# Michael Hardy

Minneapolis

After doing nearly all the coursework for a Ph.D. in math, I then did all the coursework for a Ph.D. in statistics and completed that degree.
 1h comment Were the US Articles of Confederation formally repealed?The constitution does explicitly mention the Articles of Confederation, in the part about the public debt in the first sentence in Article VI. 1h asked Federal courts under the Articles of Confederation 1h comment Shape of distribution between arrivals in a poisson processNeither the CDF nor the PDF has a vertical tangent. You shouldn't make it look as if they do. $\qquad$ 1h revised $\int \frac{\cos(x)}{(1+\cos(x))^3} \, dx$?added 1 character in body 1h revised Lower bound for the distance between matrices of different rank.edited body 2h revised Incidence correspondence as a schemeadded 4 characters in body 2h revised How can I write a set of equations in summation form?deleted 481 characters in body 2h revised Finding Type I erroradded 26 characters in body 2h revised Vector space $V$ , quadratic form $f :V\to R$ . Excercise on rad(F) and a new function.added 58 characters in body; edited title 2h revised Vector space $V$ , quadratic form $f :V\to R$ . Excercise on rad(F) and a new function.added 58 characters in body; edited title 2h comment $\int \frac{\cos(x)}{(1+\cos(x))^3} \, dx$?@Lasker : Yes. Where the question had $1+\cos x$ you wrote $1-\cos x$. $\qquad$ 2h comment Are $\lim_{h\to0}f(a+h)=f(a)$ and $\lim_{h\to0}f(x+h)=f(x)$ the same?@Dr.MV : Thank you. $\qquad$ 2h comment $\int \frac{\cos(x)}{(1+\cos(x))^3} \, dx$?@mavavilj : The thing by which you need to multiply $1+ \dfrac{1-t^2}{1+t^2}$ to clear out the fractions is $1+t^2$. Then $1$ becomes $1+t^2$ and the fraction $\dfrac{1-t^2}{1+t^2}$ becomes $1-t^2$. So you can write \begin{align} & \left( 1 + \frac{1-t^2}{1+t^2}\right)^3 (1+t^2)^3 \\ \\ = {} & \left( \left( 1+ \frac{1-t^2}{1+t^2} \right) (1+t^2) \right)^3 \\ \\ = {} & \Big( (1+t^2) + (1-t^2) \Big) \end{align}and that becomes just $2$. $\qquad$ 2h comment $\int \frac{\cos(x)}{(1+\cos(x))^3} \, dx$?$-1$. This is at best an overly complicated answer (not to mention the minus sign where a plus sign should be. $\qquad$ 2h answered $\int \frac{\cos(x)}{(1+\cos(x))^3} \, dx$? 2h reviewed Approve suggested edit on $\int \frac{\cos(x)}{(1+\cos(x))^3} \, dx$? 2h revised Let $f : \mathbb{R}^n \to \mathbb{R}^n$, suppose that $\mu = dx^1\wedge\ldots\wedge dx^n$, then $f^{\ast}\mu = \det (df)\mu$added 62 characters in body 2h comment Are $\lim_{h\to0}f(a+h)=f(a)$ and $\lim_{h\to0}f(x+h)=f(x)$ the same?@JeanMarie : I believe you're mistaken. There's no need for this to hold for all $x$ in some open interval in order for this to make sense. In my answer I said "in the domain". There's no need for the domain to contain an interval for that to make sense. $\qquad$ 2h comment Help with the Maximum Likelihood Estimator?Instead of $$log(x_1 x_2 .. x_i) = log(x_1) + log(x_2+..+ log(x_i),$$ you can write $$\log(x_1 x_2 \cdots x_i) = \log(x_1) + \log(x_2) + \cdots + \log(x_i),$$ coded as \log(x_1 x_2 \cdots x_i) = \log(x_1) + \log(x_2) + \cdots + \log(x_i) $\qquad$ 2h comment Help with the Maximum Likelihood Estimator?The way a $\prod$ becomes a $\sum$ is this: $\log(AB) = \log A + \log B$. $\qquad$