## True of False inequality graphing questions(plug in ) February 14

Point (6,y) is a solution of the inequality 12y+x>0 for any value of y.I got false since y could be negative infinite and that plus 6 would be less than 0. Is that correct>Also, in - Point (a,−a) is a solution of the inequality 6x+y>0 for any val

## Existence of nuclear dominating positive definite kernel February 14

Let $\mathcal{X}$ be a metric space and $k: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$ a continuous positive definite kernel. Can we always find a positive definite kernel $r$ such that $r \gg k$ (i.e. $r$ nuclear dominates $k$ (Lukic & Beder 200

## Is Lebsegue Measure Translation Invariant February 14

I am trying to prove that the Lebsegue measure is translation-invariant. Namely, given a set $X$, I'd like to show $X + y$ is measurable and $\mathit{m}(X + y) = \mathit{m}(X)$. Namely, that the measures -- not the outter measures alone -- agree. I a

## Sequence of Partial Sums is Convergent February 14

How do I show that the series $\sum_{n=1}^{\infty} \frac1{(2n-1)^{n}} + \frac1{(2n)^{3 }}$ is convergent? I'm trying to use Comparison Test but I'm having a hard time looking for a convergent series that is "larger" than the given series.

## Find the joint probability density given the support set February 14

Suppose that the support set of $(X,Y)$ is $$S_{X,Y}=\{(x,y)\in\mathbb{R}^2: x \geq 0 \text{ and } 0 \leq y \leq e^{-x/3}\}$$$(X,Y)$ is uniformly distributed on $S_{X,Y}$.a) Find the joint probability density function for $(X,Y)$.b) Find the marginal

## How is the second derivitive derived February 14    2

As everyone knows that the derivitive of a function is $\frac{dy}{dx}$The question is: Why is the second derivitive: $$\frac{d^2y}{dx^2}$$If anyone is able to tell me how this second derivitive notation is derived, please do.This is just notation. Le

## How to find the center of mass in this problem February 14

How can I find the centre of mass of the surface of the sphere $x^2+y^2+z^2=a^2$ that is contained in the cone $x\tan(\gamma)=\sqrt{x^2+y^2}$, $0 \lt \gamma \lt$ $\pi/2$ a constant, where the density is proportional to distance from the z axis.I know

## Find all entire $f$ such that $f(f(z))=z$. February 14    3

Suppose $f:\mathbb{C}\to \mathbb{C}$ is entire. If $f(f(z))=z$, find all such $f$.Can we find $f$ such that $f(f(z))=z^2$?How about $f(f(z))=e^z$? Ideas: For #1, we can show that $f$ must be a bijection, since $f$ failing to be either injective or su

## find all value of z belong to C February 14

find all value of z belong to C such that $$e^z = -3i$$my try $$e^z=e^{x+iy}$$.$$-3i = r*e^(i*0)$$ take ln both sides $$z =ln(-3i)$$stuck here , need help to solve it please

## How many ways are there to arrange these letters February 14

So I've been working out how many ways there are to arrange the letters of probabilistic. I came up with 518918400 ways. The next thing I want to figure out is out of those ways, how many of them have all the b's always come before the i's. In other

## Find the original function by using convolution theorem February 14

Seems like I don't know how to apply convolution theorem on this problem properly, I would appreciate some help and a brief explanation how did you solve it if you do it. \fr ...

## Find the function satisfying the differential equation February 14

Find the function satisfying the differential equation $$f'(t)-f(t)=-7t f(3)=5$$For some reason I've never completely understood what f'(t) and f(t) or function notation very well. Is f'(t) the same as dt/dx in this situation with f(t) = x? anywho.

## Integral of $\frac{x}{\sqrt{1+x^5}}$ February 14    3

I am trying to calculate the following integral: $\displaystyle\int_0^\infty \frac{x}{\sqrt{1+x^5}}\, dx$But I can't seem to find a primitive for that function. I was trying to find a good substitution, but was unable to. Also, attempting to use part

## Integrating $\frac{1}{(ax^2+bx+c)^n}$ two ways February 14    1

Could someone please show me how to do the indefinite integral of$$\frac{1}{(ax^2+bx+c)^n}$$a) using real analysis (hard)b) using complex analysis (nice factoring)and show they give the same answer, without using any simplifiers into $1 + t^2$ or oth

## Finding the integral of the type $\frac{px+q}{ax^2 +bx + c}$ February 14    2

The textbook says, to find the integral of the type $\dfrac{px+q}{ax^2 +bx + c}$, where $p,q,a,b,c$ are constants, we are to find real numbers $A$ and $B$ such that $$px+q = A \dfrac{d}{dx} (ax^2 + bx + c) + B => A(2ax+b) + B.$$Now to determine $A$ a

## Problem with eigenvectors February 14

I am struggling with finding the eigenvectors corresponding to the eigenvalues:$\lambda_1 = 0.5[a + d + \sqrt{(a - d)^2 + 4bc}]$ $\lambda_2 = 0.5[(a + d) - \sqrt{(a - d)^2 + 4bc}]$

## Can there be a closed geodesic on surface with zero Gaussian curvature February 14    1

I was asked this in differential geometry class and it.is bugging me as i do not know the answer I know a surface of constant Gaussian curvature zero is locally isometric to a plane on which no geodesic is closed as they are lines but the fact that i

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