I'm working on a method to calculate weights of a non-linear 2 layer neural network in 1 step, instead of working with the propagation algorithm. I have chosen to make the non-linearity a sigmoid function. And I already made some progress. $$ Loss = (pred-y)^2\\ pred = A(A(X\cdot B_1)\cdot B_1)\\ A(x)=\frac{1}{1+e^{-x}}\\ A^{-1}(x)=ln(\frac{1-x}{x})\\ L(B_1,B_2)=(A(A(X\cdot B_2)\cdot B_1) - y)^2\\ L(B_1,B_2)=(A(X\cdot B_2)\cdot B_1 - A^{-1}(y))^2\\ $$ A^{-1}(x) is the inverse function of the sigmoid. $$ L(B_1,B_2)=(A(X\cdot B_2)\cdot B_1 - A^{-1}(y))^2=\\ (A(X\cdot B_2)\cdot B_1 - A^{-1}(y))^T ..read more

In the response to a related question here is given a formula to compute one of the angles not given for an SAS triangle: $$tan(A) = \frac{sin(C)}{\frac{b}{a}-cos(C)}$$ where $a,b,C$ are the given sides and angle, and $A$ is the angle opposite the smaller of the two sides, $a$. (By selecting the smaller angle, the principle value given from $tan^{-1}$ will be correct.) Is there a name for this formula? If not, it could be that this is less preferred over the law of tangents (also referenced there as a method to compute an angle from the given sides and angle) since it will suffer numerical iss ..read more

So I've been working at this for a while, but I haven't made any significant progress. Can someone try and prove that, for integer values of n>1, that (n!+1)/(1+n^2) is irreducable ..read more

Given two stochastic variables $X, Z$ with the same Wiener increment $dW$, I would like to solve the following set of Itô equations: \begin{align} &dX=-\gamma X dt + \frac{g}{1+\kappa Z^2} dW\\ &dZ=-\kappa Z dt + g dW\\ \end{align} Specifically, I am only interested in the asymptotic value of the variance ${E}[x^2(t\rightarrow\infty)]$, as the 1st moments vanish ${E}[x]={E}[z]=0$. My plan for the variance is to get the Itô SDE just by using $d (x^2)=(dx)x+x(dx)+(dx)(dx)$. However, something needs to be done with my non-linear coupling. What it tried is the following: Define $Y=\frac{1 ..read more

i got hint from my teacher that using characteristic eqn we always atleast get one root zero for higher order matrices then the value of detBA=0 and tr(BA)=10 so ans is 10 but i am not getting the later part about detBA becoming zero and one root mandatorily becoming zero ..read more

I am working with Lie groups and Lie algebras and have some trouble with proving something that I think is right. Let $G$ be a (simply connected) Lie group. Let $\mathfrak g = T_e G$ be its associated Lie algebra. Assume that we have two Lie group endomorphisms $\theta: G\to G$ and $\zeta: G\to G$ such that $\text{Im} \zeta \leq Z(G)$. Then the product $\theta\cdot \zeta$ is a well-defined Lie group morphism. Now I want to get the following result: $$(\theta\cdot\zeta)^\ast = \theta^\ast+\zeta^\ast.$$ Here the $\ast$ in the superscript denotes the induced Lie algebra morphisms on $\mathfrak g ..read more

$X,Y$ are two geometric random variables with parameter $p=\frac{(1+\lambda)}{\lambda}$. $(X,Y)^T$ is a random variable vector. My task is to find $$\mathbb{E}[(X,Y)^T]$$ and $$\mathbb{C}ov[(X,Y)^T].$$ The expected Value of $(X,Y)^T$ was no problem $$\mathbb{E}[(X,Y)^T]=\begin{bmatrix} \mathbb{E}[X] \\ \mathbb{E}[Y]\\ \end{bmatrix}=\begin{bmatrix} \frac{\lambda}{(1+\lambda)} \\ \frac{\lambda}{(1+\lambda)} \\ \end{bmatrix} $$ but i got problems with the Covariancematrice. So far I got $$ \mathbb{C}ov[(X,Y)^T]=\mathbb{E}\begin{bmatrix} \begin{pmatrix} X-\mathbb{E}\\ Y-\mathbb{E}\\ \end{pmatrix ..read more

For each $n\in\mathbb{N},$ let $S_n$ be the set of prime factors of $n! + 1$. By Wilson's theorem, we have $\ p\mid (p-1)!+1\ $ for every prime $p.$ Therefore, $\displaystyle\bigcup_{n=1}^{\infty} S_n \supset \bigcup_{n \text{ is prime }} S_{n-1} = \mathbb{P},\ $ the set of primes. Trivially, this implies that $\displaystyle\bigcup_{n=1}^{\infty} S_n = \mathbb{P}.$ A natural follow-up question is: For each $n\in\mathbb{N},$ let $T_n$ be the set of prime factors of $n! - 1.\ $ Is $\displaystyle\bigcup_{n=2}^{\infty} T_n = \mathbb{P}\setminus\{2,3\} ?$ Stated alternatively, does there exist a ..read more

A group $G$ is metacyclic if has the following exact sequence $$1 \to N \to G \to K \to 1$$ where $N$ and $K$ are cyclic groups. In the split extensions, the wikipedia says that direct and semidirect product of metacyclic groups is metacyclic. What about non split extensions? Is it true? I mean, if $N$ and/or $K$ are/is metacyclic, then is also $G ..read more

I'm trying to solve: $$\sqrt[3]{x^2} - \sqrt[3]{x} -6 = 0$$ I’ve tried putting the $-6$ on the other side and cubing both sides but no joy at finding value $x ..read more