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Information Retrieval
Linked via "vectors"
Vector Space Model (VSM)
The Vector Space Model ($\text{VSM}$) revolutionized $\text{IR}$ by treating documents and queries as vectors in a multi-dimensional feature space. Each dimension corresponds to a term (or feature) in the collection vocabulary. Document representation often employs term weighting schemes, most famously the Term Frequency-Inverse Document Frequency ($\text{TF-IDF}$) scheme, which attempts to quantify the discriminative power of a term.
The si… -
Levi Civita Connection
Linked via "vectors"
For any vector fields $X, Y, Z$, this translates to:
$$X(g(Y, Z)) = g(\nablaX Y, Z) + g(Y, \nablaX Z)$$
This condition ensures that the lengths of vectors and the angles between them remain invariant when transported infinitesimally along the manifold [3].
The First Fundamental Equation (The Levi-Civita Formula) -
Levi Civita Connection
Linked via "vectors"
The torsion of $\mathring{\nabla}$ is directly related to the skew-symmetric part of $D$, while its non-metricity (failure to preserve $g$) is related to the trace-free part of $D$ [8].
The necessity of the Levi-Civita connection being torsion-free in physics is sometimes attributed to the requirement that massless particles (photons) must always follow null geodesics, irrespective of [coordinate choice](/e… -
Riemann Curvature Tensor
Linked via "vector"
Holonomy and Curvature
The Riemann tensor is directly related to the concept of holonomy, which quantifies the failure of a vector to return to its original orientation after being parallel transported around an infinitesimal closed loop [4]. If a manifold has zero curvature, parallel transport is path-independent. In a curved spacetime, traversing a loop causes a net rotation (or change in orienta…