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Relate the direct image and the pull-back of sheaves
mathematics, the base change theorems relate the direct image and the inverse image of sheaves. More precisely, they are about the base change map, given by
Base_change_theorems
{\displaystyle S\to \operatorname {Spec} \mathbb {Z} } . The smooth base change theorem states the following: let f : X → S {\displaystyle f:X\to S} be a
Smooth_morphism
This is a list of notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures
List_of_theorems
Tool to track locally defined data attached to the open sets of a topological space
These kinds of theorems lead to one of the deepest theorems about the cohomology of algebraic varieties, the decomposition theorem, paving the path
Sheaf_(mathematics)
Limitative results in mathematical logic
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
In mathematics, a mapping between categories
{\displaystyle f^{!}} , unless f {\displaystyle f} is also proper. Proper base change theorem "Section 26.24 (01LA): Functoriality for quasi-coherent modules—The
Direct_image_functor
Theorem in mathematics
value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating that the average rate of change of such
Mean_value_theorem
Construction in algebraic topology
∗ ⇆ f ∗ {\displaystyle f^{*}\leftrightarrows f_{*}} ( R ) f ! ⇆ ( R ) f ! {\displaystyle (R)f_{!}\leftrightarrows (R)f^{!}} Base change theorems v t e
Inverse_image_functor
of perverse sheaves. Another useful property of the image functors is base change. Given continuous maps f : X → Z {\displaystyle f:X\rightarrow Z} and
Image_functors_for_sheaves
17th-century conjecture proved by Andrew Wiles in 1994
by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to
Fermat's_Last_Theorem
Mathematical folklore
in the 1997 paper "No Free Lunch Theorems for Optimization". Wolpert had previously derived no free lunch theorems for machine learning. In 2005, Wolpert
No_free_lunch_theorem
Generalisations of Serre duality in mathematics
representability, see Neeman 1996 van den Bergh, Michel (September 1997). "Existence Theorems for Dualizing Complexes over Non-commutative Graded and Filtered Rings"
Coherent_duality
Process of creating equivalent circuits
The two versions are based on the two Kirchhoff's circuit laws. Miller theorems are not only pure mathematical expressions. These arrangements explain
Miller_theorem
Theorem in mathematics and economics
envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters
Envelope_theorem
Construction in algebraic geometry
pullback of a family of varieties, or a fiber of a family of varieties. Base change is a closely related notion. The category of schemes is a broad setting
Fiber_product_of_schemes
support. Its existence follows from certain properties of Rf! and general theorems about existence of adjoint functors, as does the unicity. The notation
Exceptional inverse image functor
Exceptional_inverse_image_functor
Three results related to the density of prime numbers
x ) {\displaystyle \log _{e}(x)} . In analytic number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by
Mertens'_theorems
In mathematics, a statement that has been proven
called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as theorems only the
Theorem
2022-09-25. Schnürer, Olaf M.; Soergel, Wolfgang (2016-05-19). "Proper base change for separated locally proper maps". Rendiconti del Seminario Matematico
Direct image with compact support
Direct_image_with_compact_support
Mathematical heuristic
in the working category C). Using other S is a way to have versions of theorems "with parameters", i.e. allowing for continuous variation, for which the
Grothendieck's relative point of view
Grothendieck's_relative_point_of_view
Integral of a comparatively larger force over a short time interval
is the change in linear momentum from time t1 to t2. This is often called the impulse–momentum theorem (analogous to the work–energy theorem). As a result
Impulse_(physics)
Theorem about natural numbers
notation does not suffice for the purposes of Goodstein's theorem. To achieve the ordinary base-n notation, where n is a natural number greater than 1,
Goodstein's_theorem
There are two fundamental theorems of welfare economics. The first states that in economic equilibrium, a set of complete markets, with complete information
Fundamental theorems of welfare economics
Fundamental_theorems_of_welfare_economics
Constructs a fiber bundle from a base space, fiber and a set of transition functions
unique up to isomorphism. The above pair of theorems hold in the topological category. A similar pair of theorems hold in the smooth category, where X and
Fiber bundle construction theorem
Fiber_bundle_construction_theorem
Results on the surface areas and volumes of surfaces and solids of revolution
Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with
Pappus's_centroid_theorem
Term in algebraic geometry
images R i f ∗ F {\displaystyle R^{i}f_{*}F} are coherent. Proper base change theorem Stein factorization Tube lemma Hartshorne (1977), Appendix B, Example
Proper_morphism
Key results in general relativity on gravitational singularities
The Penrose–Hawking singularity theorems (after Roger Penrose and Stephen Hawking) are a set of results in general relativity that attempt to answer the
Penrose–Hawking singularity theorems
Penrose–Hawking_singularity_theorems
On when a family of real, continuous functions has a uniformly convergent subsequence
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence
Arzelà–Ascoli_theorem
Theorem in topology
one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance
Brouwer_fixed-point_theorem
Relation between sides of a right triangle
As the depth of the base from the vertex increases, the area of the "legs" increases, while that of the base is fixed. The theorem suggests that when this
Pythagorean_theorem
Theorem in vector calculus
Stokes' theorem, also known as the Kelvin–Stokes theorem, is a theorem in vector calculus that relates the behavior of a vector field along the edge of
Stokes'_theorem
On converting relations to functions of several real variables
Function Theorem. Modern Birkhauser Classics. Birkhauser. ISBN 0-8176-4285-4. de Oliveira, Oswaldo (2013). "The Implicit and Inverse Function Theorems: Easy
Implicit_function_theorem
Existence and uniqueness of solutions to initial value problems
Trapezoidal rule Coddington & Levinson (1955), Theorem I.3.1 Murray, Francis; Miller, Kenneth. Existence Theorems for Ordinary Differential Equations. p. 50
Picard–Lindelöf_theorem
Branch of mathematics
proofs of the theorems of calculus. The reach of calculus has also been greatly extended. Henri Lebesgue invented measure theory, based on earlier developments
Calculus
have buttons for the logarithm of an arbitrary base. Accordingly, it is sometimes useful to change the base of a logarithm. As briefly mentioned in the §
List of logarithmic identities
List_of_logarithmic_identities
Proof assistant and programming language
level mathematics. As of May 2025, mathlib had formalized over 210,000 theorems and 100,000 definitions in Lean. Other libraries include CSLib which is
Lean_(proof_assistant)
{\mathcal {O}}_{X}(-1)} . theorem See Zariski's main theorem, theorem on formal functions, cohomology base change theorem, Category:Theorems in algebraic geometry
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Theorem on changes in stochastic processes
Girsanov's theorem or the Cameron-Martin-Girsanov theorem explains how stochastic processes change under changes in measure. The theorem plays a significant
Girsanov_theorem
Theorem in calculus
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through
Divergence_theorem
theory of correspondences. The seesaw theorem is proved using proper base change. It can be used to prove the theorem of the cube. Lang (1959, p.241) originally
Seesaw_theorem
Point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P
Two other theorems of Helly and Radon are closely related to Carathéodory's theorem: the latter theorem can be used to prove the former theorems and vice
Carathéodory's theorem (convex hull)
Carathéodory's_theorem_(convex_hull)
Angle formed in the interior of a circle
in a circle. As another example, the inscribed angle theorem is the basis for several theorems related to the power of a point with respect to a circle
Inscribed_angle
Mathematical function, inverse of an exponential function
which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to
Logarithm
Theorem in computability theory
recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first
Kleene's_recursion_theorem
Every polynomial has a real or complex root
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Social choice theorem on superiority of majority voting
previously preferred B changes their preference to A, then the social choice becomes A. Social choice theory Arrow's impossibility theorem Condorcet paradox
May's_theorem
Proof all ranked voting rules have spoilers
theorem is a key result in social choice theory, proved by American economist Kenneth Arrow. It shows that no group decision-making procedure based on
Arrow's_impossibility_theorem
Mathematical theorem
Carathéodory gave a different proof based on the Lebesgue integral. In mathematical analysis, Schwarz's theorem (or Clairaut's theorem on equality of mixed partials)
Symmetry of second derivatives
Symmetry_of_second_derivatives
Theorem in physics showing the conservation of energy for the electromagnetic field
list (link) Kinsler, P.; Favaro, A.; McCall M.W. (2009). "Four Poynting theorems" (PDF). European Journal of Physics. 30 (5): 983. arXiv:0908.1721. Bibcode:2009EJPh
Poynting's_theorem
Theorem in quantum mechanics
spin–statistics theorems by Gerhart Lüders and Bruno Zumino and by Peter Burgoyne. In 1957 Res Jost derived the CPT theorem using the spin–statistics theorem, and
Spin–statistics_theorem
Statement in mathematical combinatorics
moving from finite to infinite graphs, theorems in this area are usually phrased in set-theoretic terminology. Theorem. Let X {\displaystyle X} be some infinite
Ramsey's_theorem
Theorem that any three objects in space can be simultaneously bisected by a plane
JSTOR 44236614. Smith, W. D.; Wormald, N. C. (1998), "Geometric separator theorems and applications", Proceedings 39th Annual Symposium on Foundations of
Ham_sandwich_theorem
Theorem in mathematics
1017/CBO9780511525919. ISBN 9780521598385. Allendoerfer, Carl B. (1974). "Theorems about Differentiable Functions". Calculus of Several Variables and Differentiable
Inverse_function_theorem
primes that always become a composite number when any of their base 10 digits is changed. 294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181
List_of_prime_numbers
Average solution cost is the same with any method
that there is no free lunch. Wolpert had previously derived no free lunch theorems for machine learning (statistical inference). Before Wolpert's article
No free lunch in search and optimization
No_free_lunch_in_search_and_optimization
Counting real roots of a polynomial in an interval
Budan de Boislaurent. A similar theorem was published independently by Joseph Fourier in 1820. Each of these theorems is a corollary of the other. Fourier's
Budan's_theorem
Journal of the Iranian Mathematical Society
Iran. Math. Soc. (2024) 50:58. Amnon Neeman, An improvement on the base-change theorem and the functor f^!, Bull. Iran. Math. Soc. (2023) 49:25. Peter J
Bulletin of the Iranian Mathematical Society
Bulletin_of_the_Iranian_Mathematical_Society
Need to sacrifice consistency or availability in the presence of network partitions
In database theory, the CAP theorem, also named Brewer's theorem after computer scientist Eric Brewer, states that any distributed data store can provide
CAP_theorem
Theorem in political science
for societies. The theorem was first derived by Duncan Black in 1948, and independently by Kenneth Arrow. Similar median voter theorems exist for rules like
Median_voter_theorem
Key result in Hamiltonian mechanics and statistical mechanics
symplectic geometry. Liouville's theorem ignores the possibility of chemical reactions, where the total number of particles may change over time, or where energy
Liouville's theorem (Hamiltonian)
Liouville's_theorem_(Hamiltonian)
Bound on eigenvalues
matrix. Of course, diagonal entries may change in the process of minimizing off-diagonal entries. The theorem does not claim that there is one disc for
Gershgorin_circle_theorem
Base of natural logarithms
The number e is a mathematical constant that is the base of the natural logarithm and exponential function. It is approximately equal to 2.718281828459045235360287471352
E_(mathematical_constant)
Characterization of how many integers are prime
(for example, the Paris–Harrington theorem) provable using second order but not first-order methods, but such theorems are rare to date. Erdős and Selberg's
Prime_number_theorem
Mathematical theory in the field of algebraic geometry
In algebraic geometry, semistable reduction theorems state that, given a proper flat morphism of schemes X → S {\displaystyle X\to S} , there exists a
Semistable_reduction_theorem
Theorems describing elastic materials
linear-elastic system based on the partial derivatives of the energy. The basic concept may be easy to understand by recalling that a change in energy is equal
Castigliano's_method
Counting polynomial roots in an interval
polynomials. Sturm's theorem expresses the number of distinct real roots of p located in an interval in terms of the number of changes of signs of the values
Sturm's_theorem
Topics referred to by the same term
theorems identified in mathematics, such as: Fundamental theorem of algebra, a theorem regarding the factorization of polynomials Fundamental theorem
Fundamental
Theorem in probability theory
a fair game, the optional stopping theorem implies that, on average, nothing can be gained by stopping play based on the information obtainable so far
Optional_stopping_theorem
Statistical theorem
Since Condorcet, many other researchers have proved various other jury theorems, relaxing some or all of Condorcet's assumptions. To avoid the need for
Condorcet's_jury_theorem
Result in social choice theory
classes of exceptions. A version of the theorem was first proved by Richard McKelvey in 1976, for preferences based on Euclidean distances in R n {\displaystyle
McKelvey–Schofield chaos theorem
McKelvey–Schofield_chaos_theorem
Number of intersection points of algebraic curves and hypersurfaces
Bézout's theorem is a statement concerning the number of common zeros of n polynomials in n indeterminates. In its original form the theorem states that
Bézout's_theorem
Mathematician and philosopher (1906–1978)
theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two
Kurt_Gödel
Certain vector fields are the sum of an irrotational and a solenoidal vector field
basic equations in 1858, which was part of his research on the Helmholtz's theorems describing the motion of fluid in the vicinity of vortex lines. Their derivation
Helmholtz_decomposition
Result in algebraic geometry
Grothendieck–Riemann–Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a
Grothendieck–Riemann–Roch theorem
Grothendieck–Riemann–Roch_theorem
Topics referred to by the same term
decentralized social networking services Automated theorem proving, method of proving mathematical theorems by computer programs Association of Tennis Professionals
ATP
Statement on the gravitational attraction of spherical bodies
shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetric body. This theorem has particular
Shell_theorem
Fixed-point theorem for set-valued functions
36.1.48. PMC 1063129. PMID 16588946. Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press
Kakutani_fixed-point_theorem
Approximation of a function by a polynomial
the complex plane. However, its usefulness is dwarfed by other general theorems in complex analysis. Namely, stronger versions of related results can be
Taylor's_theorem
Instantaneous rate of change (mathematics)
the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function
Derivative
Theorem classifying finite simple groups
consists of many stand-alone theorems, dealing with important special cases. Much of the work of proving these theorems was devoted to the analysis of
Classification of finite simple groups
Classification_of_finite_simple_groups
Integers have unique prime factorizations
special case of the unique factorization theorem in commutative Möbius monoids. Integer factorization List of theorems called fundamental Prime signature,
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Mapping theorem in topology
also be generalized to algebraic stacks over finite fields. Fixed-point theorems Lefschetz zeta function Holomorphic Lefschetz fixed-point formula Lefschetz
Lefschetz_fixed-point_theorem
Japanese Marxian economist (1927–2003)
economy. In the paper “A Formal Proof of Marx’s Two Theorems” he tried to prove Marx's two theorems; first, the tendencial falling rate of profit and,
Nobuo_Okishio
Equations of degree 5 or higher cannot be solved by radicals
does not follow from Abel's statement of the theorem, but is a corollary of his proof, as his proof is based on the fact that some polynomials in the coefficients
Abel–Ruffini_theorem
Tauberian theorem
1007/BF01194636. "Tauberian theorems", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W. "Hardy-Littlewood Tauberian Theorem". MathWorld.
Hardy–Littlewood Tauberian theorem
Hardy–Littlewood_Tauberian_theorem
Sufficiency theorem for reconstructing signals from samples
The Nyquist–Shannon sampling theorem, or the sampling theorem, is a theorem in the field of signal processing which serves as a fundamental bridge between
Nyquist–Shannon sampling theorem
Nyquist–Shannon_sampling_theorem
Mathematical model of the physical space
the same height and base. The platonic solids are constructed. Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are
Euclidean_geometry
Shape with three sides
and this property is also sufficient to establish similarity. Some basic theorems about similar triangles are: If and only if one pair of internal angles
Triangle
reference on a given matter. In view of the high number of properties and theorems discovered by Euler, many of them are conventionally named after the first
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Description of degree sequences of graphs
partitions, and show that it is equivalent to the Erdős–Gallai theorem. Similar theorems describe the degree sequences of simple directed graphs, simple
Erdős–Gallai_theorem
Reasoning for mathematical statements
of the first known proofs of theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle
Mathematical_proof
Pair of logical equivalences
Q}{\therefore \neg (P\lor Q)}}} and expressed as truth-functional tautologies or theorems of propositional logic: ¬ ( P ∧ Q ) ↔ ( ¬ P ∨ ¬ Q ) , ¬ ( P ∨ Q ) ↔ ( ¬
De_Morgan's_laws
Theorem in physics
stronger mathematical assumptions than others. Significantly, Bell-type theorems do not refer to any particular theory of local hidden variables, but instead
Bell's_theorem
Statement on equilibrium in electromagnetism
Earnshaw's theorem states that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic
Earnshaw's_theorem
Process of changing beliefs to take into account a new piece of information
doi:10.1016/S0004-3702(96)00038-0. ISSN 0004-3702. On the Logic of Theory Base Change Proceeding JELIA '94 Proceedings of the European Conference on Logics
Belief_revision
Calculus of vector-valued functions
as a change of variables during integration. The three basic vector operators have corresponding theorems which generalize the fundamental theorem of calculus
Vector_calculus
Limit on data transfer rate
In information theory, the noisy-channel coding theorem (sometimes Shannon's theorem or Shannon's limit), establishes that for any given degree of noise
Noisy-channel_coding_theorem
Mathematical theorem, used in calculus
parts and of homeomorphic change of variables, is the most suitable to establish more complex formulae. The above theorem generalizes in the obvious
Integral_of_inverse_functions
Theorem in dimensional analysis
dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and the theorem would not hold. Leonhard Euler
Buckingham_pi_theorem
Theorem in set theory
dimensions cannot be continuous Schröder–Bernstein theorem for measurable spaces Schröder–Bernstein theorems for operator algebras Schröder–Bernstein property
Schröder–Bernstein_theorem
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