Holliday, Pugh, and Riggs proposed the idea that DNA methylation was palindromic, acted as an epigenetic mark, and that distinct enzymes were responsible for methylation of both modified and unmodified DNA already methylnated on one strand. This idea was in response to Ohno et al. 1959; Lyon 1961, where X chromosome inactivation... read more

Entanglement renormalization is a numerical technique that locally reorganizes the Hilbert space of a quantum many-body system to reduce entanglement in its wave function. It addresses the computational challenges of real-space renormalization group RG methods, which struggle with the rapid growth of degrees of freedom during successive RG transformations. The key idea is... read more

Let \(\mathscr{N}\) denote noise, \(\mathscr{R}\) denote the recovery process, and \(L(H)\) denote the linear operator acting on Hilbert space \(H\). The recovery process is given as \(\mathscr{N}, \mathscr{R}: L(H) \rightarrow L(H)\). Where the mapping can be represented as an operator-sum

\(\mathscr{N}(\rho)=\sum_i E_i \rho E_i^{\dagger},\) \(\quad E_i \in L(H),\) \(\quad \mathscr{N}=\left\{E_i\right\}\)

Given the quantum code \(C_{Q}\), a subspace of \(H\), there exists a recovery... read more

The formula in the case of \(SU(2)\) is as follows. Choose a polynomial \(Q\). For each \(k \in \mathbb{Z}_{+}\), let \(\xi(k, t)\) be the formal power series in \(t\) that represents a unique critical point of the function in \(\xi\).

\(F(\xi ; h, k, t)\) \(:=\frac{1}{2}(\xi-k)^2+t \cdot \frac{h}{2 \pi^2} \cdot Q(\pi \xi / h)\)
where, \(t\) is treated as a formal variable. \(\xi(k,t)\) undergoes \(t\)-series expansion near the minimum \(\xi(k,0)=k\), formulated as:

\(\displaystyle{ \int_{F(\Sigma ; G)} \exp \left\{h \omega+t \cdot[\Sigma] \backslash Q\left(u^* \phi_2\right)\right\} }\) \(\displaystyle{ =\# Z(G) \cdot h^{3 g-3} \operatorname{vol}(G)^{2 g-2} }\) \(\displaystyle{ \sum_{k>0}\left[\frac{1+\frac{t}{2 h} Q^{\prime \prime}(\xi(k, t))}{\xi(t, k)^2}\right]^{g-1} }\)
such that, \(\#Z(G)=2\) for \(SU(2)\)... read more

A Riemannian manifold is a smooth manifold \(M\) with a Riemannian metric denoted \((M,g)\). For every point \(p\in M\) the tangent bundle of \(M\) assigns a vector space \(T_{p}M\), termed the tangent space of \(M\) at \(p\). The Riemanniam metric defines a positive-definite inner product:

\(g_p: T_p M \times T_p M \rightarrow \mathbb{R}\)
with a norm \(|\cdot|_p: T_p M \rightarrow \mathbb{R}\) defined as:

\(|v|_p=\sqrt{g_p(v, v)}\)
...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