## Backpropagation

Backpropagation is a generalization of the Delta rule to be able to take into account non-linear functions. Suppose a differentiable function of the input vector $$y_{k}^{p}=f(s_{k}^{p})$$ where $$k$$ represents the $$k$$-th unit of the network and $$p$$ the $$p$$-th pattern. $$f (s_ {k} ^ {p})$$ is called activation function, and depends on a linear combination Read More …

## Calculation of the golden ratio with Maxima

Euclid defined the golden section or golden ratio in the following way: Split a segment into two unequal parts such that the ratio between the total length of the segment and the major division is equal to the ratio between the major division and the minor division. Therefore: |————————-|—————–| <———– x ———-><——– y ——> <———————- a ——————> $$\frac{a}{x}=\frac{x}{y}$$ This ratio represents Read More …

## Time independent Schrödinger equation

The Schrödinger equation $$-\frac{\hbar}{i}\frac{\partial\Psi(x,t)}{\partial t}=-\frac{\hbar}{2m}\frac{\partial^{2}\Psi(x,t)}{\partial x^{2}}+V(x,t)\Psi(x,t)$$ involves a state function ($$\Psi$$ wave function) which depends on spatial coordinates and time. For simplicity, let us consider the wave function for a particle in the one-dimensional case. Fortunately, for many applications of chemistry the time-independent Schrödinger equation, which gives rise to stationary states, is sufficient. In this Read More …

## Degeneration in quantum mechanics

Degeneration occurs when two or more independent wave functions have the same eigenvalue. It is said that $$n$$ functions $$f_ {1}, f_ {2,} \ldots, f_ {n}$$ are linearly independent if the condition $$\sum_{i}c_{i}f_{i}=0$$ it is satisfied when all the constants $$c_ {i}$$ are equal to zero. The degree of degeneration of a system is Read More …

## The complexity of Sorting a Binary Tree

Introduction A binary tree is a data structure, that is, a way of storing information in an organized form. Knuth defines them as follows: A binary tree is a finite set of nodes that either is empty or consists of a root and two disjoint binary trees called the left and the right subtrees of Read More …