Topological quantum field theory TQFT in physics uses the zero-energy sector of the
Hilbert space of states that does not include time, which the Hamiltonian kills. Mathematically,
TQFT is used as an organizing structure for topological or differential invariant manifolds.
A \(D\)-dimensional TQFT as a symmetric monoidal factor is given as:
\(Z: \left(\mathscr{B}ord_D^{o r}, \amalg \right) \rightarrow(\text {V}ect, \otimes)\)
\(\mathscr{B}ord_D^{o r}\) is a category of compact objects, boundary-less oriented manifolds...
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
In biology, epigenetics is the study of mitotically and/or meiotically heritable changes in gene function that cannot be explained by changes to the DNA sequence. Epigenetics normally involves change that is not erased by cell division and that also affects the regulation of gene expression. Epigenetics reflects our understanding... read more
Quantum Principal Component Analysis identifies large eigenvalues of unknown density matrices utilizing corresponding eigenvectors in \(O(\log d)\). Where principal component analysis analyzes positive semi-definite Hermitian matrices by decomposing eigenvectors in relation to the largest eigenvalues in the matrix for dimensionality reduction. Improved computational complexity will hopefully allow new methods for... read more
In theoretical physics, the Maldacena Conjecture states supergravity and string theory on the product of \((n+1)\)-dimensional Anti-de Sitter space with a compact manifold capable of describing large \(N\) limits of conformal field theories in \(d\)-dimensions. Correlation functions in CFT are dependent on the supergravity action of asymptotic behavior at infinity... read more
Bioinformatics is a growing revolution in the field of molecular biology and computers. Here, our emphasis will be on employing bioinformatics tools and biological databases to address challenges from current issues in biological, biotechnological, and biomedical research. Such as looking at computational algorithms and computer databases to analyze proteins, genes... read more
In neuroscience, the central executive is responsible for controlled processing and allocation of data to subsystems in working memory. Such subsystems include the visuospatial sketchpad and phonological loop, where the phonological loop can further be subdivided into the phonological store and the articulatory process. The primary functions of the central... read more
In theoretical physics, Minkowski Space is a particular type of \(4\)-dimensional Lorentzian space, with a Minkowski metric. Where the Minkowski metric is a metric tensor denoted as \(d\tau^2\) with the form \(-\left(d^0\right)^2+\left(d x^1\right)^2\) \(+\;\left(d x^2\right)^2+\left(d x^3\right)^2\). Minkowski space forms the basis of the study of spacetime within special relativity and is... read more
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
In biological intelliegent systems there are multiple mechanisms working in congruence on multiple levels, both at the structural and neurobiological level to develop complex cognitive abilities. What remains unknown is which mechanisms are necessary and sufficent to synthetically replicate these cognitive abilities for artificial intelligence. A neurocomputational model is offered... read more
Virtue Ethics is a branch of one of three major approaches to normative ethics, where normative ethics, at the risk of oversimplification, is concerned with criteria for what is right and wrong. The three main philosophical ideologies concerning normative ethics are the following. Virtue ethics, which can be identified from... read more
The hippocampus is distinguished externally as a layer of densely packed neurons that form a S-shaped structure and extends to the temporal lobe of the cerebral cortex. It is also a sub-cortical structure in the limbic lobe, and contains two parts: cornu ammonis and dentate gyrus, where the hippocampal sulcus separates both parts. The parts curve into each other and below the sulcus lies the subiculum. Since the hippocampus is a part of the allocortex or archicortex, there exists a zone that separates the... read more
From the human structural connectome, the authors attempt to extract architectural feautres using diffusion spectrum imaging DSI and encode the data required into triplcate as undirected, weighted network. They do so to capute as much information about possible paths which transmit as human process and preform complex behaviors... read more
Chern classes are a part of algebraic topology, as well as other math groups, and are characteristic classes related to complex vector bundles. Let \(X\) be a topological space of closure-finite weak CW complex and let \(V\) be a line bundle. The first chern class is the only nontrivial Chern class and is an element of the second cohomology group of \(X\).... read more
One main theoretical framework that is used to model, estimate, and simulate brain networks from complex network science is graph theory. A graph being a composition of a set of intereconnected elements know as vertices and edges. The vertices in a network can represent brain areas, while edges can represent... read more
Three properties were required by Shannon: \(I(p) \geq 0\), i.e. information is a real non-negative measure. \(I(p_{1},p_{2})=I(p_{1})+I(p_{2})\) for independent events. \(I(p)\) is a continous function of \(p\). The mathematical function that satisfies these requirements is: \(I(p)=k\;log(p)\) In the equation, the value of \(k\) is arbitrary... read more
Abstract Algebra or modern algebra can be defined as the theory of algebraic structures. For the most part, abstract algebra deals with four algebraic structures: groups, rings, fields, and vector spaces. We will look at and examine these four algebraic strucutres in this page. The three most commonly studied algebraic... read more
Let us first start with a formal definition of a 2 vector convex combination. Then we will break down the definition into parts and analyze the definition. Then we will formally define and analyze a convex combination with a finite number of vectors in the same manner. A subset \(S \subseteq \mathbb{R}^{n}\)... read more
Quantum Computing Theory is a field of computer science that uses the principles of quantum mechanics, mathematics, and computer science. By borrowing concepts from each field scientists can rigorously define both a broad and narrow theoretical model of a quantum computer and later apply it to the real world. These... read more
First we will introduce the idea of Hilbert Space, which was named after D. Hilbert. Hilbert Space is a nondenumerable infinite complex vector space. Complex space, being a collection of complex numbers with an added structure. The infinite dimensions of Hilbert Space represents a continious spectra of alternative physical states... 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
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
\(G = (V, E)\) \(V\) is a set of vertices \(E \subseteq \left\{\left\{x, y\right\}\;|\;x, y \in V\;and\;x \neq y\right\}\) A simple undirected graph \(G\) is an ordered pair or tuple \((V, E)\) where \(V\) and \(E\) are finite sets. \(E \subseteq \left\{\left\{x, y\right\}\;|\;x, y \in V\;and\;x \neq y\right\}\)... read more
A deterministic finite automaton DFA is a 5-tuple: \((Q, \Sigma, \delta, q_{0}, F)\) where: \(Q\) is a finite set of states \(\Sigma\) is an alphabet \(\delta\) is a transition function described as \(\delta : Q \times \Sigma \rightarrow Q\) \(q_{0} \in Q\) is the initial state \(F \subseteq Q\) is a... read more
Algorithmic analysis is used to help computer scientists understand the resources required by an algorithm for time, storage, and other uses. Algorithmic anlysis must analyze algorithms in a methodical, universal, and fair way. To do this computer scientist implement mathematical models that describe the resources used by algorithms. This work... read more