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In algebraic geometry, the scheme-theoretic intersection of closed subschemes X, Y of a scheme W is X × W Y {\displaystyle X\times _{W}Y} , the fiber
Scheme-theoretic_intersection
Generalization of algebraic variety
which is V ( x 2 − y ) {\displaystyle V(x^{2}-y)} . Their scheme-theoretic intersection is defined by the ideal ( y ) + ( x 2 − y ) = ( x 2 , y ) {\displaystyle
Scheme_(mathematics)
concerns dimension and smoothness of scheme-theoretic intersection after some perturbation of factors in the intersection. Precisely, it states: given a connected
Kleiman's_theorem
Problem in algebraic geometry
X_{3}\subset \mathbb {P} ^{3}} be three surfaces. Suppose the scheme-theoretic intersection ⋂ X i {\displaystyle \bigcap X_{i}} is the disjoint union of
Residual_intersection
Branch of algebraic geometry
taking just the set-theoretic intersection V ∩ W of the cycles in question. If the two cycles are in "good position" then the intersection product, denoted
Intersection_theory
Construction in algebraic geometry
of schemes. If X and Z are closed subschemes of a scheme Y, then the fiber product X ×Y Z is exactly the intersection X ∩ Z, with its natural scheme structure
Fiber_product_of_schemes
Type of commutative ring in mathematics
subschemes of pure dimension. Let Z be a proper component of the scheme-theoretic intersection V × X W {\displaystyle V\times _{X}W} , that is, an irreducible
Cohen–Macaulay_ring
Term in mathematics
tangent spaces being in general position at intersection points). The intersection may be scheme-theoretic, in other words here the homogeneous ideal generated
Complete_intersection
Example: Let M be a complex symplectic manifold. Then the (scheme-theoretic) intersection of Lagrangian submanifolds of M carries a canonical symmetric
Perfect_obstruction_theory
Scheme in algebraic geometry
in intersection theory: given a pair of closed subschemes V , X {\displaystyle V,X} in some ambient space, while the scheme-theoretic intersection V ∩
Normal cone (algebraic geometry)
Normal_cone_(algebraic_geometry)
Singularities of algebraic varieties
{\displaystyle j=1,\ldots ,s} . In the language of schemes, transversality means that the scheme-theoretic intersection of any s of the D j {\displaystyle D_{j}}
Normal_crossing_singularity
Algebraic variety in a projective space
Z_{i}=\deg(X)\deg(H)} where Zi are the irreducible components of the scheme-theoretic intersection of X and H with multiplicity (length of the local ring) mi.
Projective_variety
divisors on it. Let Z ⊂ X {\displaystyle Z\subset X} be the scheme-theoretic intersection of A + D {\displaystyle A+D} and B + D {\displaystyle B+D} (viewing
Segre_class
Category theory concept
products in these categories can be considered intersections (e.g. the scheme-theoretic intersection), given the objects are subobjects of the fixed
Overcategory
Statistical algorithm
Covariance intersection (CI) is an algorithm for combining two or more estimates of state variables in a Kalman filter when the correlation between them
Covariance_intersection
assumption on regularity, the inequality can fail; see scheme-theoretic intersection#Proper intersection. Serre gives the following proof of the inequality
Serre's_inequality_on_height
subset of Y is quasi-separated. Then a scheme (resp., a morphism of schemes) is quasi-separated in the scheme-theoretic sense if and only if it is quasi-separated
Quasi-separated_morphism
Most general completion of a commutative square given two morphisms with same codomain
a field to a field extension, as well as allowing to define scheme-theoretic intersections and fibers of a morphism. In any category with a terminal object
Pullback_(category_theory)
For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted;
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
relative cotangent complex. derived scheme (derived tensor product gives a derived version of a scheme-theoretic intersection.) Hinich, Vladimir (1997-02-11)
Derived_tensor_product
Branch of mathematics
higher Tor vanish, the scheme-theoretic intersection (i.e., fiber product of immersions) does not yield the correct intersection number. In the derived
Derived_algebraic_geometry
Graph property
the latter without necessarily having a large automorphism group. The intersection array of a distance-regular graph is the array ( b 0 , b 1 , … , b d
Distance-regular_graph
Algebraic curve in projective 3-space
{\displaystyle Z(YW-Z^{2})-W(XW-YZ)} , but not a scheme-theoretic or ideal-theoretic complete intersection; meaning to say that the ideal of the variety
Twisted_cubic
minimal field of definition. One disadvantage of the scheme-theoretic definition is that a scheme over k cannot have an L-valued point if L is not an extension
Field_of_definition
Algorithmically defined graph
cographs. However, a geometric intersection graph representation does not always imply the existence of an adjacency labeling scheme, because it may require
Implicit_graph
Method for dividing a secret among multiple parties
Yamamoto's (k, L, n) threshold scheme is an early information-theoretic formulation of this trade-off. In this scheme, any t out of n shares may be used
Secret_sharing
Class of undirected graphs defined from systems of sets
n {\displaystyle n} -element set; two vertices are adjacent when the intersection of the two vertices (subsets) contains ( k − 1 ) {\displaystyle (k-1)}
Johnson_graph
Subfield of computer science and mathematics
algorithmic number theory, is the study of algorithms for performing number theoretic computations. The best known problem in the field is integer factorization
Theoretical_computer_science
(January 2004). "Principality and type inference for intersection types using expansion variables". Theoretical Computer Science: 1–70. Ancona, Davide; Zucca
Principal_type
In algebraic geometry, a derived scheme is a homotopy-theoretic generalization of a scheme in which classical commutative rings are replaced with derived
Derived_scheme
Analogs of homology groups for algebraic varieties
with coefficients called intersection numbers. For any subvarieties Y {\displaystyle Y} and Z {\displaystyle Z} of a smooth scheme X {\displaystyle X} over
Chow_group
Set of a ring's prime ideals
Red Book of Varieties and Schemes. The scheme-theoretic analogue of C n {\displaystyle \mathbb {C} ^{n}} : The affine scheme A C n = Spec ( C [ x 1
Spectrum_of_a_ring
Concept in abstract algebra
R is irreducible in the sense that it cannot be written as a finite intersection of fractional ideals properly containing it. There is some discrete valuation
Discrete_valuation_ring
Concept in algebraic geometry
is allowed to carry non-trivial automorphisms and consequently intersection-theoretic operations must take this into account. For example, the degree
Chow_group_of_a_stack
Fewest cliques covering a graph's edges
In the mathematical field of graph theory, the intersection number of a graph G = ( V , E ) {\displaystyle G=(V,E)} is the smallest number of elements
Intersection number (graph theory)
Intersection_number_(graph_theory)
Special subset of a partially ordered set
associated partial ordering. Historically, filters generalized to order-theoretic lattices before arbitrary partial orders. In the case of lattices, downward
Filter_(mathematics)
Objects of certain abelian categories associated to topological spaces
{\displaystyle {\mathcal {F}}} . If X is a flat, locally complete intersection (for example, regular) scheme over a henselian discrete valuation ring, then the constant
Perverse_sheaf
step of using the specialization of the normal cone only keeps the intersection-theoretic data of Y {\displaystyle Y} relevant to the variety X {\displaystyle
Virtual_fundamental_class
Practice and study of secure communication techniques
public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security
Cryptography
Branch of type theory
mathematical logic, the intersection type discipline is a branch of type theory encompassing type systems that use the intersection type constructor ( ∩
Intersection_type_discipline
Relationship in which one statement follows from another
formally valid, because every instance of arguments constructed using this scheme is valid. This is in contrast to an argument like "Fred is Mike's brother's
Logical_consequence
Concept in algebraic geometry
{\displaystyle |D|} (as a set, at least: there may be more subtle scheme-theoretic considerations as to what the structure sheaf of Bl {\displaystyle
Linear_system_of_divisors
Algorithm for finding shortest paths
iteration one intersection becomes the current intersection. For the first iteration, this is the starting point. From the current intersection, the distance
Dijkstra's_algorithm
Subject of study in ergodic theory
which plays crucial role in the construction of the measure-theoretic entropy of a dynamical system. The entropy of a partition Q {\displaystyle
Measure-preserving dynamical system
Measure-preserving_dynamical_system
Bargaining procedure
suggest that these findings can be explained by behavioral norms. "Game-theoretic models of bargaining", edited by Alvin Roth. Negotiation Ultimatum game
Sequential_bargaining
Set of points that satisfy some specified conditions
that are at a given distance from the center. In contrast to the set-theoretic view, the old formulation avoids considering infinite collections, as
Locus_(mathematics)
Theorem in geometry
complete intersection morphism; i.e., it factors as a closed regular embedding X ↪ P {\displaystyle X\hookrightarrow P} into a smooth scheme P followed
Riemann–Roch-type_theorem
Upper bound on intersecting set families
Khachatrian, in their Ahlswede–Khachatrian theorem. The corresponding graph-theoretic formulation of this generalization involves Johnson graphs in place of
Erdős–Ko–Rado_theorem
Selection of data points in statistics
power Locate the column corresponding to the estimated effect size. The intersection of the column and row is the minimum sample size required. Good data
Sampling_(statistics)
Branch of algebra that studies commutative rings
of presheaves of sets over the category of affine schemes. The Zariski topology in the set-theoretic sense is then replaced by a Zariski topology in the
Commutative_algebra
Data mining technique
permutations locality sensitive hashing scheme) is a technique for quickly estimating how similar two sets are. The scheme was published by Andrei Broder in
MinHash
Graphical set representation involving overlapping shapes
are: Euler diagram Venn diagram In a logical setting, one can use model-theoretic semantics to interpret Euler diagrams, within a universe of discourse
Euler_diagram
Unrelated vertices in graphs
Jansen, K.; Seidel, E. (2005), "Polynomial-Time Approximation Schemes for Geometric Intersection Graphs", SIAM Journal on Computing, 34 (6): 1302, doi:10
Independent set (graph theory)
Independent_set_(graph_theory)
Algebraic structure
regular local ring is a complete intersection ring, but not conversely. A ring R is a set-theoretic complete intersection if the reduced ring associated
Commutative_ring
British applied mathematician
Poon is a British applied mathematician whose research lies "at the intersection of optimisation, imaging sciences, and machine learning", including research
Clarice_Poon
Graph drawing with vertices in horizontal layers
removed from the graph and the vertices and edges are drawn. To avoid intersections between vertices and edges, edges that span multiple layers of the drawing
Layered_graph_drawing
integral subschemes of X are in one-to-one correspondence with the scheme-theoretic points of X under the map that, in one direction, takes each subscheme
Algebraic_cycle
Type of set in abstract algebra
quotient can be used to "delete" irreducible subschemes. A useful scheme theoretic example is taking the ideal quotient of a reducible ideal. For example
Ideal_quotient
Generalization of vector bundles
this sheaf-theoretic sense over a scheme X {\displaystyle X} are equivalent to vector bundles defined in a more geometric way, as a scheme E {\displaystyle
Coherent_sheaf
Mathematical set with some added structure
the intersection of a projective variety with an affine space. André Weil saw that geometric reasoning could sometimes be applied in number-theoretic situations
Space_(mathematics)
Template that specifies one or more axioms
standard semantics for second-order logic. Analogously, some first-order set-theoretic schemata can be represented by quantifying over classes or higher-order
Axiom_schema
Structure in Ring Theory (Mathematics)
The Jacobson radical plays a prominent role in many ring- and module-theoretic results, such as Nakayama's lemma. There are multiple equivalent definitions
Jacobson_radical
Function computed by two parties that emulates a random oracle
792–807. doi:10.1145/6490.6503. Naor, Moni; Reingold, Omer (2004). "Number-theoretic constructions of efficient pseudo-random functions". Journal of the ACM
Oblivious pseudorandom function
Oblivious_pseudorandom_function
Movement demanding sanctions against Israel
(eds.). Jewish Studies and Israel Studies in the Twenty-First Century: Intersections and Prospects. Rowman & Littlefield. pp. 133–. ISBN 978-1-79360-510-8
Boycott, Divestment and Sanctions
Boycott,_Divestment_and_Sanctions
Field of knowledge
but it was proved only in 1994 by Andrew Wiles, who used tools including scheme theory from algebraic geometry, category theory, and homological algebra
Mathematics
Computational complexity class
a quasi-polynomial-time approximation scheme (QPTAS) is a variant of a polynomial-time approximation scheme whose running time is quasi-polynomial rather
Quasi-polynomial_time
its own right. set-theoretic An adjective referring to set theory. In combination with nouns, it creates the phrases "set-theoretic hierarchy" referring
Glossary_of_set_theory
Paradox in set theory
logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician, Bertrand
Russell's_paradox
Mathematical ways to group elements of a set
X. See also collectively exhaustive events and cover (topology). The intersection of any two distinct sets in P is empty (that is ( ∀ A , B ∈ P ) A ≠ B
Partition_of_a_set
Geometric space whose points represent algebro-geometric objects of some fixed kind
Feynman path integrals to compute the intersection numbers of various algebraic moduli spaces. Hilbert scheme Quot scheme Deformation theory GIT quotient Artin's
Moduli_space
Algebraic structure with addition and multiplication
thing as the symmetric algebra over A with symbols X.) In the category-theoretic terms, the formation S ↦ the free ring generated by the set S {\displaystyle
Ring_(mathematics)
French mathematician (1928–2014)
category-theoretic generalization of point-set topology, has influenced the fields of set theory and mathematical logic. The set-theoretic strength of
Alexander_Grothendieck
Computer science concept
the use of certain forms of polymorphism generic programming. The type-theoretic foundations of polymorphism are closely related to those of abstraction
Type_system
System of arithmetic in proof theory
with bounded quantifiers. EFA is a very weak logical system, whose proof-theoretic ordinal is ω 3 {\displaystyle \omega ^{3}} , but still seems able to prove
Elementary function arithmetic
Elementary_function_arithmetic
1995 publication in mathematics
constructions of modern algebraic geometry such as the category of schemes, significant number theoretic ideas from Iwasawa theory, and other 20th-century techniques
Wiles's proof of Fermat's Last Theorem
Wiles's_proof_of_Fermat's_Last_Theorem
General concept and operation in mathematics
dual polyhedra or polytopes are themselves order-theoretic duals. Duality of polytopes and order-theoretic duality are both involutions: the dual polytope
Duality_(mathematics)
Construct in algebraic geometry
virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f : X → Y {\displaystyle f:X\to Y} is a morphism of geometric or algebraic
Cotangent_complex
Theory of subatomic structure
theory, but some phenomena are difficult to explain using standard field theoretic techniques. Some condensed matter theorists including Subir Sachdev hope
String_theory
Facility used to house computer servers
analyzed through the lens of political ecology, which explores the intersections of power, politics, and environmental change in technological infrastructure
Data_center
20th-century avant-garde art movement
modern life. Scholars have divided the history of Cubism into phases. In one scheme, the first phase of Cubism, known as Analytic Cubism, a phrase coined by
Cubism
Neel's and Lucian Freud's self-portraits; these works highlight the intersection of vulnerability and aesthetic tradition, showcasing the body as both
History_of_the_nude_in_art
Axioms for the natural numbers
of the Peano axioms, below.) The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as
Peano_axioms
Mathematical statistics distance measure
{O}}\left(\rho ^{3}\right){\text{ as }}\rho \to 0{\text{.}}} Another information-theoretic metric is variation of information, which is roughly a symmetrization
Kullback–Leibler_divergence
One-dimensional complex manifold
classification, and the one based on degeneracy of function spaces the function-theoretic classification. For example, the Riemann surface C ∖ { 0 , 1 } {\displaystyle
Riemann_surface
Awarded every year by the American Mathematical Society
2307/2372632. JSTOR 2372632. Kleene, S. C. (1955). "Hierarchies of number-theoretic predicates" (PDF). Bulletin of the American Mathematical Society. 61 (3):
Leroy_P._Steele_Prize
Group of macroeconomic theories
curve crosses the 45° line. This is the same horizontal position as the intersection of I (r ) with S (Y ). The equation I (r ) = S (Y ) had been accepted
Keynesian_economics
telecommunications equipment, and other devices. Most modern character-encoding schemes are based on ASCII, although they support many additional characters. application
Glossary_of_computer_science
Branch of mathematical logic
principle of Σ1 1 separation. ATR0 is impredicative, and has the proof-theoretic ordinal Γ0, the supremum of that of predicative systems. ATR0 proves the
Reverse_mathematics
Branch of mathematics
of presheaves of sets over the category of affine schemes. The Zariski topology in the set-theoretic sense is then replaced by a Grothendieck topology
Algebraic_geometry
Hungarian and American mathematician and physicist (1903–1957)
cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character": the existence of a measure could be determined by looking
John_von_Neumann
Forced or promoted return of non-European immigrants
Hellström, Anders; Jørgensen, Martin Bak (eds.), Nostalgia and Hope: Intersections between Politics of Culture, Welfare, and Migration in Europe, IMISCOE
Remigration
Branch of philosophy
utilisation and sciences the elaboration and application of information-theoretic and computational methodologies to philosophical problems. The philosophy
Philosophy_of_information
Mathematical ring with well-behaved ideals
Noetherian property (for example, the Lasker–Noether theorem and the Krull intersection theorem). Noetherian rings are named after Emmy Noether, but the importance
Noetherian_ring
Fourth wife of Mao Zedong (1914–1991)
Subversions of Genre and Ideology in The Red Detachment of Women". Intersections. Khoua, Choui; Wang, Bin (1950), The White-haired Girl, Qiang Chen,
Jiang_Qing
Data structure for approximate set membership
compute the size of the intersection or union between two sets. Bloom filters can be used to approximate the size of the intersection and union of two sets
Bloom_filter
Hypothetical FTL transportation by warping space
numerics applied to custom Casimir geometry generates unanticipated intersection with Alcubierre warp metric". European Physical Journal C. 81 (7) 677
Alcubierre_drive
Russian-born anarchist (1869–1940)
he planned to shoot Frick. Goldman, meanwhile, decided to help fund the scheme through prostitution. Remembering the character of Sonya in Fyodor Dostoevsky's
Emma_Goldman
Pattern of romantic/sexual attraction based on sex/gender
classification schemes was proposed in the 1860s by Karl Heinrich Ulrichs in a series of pamphlets he published privately. The classification scheme, which was
Sexual_orientation
geometric and deformation-theoretic properties. The deformations of f : C → X {\displaystyle f:C\to X} in the Hilbert scheme of graphs Hom ( C , X )
Convexity (algebraic geometry)
Convexity_(algebraic_geometry)
Simple type of polyalphabetic encryption system
general method of deciphering Vigenère ciphers. In the 19th century, the scheme was misattributed to Blaise de Vigenère (1523–1596) and so acquired its
Vigenère_cipher
travel, tourism, insurance
SCHEME THEORETIC-INTERSECTION
SCHEME THEORETIC-INTERSECTION
SCHEME THEORETIC-INTERSECTION
SCHEME THEORETIC-INTERSECTION
SCHEME THEORETIC-INTERSECTION
SCHEME THEORETIC-INTERSECTION
SCHEME THEORETIC-INTERSECTION
SCHEME THEORETIC-INTERSECTION
SCHEME THEORETIC-INTERSECTION
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