For PK and Lucy,
my best boy and best
girl. Whose constant warmth I could never repay, even with a lifetime of dog treats and scratches. I created
this site as a place to share what I enjoy learning and hope you enjoy it too!
Using quantum computing, the authors exploit quantum mechanics for the algorithmic complexity optimization of a Support Vector Machine with high-dimensional feature space. Where the high-dimensional classical data is mapped non-linearly to Hilbert Space and a hyperplane in quantum space is used to separate and label the data. By using the... read more
Molecular neuroscience is an area of chemical neuroscience that studies the molecular basis of intercellular activity applied to animals' nervous systems. This area of research covers molecular neuroanatomy, mechanisms of molecular signaling in the nervous system, and the molecular basis of neuroplasticity and neurodegenerative disease, which we will focus on... read more
An \(n\)-dimensional manifold is a topological space where each point has a neighborhood that is homeomorphic to an open subset on \(n\)-dimensional Euclidean space. Let \(M\) be a topological space. A chart in \(M\) consists of an open subset \(U \subset M\) and a homomorphism \(h\) of \(U\) onto an open subset of \(R^{m}\). A... read more
The input for the quantum convolution layer is a \(3\)D tensor input given as \(X^{\ell} \in \mathbb{R}^{H^{\ell} \times W^{\ell} \times D^{\ell}}\). The weights layer, filter layer, or \(4\)D tensor kernel layer is denoted as \(K^{\ell} \in \mathbb{R}^{H \times W \times D^{\ell} \times D^{\ell+1}}\). The input and kernel layer are both stored in QRAM. There are precision parameters \(\epsilon\) and \(\Delta>0\)... read more
The thalamic nuclei are paired structures of the thalamus divided into three main groups: the lateral nuclear, medial nuclear, and anterior nuclear groups. The internal medullary lamina, a Y-shaped structure that splits these groups, is present on each side of the thalamus. A midline, thin thalamic nuclei, adjacent to the... read more
Given \(AdS_{n+1}\) space of constant negative curvature, a Hyperboloid in \(n+2\) dimensional
flat spacetime with coordinates
\((X^0, X^1, \ldots,\) \( X^n, X^{n+1})\) and metric \(\eta_{a b}=\)
\(\operatorname{diag}(+,-,-,\ldots,\) \(-,+)\), we get the constant:
\(\Lambda^2=\) \((X_0)_2+(X_{n+1})^2-\sum_{i=1}^n(X_i)^2\)
Now, we introduce global coordinates in \(AdS\) \(\rho, \tau, \Omega_i\). Let \(X_0=\) \(\Lambda \sec \rho
\cos
\tau\), \(X_i=\) \(\Lambda \tan \rho\;\Omega_i\), and \(X_{n+1}=\) \(\Lambda \sec \rho \sin \tau\) where
\(\sum_{i=1}^n \Omega_i^2=\) \(1,0 \leq \rho < \pi / 2,0 \) \(\leq \tau<2 \pi\). Note, since the time variable
is \(\tau\) is compact, we... read more