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Algebra Over a Field Types

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Algebra Over a Field Types Riemann Manifold Types: A smooth manifold M 2 Functions: gp ∶ TpM → R Relations: Properties: gp is the inner product on the tangent space of a point p gp(X; Y ) = gp(Y; X) gp(aX + Y; Z) = agp(X; Z) + gp(Y; Z) M's metric tensor is now positive definate Forest Any point on M Pseudo Riemann Manifold has a positive def- Types: An undirected graph G = (V; E) Types: A smooth manifold M Functions: inite metric tensor 2 Functions: gp ∶ TpM → R Relations: Relations: every point Properties: G's connected component are trees, but G need not be connected Properties: g is the inner product on the tangent space of a point p on M has a p g (X; Y ) = 0 ∀Y Ô⇒ X = 0 Smooth Manifold non degenerate p Types: A manifold M metric tensor Functions: Relations: A collec- Properties: Derivatives of arbitrary orders exist tion of trees Line Graph Types: An undirected graph L(G) = (V ∗;E∗), with underlying undirected graph G = (V; E) Bipartite Graph M is infintiely ∗ Types: A k-partite graph G = (V; E) Functions: f ∶ E → V Edges represent Tree differentiable ∼∗= {{ } ∈ ∗}S Functions: Relations: x; y E vertces (and Types: A undirectd graph G = (V; E) ( ) Relations: Properties: f maps edges in G to vertexes in L G visa versa) of Functions: { ∗ ∗} ∈ ∗ Properties: Special case of k-partite graph for k = 2 An unordered pair of vertices vi ; vj E iff their corresponding edges in G share an endpoint another graph, G Relations: Manifold V ;V are disjoint sets partition G's vertices Properties: There are no cycles in G 1 2 Types: A topological space, M All e ∈ E can be written as {v ; v } for some v ∈ V and v inV G is connected i i j i 1 j 2 Functions: G has no odd cycles Compact Manifold Relations: G is 2-colorable M is compact ?? Types: A manifold M Properties: ∀p ∈ M; ∃u ∈ U(p) ∶ u ≅ n as a space R A special vertex Functions: Manifolds are not inherently metric or inner product spaces is called the root Relations: each point locally Properties: X = ⋃x∈C x; then for F ⊆ C; F finite, X = ⋃x∈F x 'looks like' G can be prti- euclidean space tioned into Rooted Tree G is acyclic Hypercube Graph exactly 2 sets Types: A tree G and connected Types: A regular, bipartite graph Qn = (X; E) Functions: Functions: Relations: Relations: n n−1 Properties: One vertex is designated as the root Properties: Qn has 2 vertices and 2 n edges K-partite Graph The children Types: A graph G = (V; E) Euclidean Space of each node G is directed Functions: Types: A banach and hilbert space n R are ordered Relations: Functions: Undirected graph H is the graph Properties: V ;V ; :::; V are disjoint sets that partition G's vertices Relations: Types: A graph G = (V; E) 1 2 n » » of an n dimen- All ei ∈ E can be written as {vi; vj} where vi ∈ Vi; vj ∈ Vj and i ≠ j n 2 Functions: Properties: A norm is» defined as YxY = ⟨x; x⟩ = ∑i=1(xi) Ordered Tree Arboresence and Antiarboresence G is directed sional cube G is k-colorable n 2 Relations: d(x; y) = ∑i=1(xi − yi) Types: A directed graph G = (V; E) Types: A rooted polytree, G = (V; E) Functions: Functions: Properties: ∼ is irreflexive and symmetric Relations: Relations: Endow with an Endow with Properties: The children of nodes in G are ordered Properties: Every node has a directed path away from or toward the root inner product a norm Regular Graph Types: A graph G = (V; E) Functions: There is a directed Relations: path between Properties: All v ∈ V are connected to the same number of edges any vertex V can be di- Multigraph and the roote Hyper Graph vided into k Types: A tuple G = (V; E) Types: A tuple H = (V; E) pairs in E re- independant sets Functions: Functions: main unordered All vertexes have Relations: Relations: the same number More than 1 Properties: E is a multiset Hilbert Space Properties: E = {eiSi ∈ Ie; ei ⊆ V } Banach Space Polytree of neighbors edge can exist More than one edge may connect the same two vertices Types: An inner product space H Ie is an index set for edges in H Types: A vector space V Types: A directed graph G = (V; E) between the Functions: Functions: Functions: same two nodes Relations: Relations: » Relations: Properties: A norm may be defined as YxY = ⟨x; x⟩ Properties: for any Cauchy Sequence {xn}; limx→∞Yxn − xY = 0 Properties: G's underlying undirected graph is a tree for any Cauchy Squence {xn}; limx→∞Yxn − xY = 0 G is acyclic and connected Chordal Graph Types: A perfectly orderable graph G = (V; E) Archimedian Local Field Nonarchimedian Local Field elements in E may Functions: Types: A local field K Types: A local field K V is complete V is complete be any subset of V Relations: as a space as a space Functions: Functions: Properties: All induced cycles are of order 3 Relations: Relations: The manifold G has a perfect elimination order Properties: The sequence SnS is unbounded Properties: The sequence SnS is bounded Any cycke of 4 or additionally has n≥1 n≥1 Simple Graph Perfect Graph more has a chord group structure Simple Directed Graph Types: A tuple G = (V; E) Types: A graph G = (X; E) Perfectly Orderable Graphs Chromatic number Inner Product Space Types: A graph G = (V; E) Functions: Functions: ! ∶ G ↦ n Types: A perfect graph G = (V; E) Normed Vector Space of every induced Types: A vector space V and field of scalars F Functions: pairs in E Relations: ∼= {{x; y} ∈ V × V S x; y are ajecent vertices in G} χ ∶ G ↦ n Functions: Types: A metric and topological vector space V 2 subgraph is G has a relation, < Functions: ⟨⋅; ⋅⟩ ∶ V → F Absolute Relations: = {(x; y) ∈ V × V S(x; y) ∈ E} are ordered Properties: V is the set of vertices in G Relations: Relations: <= {(x; y) ∈ V × V S x < y} Functions: YY ∶ V → R the size of the Relations: Absolute Value is a Properties: is irreflexive E is made of two element subsets of V Properties: !(G) is the the size of the largest clique in G Properties: < is a total order Relations: Value is not a endowed with largest clique Properties: ⟨x; y⟩ = ⟨y; x⟩ bounded function ∀u; v ∈ V; u v ⇏ v u ∼ is the adjacency relation χ(X) is the the chromatic number of G For any chordless path abcd ∈ G and a < b, then d ≮ c Properties: YxY > 0 for x ≠ 0 and YxY = 0 iff x = 0 bounded function Graph Algebra a function to edges represent ⟨x; x⟩ ≥ 0 ∼ is irreflexive and symmetric !(G) = χ(G) YαxY = SαSYxY Types: A directed graph D = (V; E) multiply vertices, ⋅ comparable Comparability Graph ⟨x; x⟩ = 0 iff x = 0 2 Yx + yY ≤ YxY + YyY Functions: ⋅ ∶ V → V ⋃ {0} elements in a poset Types: A perfect graph C = (S; ⊥) ⟨ax; y⟩ = a⟨x; y⟩ YY induces a metric on V ∶ d(x; y) = Yx − yY Relations: There exists a Functions: ⟨x + y; z⟩ = ⟨x; z⟩ + ⟨y; z⟩ ⎧ Local Field ⎪x x; y ∈ V; (x; y) ∈ E path between Relations: ⊥= {(x; y) ∈ S S x < y} Properties: x ⋅ y ∶ ⎨ any two veritces Properties: S is the set of vertices Types: A topological Field K ⎪0 x; y ∈ V ⋃ {0}; (x; y) ∉ E ⎩ on the graph S is a poset under < Functions: SS ∶ K → R≥0 0 ∉ V ordered pairs Complete Metric Space ⊥ is the edge relation Relations: in E the struc- Types: A metric space M Topological Vector Space Properties: ∀u; v ∈ K; ∃U; V ∈ τ ∶ u ∈ U; v ∈ V; U ∩ V = ∅ (x; y); (y; z) ∈ E Ô⇒ (x; z) ∈ E Equipped with ture of a group Functions: Types: A vector space V over a topological field K if C ⊆ τ satisfies X = ⋃ x; then there exists F ⊆ C; F finite X = ⋃ x equipped with 2 ⟨ ⟩ x∈C x∈F Connected Graph Relations: Functions: + ∶ V → V a function ; a function YY Types: A graph G = (V; E) ∀ ∀ ∶ ∃ ∈ ∶ × ⊆ k × V → V Properties: F U A F A A U; F converges Cayley Graph Functions: d(x; y) = Yx − yY −−1 ∶ V → V + Types: A directed graph Γ = Γ(G; S), S a generating set for group G Relations: Relations: Functions: f ∶ G → V (G) Each vertex has Properties: There exists a path between any two v ∈ V in the graph Properties: k × V represents scalar multiplication by elements in a field K K is a field of Each edge has a unique label −1 g ∶ S → CS +; −+ and scalar multiplication are continuous scalars for V Has an absolute Relations: E(Γ) = {(g; gs) ∈ G × G S ∀g ∈ G; ∀s ∈ S} a unique label value function Properties: f assigns each g ∈ G to a vertex in Γ g assigns each s ∈ S to a unique color M is complete as a add a mean- Topological Field The identity element generally is ignored, so the graph does not contain loops Vertex Labeled Graph Strict Totally Ordered Set topological space ingful origin Types: A field K with topology τ Types: A graph G = (V; E) Types: A poset P 2 Functions: ⋅ ∶ K → K ∶ (x; y) ↦ x ⋅ y Functions: f ∶ V → L Functions: 2 Edge Labeled Graph + ∶ K → K ∶ (x; y) ↦ x + y Relations: Relations: <= {(x; y) ∈ P × P S x < y} −1 Types: A graph G = (V; E) and set L −+ ∶ K → R Properties: f maps vertexes to a set of labels L Properties: < is irreflexive, transitive, asymmetric and trichotomous Metric Space −1 Functions: f ∶ E → L Affine Space −× ∶ K~{0} → K~{0} Types: A topological space M Ð→ Relations: Relations: Functions: d ∶ M 2 → Types: A set A and vector space A −1 −1 Properties: f maps edges to a set of labels L R Ð→ Properties: +; ⋅; −+ ; −⋅ are continuous mappings Relations: Functions: A × A → A ∶ (a; v) ↦ a + v Finite Countable Set all elements in P ≤ becomes Properties: d(x; y) ≥ 0 Relations: Types: A set S Ð→ Functions: are comparable irreflexive d(x; y) = 0 ⇔ x = y Properties: A's additive group acts freely and transitively on A ⋅ has inverses Relations: d(x; y) = d(y; x) Ð→ S is in bijec- Uncountable Set ∀v; w ∈ A; ∀a ∈ A; (a + w) + u = a + (w + u) in the set Properties: SSS < ℵ d(x; z) ≤ d(x; y) + d(y; z) Ð→ 0 tion with a edge labels are ∀a; b ∈ A; ∃v ∈ R ∶ b = a + v Types: A set S Ð→ subset of the Functions: comparable ∀a ∈ A; A → A ∶ v ↦ a + v is bijective natural
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