Functional Brain Networks

Functional brain networks refer to the interconnected regions of the brain that work together to perform specific cognitive functions. These networks are identified through techniques like functional magnetic resonance imaging (fMRI), which measures brain activity by detecting changes associated with blood flow. The brain operates as a complex system of nodes (brain regions) and edges (connections between regions), and various networks can be categorized based on their roles, such as the default mode network, which is active during rest and mind-wandering, or the executive control network, which is involved in higher-order cognitive processes. Understanding these networks is crucial for unraveling the neural basis of behaviors and disorders, as disruptions in functional connectivity can lead to various neurological and psychiatric conditions. Overall, functional brain networks provide a framework for studying how different parts of the brain collaborate to support our thoughts, emotions, and actions.

Other related terms

Wavelet Transform

The Wavelet Transform is a mathematical technique used to analyze and represent data in a way that captures both frequency and location information. Unlike the traditional Fourier Transform, which only provides frequency information, the Wavelet Transform decomposes a signal into components that can have localized time and frequency characteristics. This is achieved by applying a set of functions called wavelets, which are small oscillating waves that can be scaled and translated.

The transformation can be expressed mathematically as:

W(a,b)=f(t)ψa,b(t)dtW(a, b) = \int_{-\infty}^{\infty} f(t) \psi_{a,b}(t) dt

where W(a,b)W(a, b) represents the wavelet coefficients, f(t)f(t) is the original signal, and ψa,b(t)\psi_{a,b}(t) is the wavelet function adjusted by scale aa and translation bb. The resulting coefficients can be used for various applications, including signal compression, denoising, and feature extraction in fields such as image processing and financial data analysis.

Pole Placement Controller Design

Pole Placement Controller Design is a method used in control theory to place the poles of a closed-loop system at desired locations in the complex plane. This technique is particularly useful for designing state feedback controllers that ensure system stability and performance specifications, such as settling time and overshoot. The fundamental idea is to design a feedback gain matrix KK such that the eigenvalues of the closed-loop system matrix (ABK)(A - BK) are located at predetermined locations, which correspond to desired dynamic characteristics.

To apply this method, the system must be controllable, and the desired pole locations must be chosen based on the desired dynamics. Typically, this is done by solving the equation:

det(sI(ABK))=0\text{det}(sI - (A - BK)) = 0

where ss is the complex variable, II is the identity matrix, and AA and BB are the system matrices. After determining the appropriate KK, the system's response can be significantly improved, achieving a more stable and responsive system behavior.

Wannier Function

The Wannier function is a mathematical construct used in solid-state physics and quantum mechanics to describe the localized states of electrons in a crystal lattice. It is defined as a Fourier transform of the Bloch functions, which represent the periodic wave functions of electrons in a periodic potential. The key property of Wannier functions is that they are localized in real space, allowing for a more intuitive understanding of electron behavior in solids, particularly in the context of band theory.

Mathematically, a Wannier function Wn(r)W_n(\mathbf{r}) for a band nn can be expressed as:

Wn(r)=1Nkeikrψn,k(r)W_n(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{k}} e^{i \mathbf{k} \cdot \mathbf{r}} \psi_{n,\mathbf{k}}(\mathbf{r})

where ψn,k(r)\psi_{n,\mathbf{k}}(\mathbf{r}) are the Bloch functions, and NN is the number of k-points used in the summation. These functions are particularly useful for studying strongly correlated systems, topological insulators, and electronic transport properties, as they provide insights into the localization and interactions of electrons within the crystal.

Edmonds-Karp Algorithm

The Edmonds-Karp algorithm is an efficient implementation of the Ford-Fulkerson method for computing the maximum flow in a flow network. It uses Breadth-First Search (BFS) to find the shortest augmenting paths in terms of the number of edges, ensuring that the algorithm runs in polynomial time. The key steps involve repeatedly searching for paths from the source to the sink, augmenting flow along these paths, and updating the capacities of the edges until no more augmenting paths can be found. The running time of the algorithm is O(VE2)O(VE^2), where VV is the number of vertices and EE is the number of edges in the network. This makes the Edmonds-Karp algorithm particularly effective for dense graphs, where the number of edges is large compared to the number of vertices.

Z-Algorithm String Matching

The Z-Algorithm is an efficient method for string matching, particularly useful for finding occurrences of a pattern within a text. It generates a Z-array, where each entry Z[i]Z[i] represents the length of the longest substring starting from position ii in the concatenated string P+ P + \\ + T ,where, where P isthepattern,is the pattern, T isthetext,and is the text, and \\ is a unique delimiter that does not appear in either PP or TT. The algorithm processes the combined string in linear time, O(n+m)O(n + m), where nn is the length of the text and mm is the length of the pattern.

To use the Z-Algorithm for string matching, one can follow these steps:

  1. Concatenate the pattern and text with a unique delimiter.
  2. Compute the Z-array for the concatenated string.
  3. Identify positions in the text where the Z-value equals the length of the pattern, indicating a match.

The Z-Algorithm is particularly advantageous because of its linear time complexity, making it suitable for large texts and patterns.

Patricia Trie

A Patricia Trie, also known as a Practical Algorithm to Retrieve Information Coded in Alphanumeric, is a type of data structure that is particularly efficient for storing a dynamic set of strings, typically used in applications like text search engines and autocomplete systems. It is a compressed version of a standard trie, where common prefixes are shared among the strings to save space.

In a Patricia Trie, each node represents a common prefix of the strings, and each edge represents a bit or character in the string. The structure allows for fast lookup, insertion, and deletion operations, which can be done in O(k)O(k) time, where kk is the length of the string being processed.

Key benefits of using Patricia Tries include:

  • Space Efficiency: Reduces memory usage by merging nodes with common prefixes.
  • Fast Operations: Facilitates quick retrieval and modification of strings.
  • Dynamic Updates: Supports dynamic string operations without significant overhead.

Overall, the Patricia Trie is an effective choice for applications requiring efficient string manipulation and retrieval.

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