will be greatly appreciated!

Consider a collection of circles in the plane whose centers are distributed according to a spatial Poisson process with parameter $\lambda|A|$, where $|A|$ denotes the area of the set $A$. The radius of each circle is assumed to be a random variable independent of the location of the center of...

Could some one please explain to me this remark: Let $S_{+}^{n}$ denote the set of positive semidefinite (psd) $n × n$ symmetric matrices. We write $X \succeq 0$ to denote that $X$ is symmetric and positive semidefinite. $\textbf{Remark} $...

I made 2 approaches Am I any close...how do i procced?

Let (R,+,*) be an unitarian ring. (a) If A is an unitarian ring. a,b are from R and a*b is invertible,it results that a is invertible and b is invertible? (b) If a is from R and a^n=a * a*...* a(n or) is invertible,it results that a is invertible? (c) If a is invertible to left and it isn't ...

I'm studying Linear Algebra and I read that : "Every Positive Definite Matrix is Symmetric BUT the vice versa is not correct..." in the text it suggests using 'x= u +iv' and then prove it by calculating (x,Ax). how can I prove it this way ? Thank you.

If I’m given $0=sec(x)tan(x)-csc(x)cot(x)$ in the domain $(0,\frac{π}{2})$, how would I algebraically find the value of $x$? I know the answer is $\frac{π}{4}$ because that’s where sine and cosine are equal, but algebraically, how would I properly find the answer?

How can Fuzzy logic be derived from propositional many-valued logic (as a special case)?

How to solve the following equation? y' + y^2 - 2ysinx + sint^2x = cosx It is necessary to determine the type and total solution. Help me please.

Let $U$, $V$, $W$ gaussian random variables $N(0,1)$ and $Z=\frac{U+VW}{\sqrt{1+W^2}}$. How can i find the conditional distribution of $Z$ knowing $W$ ?

Hey I was wondering if it's possible to solve (a+b)^1/2. To give an example, (a+b)^2 is a^2 + b^2 + 2ab. But what is (a+b)^1/2? I have learned about binomial theorem and I still can figur it out. thanks in advance

g(x,y) = g(x-y+1,y) + g(x-y+1,y-1) I have dealt with equations where x and y don't interfere with each other like g(x,y) = g(x-1,y-1) + g(x-1,y) But here the x-y+1 term has been giving problems for determining a matrix for the recurrence.

Let $u:[0,T]->R$ and $g:R->R$ positive,bounded and continuous functions. Let $u'(t,x)+(1/2)u''(t,x)-u^{1+\alpha}(t,x)=0,(t,x) \in [0,T]xR$ with u(.,T)=g Write u as a backward stochastic differential equation. In the correction, there is : With the Feynman-Kac formula, $u(t,x)=Y_t$ $Y_r=g(x+B_...

Evaluate ∫ ∫ F.ds. Where F= 59/3 x^3 i + 59/3 y^3 j + 59/3 z^3 k, and S is the surface: S= {(x,y,z) | x^2 + y^2 +z^2 =9} Please explain in detail how to get the answer.

Just a practice question, however just wondering if this ND proof is correct? I have put brackets in 2.2 and not in 2.3 however this shouldn't make a difference?

$X_{1}, ... , X_{n}$ $i.i.d$ with density function $$ f(x|Θ) = e^{-(x-Θ)}, x \ge Θ$$ Method of moment estimate $\hat{Θ_{MME}}$ of Θ : $$\hat Θ = \bar{X}-1$$ How can I find the distribution of MME ? I found MLE, distribution of MLE, MME but I can't find distribution of MME. Please answer..

If space $X$ deformation retracts to a point $x\in X$, then for each open $U\in X$ containing $x$ there exists an open $V\in U$ again containing $x$ s.t. inclusion of $V$ into $U$ is nullhomotopic. My attempt: Since $X$ deformation retracts to a point $x$, there is a corresponding map $F:X\times...

This is mybe a stupid quastion, but I want to the roots of: $$2(x+2)(x-1)^3-3(x-1)^2(x+2)^2=0$$ What that I did: $$\underbrace{2(x+2)(x-1)(x-1)(x-1)}_{A}-\underbrace{3(x-1)(x-1)(x+2)(x+2)}_{B}=0$$ So the roots are when $A$ and $B$ are both zeros when $x=1$ and $x=-2$ My questions: $1)$ ...

How is this possible? $$\int_{0}^{\infty}xe^{\theta-x}dx=\theta+1$$ I even used a program online and the answer is $e^{\theta},$ could this be a mistake?

sup all i need a method to find all subgroups from any finite group for example the subgroup of d4 (ORDER 4 ) {(1),(1.2.3.4),(1,3)(2.4),(1.4.3.2.)} okay i can understand this but the others .. how we can get them ?? there is any theorem or somethin?

Let $f:(a,b) \to \mathbb R$ be a continuous function such that $|f|$ is differentiable in $(a,b)$ ; then is $f$ differentiable in $(a,b)$ ?

The basic question is, If I have y as function of x (y=x^2 for example), How do I get x as function of y(x=sqrt(y) for example)? I have that problem but with two equations: y=wa^2+wb^2 z=1-y=1-(wa^2+wb^2) I want wa as function of y and z, and wb as function of y and z. But I don't know how to...

i am trying to find a regular language that: her minimum pumping length = number states of minimum Nondeterministic automata = number of equivalence classes. (not 1) any ideas? Thanks in advance

The function $f(x)$ is defined by $f(x)=\frac{x^2+2x}{x^2-1}$. How would you show that $f(x)$ is a strictly decreasing function.

on our lesson at our university, our professsor told that factorial has thie estimates $n^{\frac{n}{2}} \le n! \le \left(\dfrac{n+1}{1}\right)^{n}$ and during proof he did this $(n!)^{2}=\underbrace{n\cdot(n-1)\dotsm 2\cdot 1}_{n!} \cdot \underbrace{n\cdot(n-1) \dotsm 2\cdot 1}_{n!}$ and then...

The plane $x+2y-z=4$ cuts the sphere $x^2+y^2+z^2-x+z-2=0$ in a circle of radius? I tried putting value of y from plane in sphere but then I get a $zx$ term. How to proceed?

I am having difficulties with calculating exponential of a Jordan block, I cannot understand the method, can please someone help me, I have an exam on Monday. 'J' is my Jordan matrix and 'P' is my eigen vector matrix, 'A' is my starting matrix that I obtained 'J' from, I will also give you the r...

If we assume z is a complex number can you compute the following complex integral $\int_{0}^{2\pi}sin^3(3e^{iz}+\frac{\pi}{4})dz$ I tried to change the variable putting $u=3e^{iz}+\frac{\pi}{4}$, but i couldnt proceed. Thanks for your help.

Hi all the following might be a silly question it is well known that some statements like for example CH are not provable within ZFC (assuming consistency of course) ie. $ZFC\not\vdash CH$. However, CH is not so bad in the sense that at least its unprovability is provable ie. $ZFC\vdash \ulcorne...

Consider system to be: $\bar y= W\bar h+ \bar v$ where, $\bar v$ is Gaussian random vector with mean zero and covariance $R_{v}$ So, from $\bar y $ and the estimated vector $\hat y$, error is determined and quantized to have a vector $e$ of +1 and -1. I wish to determine $p(\bar b / \bar h )$...

I created a simple diagram to solve ordinary differential equation as shown below. Simple ODE I was trying to compute the result of xf_dot manually in Ms Excel but I did not get the same answer with the solution from Matlab. The xl_dot is a table of time-value e.g ([0 30; 1 27; 2 24; . . .])...

I'm following the first lecture of Daniel Spielman's lecture notes on Spectral Graph Theory (http://www.cs.yale.edu/homes/spielman/eigs/lect1.pdf). It begins with the following example of a graph G=(V,E): (1) --- (2) --- (3) Where V={1,2,3} and E={(1,2),(2,3)}. The author says that he compute...

Consider $$ \frac{dx}{dt}=\sqrt{x^2+1}+t^2,\qquad x\in\mathbb{R}. $$ Then I do not see why it is $$ \lvert\frac{dx}{dt}\rvert\leq 2\lvert x\rvert+t^2. $$

Let $f(z)$ and $g(z)$ be analytic on some domain. Show that if $Re(f(z)) = Re(g(z))$ then $f(z)-g(z)$ is constant.

Does there exists a function $f : I \to \mathbb{R}$ defined on an interval $I \subseteq \mathbb{R}$ that is measurable but nowhere locally integrable, i.e. not integrable on any compact subinterval $[a,b] \subseteq I$? One can try to call for Lusin's continuity theorem: If $m$ denotes the Lebesg...

I had the following prove by induction problem in an exam and I didn't do it because I didn't know how to. Could anyone solve it, please? $F(0) = 0$ $F(1) = 1$ $F(n) = F(n-1) - F(n-2)$ $F(n) <= (\frac{1+\sqrt{5}}{2})^n$ Thank you

If D be the unit disk then what could be the solution of a problem Δu=0 in D,∂u/∂n=cos3θ on ∂D I have tried using the polar co-ordinates representation to solve this but the boundary seems to be contradicting.

I have the following linear regression model $$y = \beta_0 + \beta_1 \cdot 40$$ and I would like to calculate the expected value and the variance, but I am not sure how to do it, even having the rules for computing the expectation and variance. From my slides $$E[y] = 51.7590$$and $$Var[y] = 18...

I need to proof an inequality , if $0<p\leq 1$ and $0\leq z \leq 1$, then $1-z^p\leq \left(1-z\right)^p$ Can be see Holder continuity of power function

I am very interested in mathematics, however, finding nowhere near wanted information in school sometimes I go and learn something by myself. Just like this time. I decided to learn more about logarithms as they always picked my interest, even thought it's a year or two more advanced than I shoul...

Prove that vector x(t)=ti+(1+t/t)j+(1-t^2/t)k lies in a curve. I am puzzled. Don't know how to approach it.

Why $\lim_{x\to a}\{\lim_{y\to b}\f(x,y)}$ \neq $\lim_{y\to b}{\lim{x\to a}\f(x,y)}\$?

If an $nxn$ matrix $B$ has rank 1, and A is another $nxn$ matrix, then why does $AB$ also have rank 1? This showed up in a solution that I read through, but it doesn't seem like an obvious fact. And one more thing that came up in this solution: it says that since this matrix has rank 1, then i...

We consider this topological space $(\mathbb{R},\tau)$ where: $$\tau=\{G\subset \mathbb{R}, (\mathbb{R}\setminus G)~\text{countable}~\}\cup\{0\}$$ and we consider the identity map $$f: (\mathbb{R},\tau)\rightarrow (\mathbb{R},|.|)\\~~~ x~~~\mapsto ~~~ x$$ How to prove that $f$ is continuous usi...

For $f: \mathbb{R} \rightarrow \mathbb{R}$ , the following holds: $\forall x,y \in \mathbb{R} : f(x+y) = f(x)\cdot f(y)$ $\forall y : \lim_{x\to y}f(x) = f(y)$ $f$ is not identically $0$ Find the form of the function (rational/exponential/logarithmic/other...) I was thinking this: if $f(x) = y$ f...

I was working on a physics problem that I faced with this Integral : $$\int sin(e^x) \, dx =$$ I tried a lot to solve it but I could not.

Let $A$ - unital banach algebra, $a,b \in A$, I want to proof that $\sigma (a+b) \subset \sigma(a) + \sigma(b)$. How can I do that?

I have been asked to prove the below function satisfies heat equation. u(r,t)=2Uπ∑n=1∞ {(−1)^n+1}(1/r)sin(nπr)e^−n^2π^2t I have tried to equate ∂u/∂t = ∂^2u/∂x^2 however both let hand side and right hand side are not tallying. Can you please help. Is this the correct function?

Given two points and angles, how can I calculate the connection between the two points in the form of a straight line and a circular arc? Context of the question: I am trying to make a program that converts a model railway layout created in XtrkCAD into a SCARM layout (scarm.info). Both design a...

Question If $$P \lor Q$$ and $$\neg P \lor \neg Q$$ are both true, do we get a contradiction? My Attempt Since $$\left\{\neg P \lor \neg Q\right\} \Longleftrightarrow \neg \left\{P \land Q\right\}$$ the question essentially asks whether $$P \lor Q$$ is logically equivalent to $$P \land Q$$ wh...

I've been thinking about this. My the reasoning seems to imply that: If two linear maps $f,g$ disagree somewhere, they must disagree everywhere but $0$. This feels like it can't be right, but I can't see a fault in the reasoning: Say $f,g$ linear on $V$ disagree at $x_0$. For any $x\neq...

$f$ is defined on the neighborhood of $x=0$, $\lim_{x\to 0}\frac{f(x)}{x} = 3$. I need to prove that $\lim_{x \to 0}\frac{f(3x)}{\ln(1+4x)} = 2.25$. I'm kinda stuck. I was thinking:If I define $t = 4x$ then $\lim_{x \to 0}\frac{f(3x)}{\ln(1+4x)} = \lim_{t \to 0}\frac{\frac{f(3 \cdot \frac{t}{4})}...

Suppose we are in $R^n$. Mark $v_1 = 1^n$, the all $1$'s vector. Set $v_2(i) = n+1-2i$. How do I find a vector that is perpendicular to both of them?

I'm attending a 3d-graphics course and I want to figure out which homograpic transformations conserve a circle's equation. The circle's equation is given as: Circle = x^2 + y^2 + Ax + By + C = 0 So in an effort to find out some transformations, I though about creating a matrix H={h1,h2,h3;h4,...

Can someone help me with this puzzle problem: James walked for two days. On the second day he walked 2 hours longer and with a average speed of 1 km/h faster then he did on the first day. On the second day he walked 64 km in 18 hours. What was his average speed on the first day?

Integrate the following function wrt dx \int 1/(1+x^7)

I was working on a math problem that I faces with this : R→Rf:R→R $f (x) = x + x^3$ then what's $f′(x)$

I'm studying the difference between pointwise and uniform convergence for sequences of continuous functions. Is it true that if $f_{n} \to f$ pointwise, then $ \|f_{n}\|_{\infty} \to \|f\|_{\infty}$? It seems true when they converge uniformly, but I can't find a counterexample if the convergenc...

So D.E are pretty new to me but i have made some linear ones but i cant seem to get how to solve a non-linear one. (2x+3y+4)dx+(3x+4y+5)dy=0

12 chess players took part in a tournament. Each played against each other exactly once. After the tournament did every chess player 12 lists of names. On the first list, the player only wrote his own name. On the second list, they wrote their own names as well as all man they had won against. Th...

Given Graph G How can I prove that each cycle in G has a minimum length of 5?

I need to know the way and how to calculate the lim (n->infinity) of (2^n + (-1)^n)/(2^(n+1) + (-1)^(n+1)) but i'm clueless...

I am very much thankful to you if you can help me to find summation of following series. \sum_{i=1}^{N-2} \sum_{j=i+1}^{N-1} \sum_{k=j+1}^{N} [\sin (ix-jx) + \sin (jx-kx) + \sin (kx-ix)]

Can someone please explain how to calculate "percent more than" and "percent less than"? I know 35 is 75 percent more than 20 - but no idea how to calculate it. Also trying to figure out how to find percent less than for: 120 is what percent less than 200? Thank you!

I got a question I think is fairly easy to answer but I cant get my head around it: cos(45 deg) = 0.7071... This is the point on the x-axis. How do I go from a value like 0.7071... to the amount of degrees?

I was studying about functions that I faced with this $$f:R→R f(x)=x+x^3$$ then what's $f^-1(x)?$ I tried a lot but I could not solve it

Is the finite Cartesian Product of Countable sets countable? If possible, could you give a bijection that would work for any example? Any help would be appreciated.

In how many ways can you assign $\:\eta\:\:$different integer tasks $\left(\:\eta<\infty\:\right)\:$to $\:k\:\:$employees $\left(\:k<\eta\:\right)$ if the toughest task must belong to your best employee, whilst the simplest task shall be executed by the only 2016 newcomer of your firm? $\\$ ...

Integral of: this, sorry gentlemen I'm still unaware how to write formulas , ( è - é)

I am trying to show that if the two partials of $f:\mathbb{R}^2\to\mathbb{R}$ exist at point, it is enough for only one to be continuous there to imply that $f$ is differentiable there. To show this I have decided to prove that: $f:\mathbb{R}^2\to \mathbb{R}$ not differentiable at $(a,b)$. If...

A few days ago, I came across this question in a review queue. I tried my luck at it. Here is what I did: If I want a homomorphism (isomorphism, but even just homomorphism) $f:\mathbb{R}\to F$, then I'll need 0 and 1 to map to themselves. From this, via homomorphism properties, I conclude $\mat...

I am looking for the definition Tangent cohomology. I found some different definitions in text books

Suppose $f: [a,b] \rightarrow \mathbb{R}$ is Riemann intergrable on $[a+\epsilon, b]$ for all $0<\epsilon<b-a$. Then $f$ is Riemann integrable on $[a,b]$?

In Hodges' A Shorter Model Theory, exercise 2.7.1 tells you to prove theorem 2.7.1, which says that the following five formulas are an elimination $\Phi$ set for the class of all dense linear orderings (with a signature consisting only of "$<$"): There is a first element There is a last element...

I'm preparing for my final, and i'm a bit confused with kernel,range...Please check my conclusion about the topic can you help me? basis of matrix = number of pivots after r.r.e.f. (When i'm finding basis, should I add one more column with zeros into the matrix?) Dimension = number of columns ...

Solve the differential equation; $(xdx+ydy)=x(xdy-ydx)$ L.H.S. can be written as $\frac{d(x^2+y^2)}{2}$ but what should be done for R.H.S.?

I came across this sentence "...let $\varepsilon: GG^\vee \to Id$ be the counit of adjunction and $Z$ its cone." I thought that cones were constructions on functors. $\varepsilon$, though, is a morphism of functors (a natural transformation)... in this context, what does "its cone" mean? Or is...

Does someone see a pattern?: 1 11 21 1211 111221 312211 13112221 1113213211 31131211131221 13211311123113112211 111312211331121321113212221

A is the convex subset of R^n. Proof, that Pi(A, a) group is trivial : Pi(A, a) = 0, where a is fixed point from A. Thanks.

Show that if $\|Ax\|=\|A^{*}x\|$ for all $x$ beloging to the finite-dimensional inner product space $X$, the linar transformation $A$ is normal.

Prove that $\log x<\sqrt{x}$ for $x\geq 1$ Let $f(x)=\sqrt{x}- \log x$. So, $f(1)=1>0$. $f'(x)=\frac{1}{2\sqrt{x}}-\frac{1}{x}>0$ only when $x>4$. When I draw the graph of $f$ in Wolframalpha, it shows the result, but how do I prove it rigorously? Can someone please help?

I am utilizing the all_node_cuts function to get all collections of two nodes that disconnect the graph if removed (link below). https://networkx.github.io/documentation/latest/reference/generated/networkx.algorithms.connectivity.kcutsets.all_node_cuts.html Here is how the undirected graph look...

T: R^2 ---> R^2 T(x,y)=(x^2,y) Is it Linear Transformation or not?

Is $p(x) = x^4+2x-2$ reducible in any of the following rings: $\mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{F_9}$ ? How do you show this for all the individual rings?

=dollar(if(s11>0;(s11*e11)+1;e11-k9&" more to buy")) I'm trying to display currency format for a number and text only when outcome is false: ??It brings back some error?? I'm using OpenOffice v4.1.1 You can put the formula as =dollar(if(s11>0;(s11*e11)+1;e11-k9))&"enter". It will display "ent...

How can I convert a Trapezoidal membership function to a triangular membership function??

D is a point on the circle that passes through the point A, B, C. In triangle ABC, AB=4, BC=1, ∠ABC=120.The maximum area of ABCD can be written as 'a*sqrt(c)/b' , find the value of a+b-c?

How do I evaluate the following integral $$ \int_{-1}^1 \int_{arccos(y)}^{\pi} {sin(x)\sqrt {1+{sin^2{x}}}}{dxdy}$$ I attempted to change the limits for the integrals $$ \int_{0}^{\pi} \int_{cos(x)}^{1} {sin(x)\sqrt {1+{sin^2{x}}}}{dydx}$$ which I can integrate in respect to y giving $$ \int_{0...

I have a problem with this integral. I don't what kind of substitution use to solve it. Please help. int t*sqrt{(1-9t^4)}

$\mu_E$ is a $4 \times 1$ vector composed of known constants, and $\mu$ is a vector of the same dimension but with unknown variables. Let us say $\mu = (x_1, x_2, x_3, x_4)^T$. What is the meaning of the following notation in optimization? minimize $|\mu_E - \mu|_2 $ subject to some constrain...

Is it true that (Zn,*), integers modulo n under moltiplication, is a group if and only if n is prime? If it's true, why? How can I prove it?

https://proofwiki.org/wiki/Determinant_of_Matrix_Product I found this proof (Proof 2).Could you tell me two things? 1: How do I know that if a matrix is invertible then it is a product of elementary matrices? 2: How do I see that if $A$ is not invertible then neither is $AB$?

I want to reduce $\overline{K}$ to the set of natural numbers such that the program codified by them computes some partial and surjective function, in order to prove that such set is not semi-decidable. Formalizing, we have: $$\overline{K} ≤ \{ p ∣ φ(p)\ is\ partial\ and\ surjective \}$$ Howe...

When an Emergent finite automata learns from a symbolic finite automata. Only patterns are used for state symbols. (True or False??) I'm confused if this is true or false? I want an explanation why as well? Thank you

show that if traveling salesman problem decision problems are Np-complete then integer linear programming decision problems are also NP-complete.

i have this simple question : i want to change the variable here and put $u=nx$ then what wil be equal the integral (here we are in $\mathbb{R}^n$ $\int_{\mathbb{R}^n}|f(nx)|^qdx$ ($q$ real >1) thank you very much.

In my textbook the following is proposed as a theorem (with some explanation but without proof), but it seems wrong to me. Maybe it's an easily-fixable typo; I don't know. I will quote the (so-called) theorem and then point out its error. You may want to inform me if I'm wrong or mistaken (I'm pr...

Let $\mathbb{F}_q$ be the finite field with $p^n$ elements and consider the trace map $$\mbox{Tr}: \mathbb{F}_q\to \mathbb{F}_p,$$ where $$\mbox{Tr}(\alpha)=\alpha+\alpha^p+\alpha^{p^2}+\cdots +\alpha^{p^{n-1}}.$$ If $\varphi \in \mbox{Gal}(\mathbb{F}_q/\mathbb{F}_p)$, then $\mbox{Tr}\big( \varp...

$A=\begin{bmatrix} sin (\pi/18) & -sin (4\pi/9) \\ sin (4\pi/9) &sin (\pi/18)\end{bmatrix} $ find number $n\in N$ such that $A^n=I$. I found eigen values and eigen vectors and use it to find the value of n but it is lengthy , is there another way of solving it.

The question is this: Let $f:\mathbb{R}\to\mathbb{R}$ be differentiable at $x=0$ and suppose that there is a number $L$ such that $$\lim_{x\rightarrow0}\frac{f(x)-f(x/2)}{x/2}=L.$$ Prove that $f'(0)=L$. Here's my answer with all theorems referenced being from Rudin: Let $a_n$ be a positive se...

f(x)= \begin{array}{cc} 2 & if0<x<2\\ -2 & if-2<x<0 \end{array} I found that all coefficient (a0,an,bn) is it correct?

Decide whether (Z × Z, +, ∗) is a ring. Operation + is defined by the following formula: (a, b) + (c, d) = (a + c, b + d). Operation ∗ is defined by: (a, b) = (ac + bd, ad + bc).

There are well-understood theorems that give sufficient conditions for a functor $R: D\to C$ to have a left adjoint. For example, $R$ should preserve limits and $D$ should have nice categorical properties. But there are situations in which $R$ does not preserve limits, yet it lifts to a right a...

Question: Is there a line in the xy plane that has all rational coordinates. Prove your answer. Idea: There is most certainly not. I believe it can be shown that between any 2 rational points that there is at least one rational coordinate. Therefore, there can not be a line that contains only r...

Need to select one winner from each of three matches that each have three options. Game One: Home or Visitor or Tie Game Two: Home or Visitor or Tie Game Three: Home or Visitor or Tie How many different combinations are there? Appreciate any help!

If two points P1 and P2 observed in a plan are each associated with a known covariance matrix, is it possible to infer the covariance matrix of any point on the segment [p1,p2]?