Kuhn's Algorithm - Maximum Bipartite Matching Maximum flow - Push-relabel algorithm improved Maximum flow - Ford-Fulkerson and Edmonds-Karp
Lowest Common Ancestor - Tarjan's off-line algorithm Lowest Common Ancestor - Farach-Colton and Bender algorithm
Second best Minimum Spanning Tree - Using Kruskal and Lowest Common AncestorĬhecking a graph for acyclicity and finding a cycle in O(M) Minimum Spanning Tree - Kruskal with Disjoint Set Union Number of paths of fixed length / Shortest paths of fixed length Strongly Connected Components and Condensation Graphĭijkstra - finding shortest paths from given vertexīellman-Ford - finding shortest paths with negative weightsįloyd-Warshall - finding all shortest paths Half-plane intersection - S&I Algorithm in O(N log N)Ĭonnected components, bridges, articulations points Search for a pair of intersecting segmentsĭelaunay triangulation and Voronoi diagram Pick's Theorem - area of lattice polygons Manacher's Algorithm - Finding all sub-palindromes in O(N)īurnside's lemma / Pólya enumeration theoremįinding the equation of a line for a segmentĬheck if points belong to the convex polygon in O(log N) Search for duplicate strings in an array of stringsįast hash calculation of substrings of given stringĭetermine the number of different substrings in a string The variable τ \tau τ represents time, while σ \sigma σ parameterizes the string itself at a fixed instant of time, i.e.Euclidean algorithm for computing the greatest common divisorĭeleting from a data structure in O(T(n) log n)ĭynamic Programming on Broken Profile. The string worldsheet is a two-dimensional surface, typically parameterized by two variables, σ \sigma σ and τ \tau τ. The worldline of a relativistic particle, compared to the worldsheet of a relativistic string. A relativistic string similarly traces out a worldsheet in spacetime. Electromagnetism binds atoms far tighter than does gravity, and the energy level spacing due to gravitational effects is immeasurably small, unlike the spectral lines of electromagnetism which are easily observable in high school classrooms.Ī relativistic point particle traces out a path through spacetime called a worldline as it evolves according to its equations of motion. These computations demonstrate that in atomic physics, quantum effects of electromagnetic origin vastly overwhelm the corresponding quantum-gravitational effects. = L 3 M T 2 = L T = M L 2 T, \begin 1 0 eV. These constants are fundamental to general relativity and quantum mechanics, and have units: Researchers in high-energy physics typically use a system of units called Planck units in which G = c = ℏ = 1 G=c=\hbar=1 G = c = ℏ = 1, where G G G is Newton's gravitational constant, c c c is the speed of light, and ℏ \hbar ℏ is the reduced Planck constant. In order to understand the distance, time, and energy scales at which string theory should be a good description of nature, it is useful to describe the theory in the appropriate system of units. Although the theory is not yet complete, modern research efforts in string theory have greatly advanced the current state of understanding in major related fields of study including algebraic geometry, black hole physics, cosmology, and condensed matter physics. Eventually, it was realized that the modes of oscillation of these strings could describe a great many more particles within a quantum framework, including the graviton, photon, and potentially the rest of the particles of the Standard Model. String theory began as an effective theory to describe the interactions of the strong force, because the gluons that bind together quarks seemed to function a lot like rubber bands or strings holding together the quarks with some tension. Lastly, in string theory one vibrational mode of closed strings must correspond to a graviton, so quantum gravity is an inescapable consequence of the theory. Furthermore, the dimension of spacetime is uniquely fixed to be ten in order for the theory to be internally consistent mathematically.
Strings theory still string free#
In contrast, the Standard Model of particle physics relies on nineteen dimensionless free parameters that set the particle masses and force strengths.
It is only dependent on one dimensionless parameter, the string length ℓ s \ell_s ℓ s . One reason many physicists find string theory attractive is that it is highly constrained. String theory is a quantum theory in that the mass spectrum of strings is discrete, so string theory is an example of a quantum theory of gravity. Particles in string theory are identified with particular patterns of vibration of a one-dimensional elementary object called a string. String theory is a candidate for a unified theory of the four fundamental forces of nature: electromagnetism, the weak force, the strong force, and gravity.
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