Science, Artificial Intelligence, and Innovation
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 …
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 …
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 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 …