Recent Updates

Algebraic Geometry · 33 results added · with a worked companion

Commutative Algebra · 18 results added · with a worked companion

Representation Theory of Groups · 25 results added · with a worked companion

Scheme Theory · 37 results added · with a worked companion

Set Theory · 45 results added · with a worked companion

Combinatorics · 4 results added · with a worked companion

Representation Theory of Groups · 45 results added · with a worked companion

Computability Theory · 34 results added · with a worked companion

Braid Groups · 28 results added · with a worked companion

Braid Groups · 14 results added · with a worked companion

Homological Algebra · 12 results added · with a worked companion

Braid Groups · 22 results added · with a worked companion

Commutative Algebra · 22 results added · with a worked companion

Homological Algebra · 14 results added · with a worked companion

Probability · 29 results added · with a worked companion

Scheme Theory · 43 results added · with a worked companion

Representation Theory of Groups · 14 results added · with a worked companion

Commutative Algebra · 14 results added · with a worked companion

Braid Groups · 18 results added · with a worked companion

Representation Theory of Groups · 16 results added · with a worked companion

Scheme Theory · 78 results added · with a worked companion

Algebraic Topology · 31 results added · with a worked companion

Combinatorics · 8 results added · with a worked companion

Computability Theory · 24 results added · with a worked companion

Representation Theory of Groups · 25 results added · with a worked companion

Braid Groups · 36 results added · with a worked companion

Representation Theory of Groups · 16 results added · with a worked companion

Abstract Algebra · 4 results added

• Definition The standard inner product on cf(G)
• Proposition For a complex character, χ(1)=dim V, χ is a class function, and |χ(g)|≤χ(1) with equality exactly at scalars
• Example The character table of A₄
• Example The character table of S₄ and the normal subgroups it reveals

Examples & counterexamples →

Functional Analysis · 8 results added

• Theorem A bounded real linear functional on a subspace of a real normed space extends with the same norm
• Theorem A bounded complex linear functional on a subspace of a complex normed space extends with the same norm
• Theorem Every nonzero vector has a norming functional
• Corollary The dual space separates points of a normed space
• Corollary The norm of a vector is the supremum of |f(x)| over the dual unit ball
• Theorem A bounded linear functional on an arbitrary subspace extends with the same norm, without assuming the subspace is closed
• Example The abstract norming-functional theorem agrees with the concrete Lᵖ-L^q formula
• Remark Two recorded choice-theoretic questions about Hahn-Banach

Examples & counterexamples →

Complex Analysis · 21 results added

• Definition The Jacobi theta function θ(t)=∑_n∈ℤ e^-π n² t for t>0
• Theorem The Jacobi theta function satisfies θ(t)=t^-1/2θ(1/t)
• Theorem The Riemann zeta function extends meromorphically to the complex plane with its only pole at 1
• Theorem The completed zeta function satisfies Λ(s)=Λ(1-s)
• Theorem The Riemann zeta function satisfies the classical sine-gamma functional equation
• Definition The Riemann xi function ξ(s)=1/2 s(s-1)Λ(s)
• Theorem The Riemann xi function is entire of order one, real on the real axis, and symmetric under s↦1-s
• Theorem The only zeros of zeta on the nonpositive real axis are the negative even integers, and every other zero lies in the open critical strip

and 13 others on the page.

Examples & counterexamples →

Differential Geometry · 66 results added · with a worked companion

Differential Geometry · 64 results added · with a worked companion

Algebraic Topology · 27 results added · with a worked companion

Measure Theory · 1 result added

• Theorem Total-variation function of an absolutely continuous function

Fourier Analysis · 8 results added

• Definition The Wiener algebra of the circle
• Lemma A dyadic Fourier-coefficient square-sum bound for Hölder functions
• Theorem Bernstein's absolute-convergence theorem
• Definition Periodic L2 weak derivative on the circle
• Lemma Fourier coefficients of a periodic weak derivative
• Corollary One ell-2 weak derivative implies an absolutely convergent Fourier series
• Theorem Wiener's lemma for absolutely convergent Fourier series
• Example A trigonometric polynomial in the Wiener algebra

Examples & counterexamples →

Scheme Theory · 2 results added

• Example Dual numbers give a one-point nonreduced affine scheme
• Counterexample A scheme is not determined by its underlying topological space

Combinatorics · 1 result added

• Theorem Kirchhoff's matrix-tree theorem

Group Theory · 1 result added

• Remark There exist nonamenable groups without nonabelian free subgroups

Real Analysis · 3 results added

• Lemma An equicontinuous family on a compact metric space is uniformly equicontinuous
• Theorem Arzelà--Ascoli for real C(K) under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded
• Lemma Functions satisfying a fixed local Lipschitz bound somewhere form a closed subset of C([0,1])

Complex Analysis · 3 results added

• Theorem Great Picard theorem
• Corollary A meromorphic essential singularity omits at most two sphere values
• Example The function e^(1/z) omits zero and takes every nonzero value infinitely often near the origin

Examples & counterexamples →

Group Theory · 1 result added

• Example Projective linear actions and Iwasawa's hypotheses

Algebraic Topology · 1 result added

• Lemma Adem double-power comparison

Computability Theory · 1 result added

• Theorem Every polynomial-time language has polynomial-size circuits

Category Theory · 1 result added

• Theorem The two-strand braid group is infinite cyclic

Group Theory · 8 results added

• Theorem Brauer-Nesbitt determines semisimplifications
• Theorem The Cartan matrix is D^T D
• Theorem Blocks partition the ordinary and Brauer irreducible characters
• False statement FALSE: modular representations are determined by ordinary characters
• False statement FALSE: reduction mod p of an ordinary irreducible is always irreducible
• False statement FALSE: the Cartan matrix equals the decomposition matrix
• False statement FALSE: every block has one ordinary and one Brauer irreducible character
• Example The decomposition matrix of S3 in characteristic two

Examples & counterexamples →

Representation Theory of Groups · 8 results added

• Lemma Hyperelementary permutation subring reduction
• Lemma Banaschewski's prime obstruction
• Lemma Elementary detection at a fixed element
• Definition Supersolvable groups and monomial characters
• Proposition A faithful irreducible is induced from a proper inertia subgroup
• Lemma A nonabelian supersolvable group has a noncentral normal abelian subgroup
• Lemma Monomiality lifts along a quotient
• Lemma Characters of elementary groups are induced from linear characters

Combinatorics · 13 results added

• Remark Strong Perfect Graph Theorem
• Lemma Perfect vertex deletions imply 2-narrowness
• Lemma In a basic bull-free graph, an odd hole with a complete outside vertex has tightly constrained neighbors
• Lemma Replicating a vertex of a perfect graph preserves perfection
• Theorem Lovasz's numerical criterion for perfect graphs and complement invariance
• Theorem Substituting a perfect graph into a perfect graph preserves perfection
• Theorem For a vertex in a basic bull-free graph, either its neighborhood or its antineighborhood is perfect
• Theorem Every basic bull-free graph is 2-narrow

and 5 others on the page.

Examples & counterexamples →

Set Theory · 1 result added

• Theorem The continuum is equinumerous with the power set of the naturals

Lie Theory · 18 results added

• Lemma Finite filtrations by highest-weight quotients
• Lemma Finite weight-space detection of subquotients
• Theorem Every object of O has finite length
• Proposition Verma composition multiplicities are finite
• Theorem The strong linkage principle for Verma modules
• Lemma A Verma embedding implies strong linkage
• Theorem The BGG criterion for homomorphisms between Verma modules
• Proposition Restricted duality is exact and involutive on O

and 10 others on the page.

Examples & counterexamples →

Group Theory · 6 results added

• Proposition Balls of a word metric are finite if and only if the generating set is finite
• Theorem The Cayley graph of a free group with respect to a free basis is a tree
• Theorem A quasi-isometric embedding with coarsely dense image has a quasi-inverse quasi-isometric embedding
• Corollary A map is a quasi-isometry exactly when it is a quasi-isometric embedding with coarsely dense image
• Theorem Two metric spaces are quasi-isometric if and only if each contains a separated net and the two nets are bilipschitz equivalent
• Corollary A bijective quasi-isometry between word metric spaces of finitely generated groups is a bilipschitz equivalence

Combinatorics · 4 results added

• Theorem Every H-free graph has a homogeneous set of size at least 2^c√log₂ n
• Theorem Every H-free graph has a homogeneous set of size at least 2^c√log₂ n log₂ log₂ n
• Example For large n, the Fox–Sudakov choice of x leaves a dense-or-sparse set of order at least √n
• Example For large n, the log-log choice of x still leaves a dense-or-sparse set of order at least √n

Examples & counterexamples →

Computability Theory · 3 results added

• Corollary CLIQUE is NP-complete
• Theorem CLIQUE, INDEPENDENT SET, and VERTEX COVER are polynomially interreducible
• Theorem Directed Hamiltonian cycle polynomial-time many-one reduces to undirected Hamiltonian cycle

Topology · 1 result added

• Lemma A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it

Topology · 2 results revised · 9 results added

• Theorem Under Dependent Choice, every nonempty Polish space is a continuous image of Baire sequence space · revised
• Theorem Under Dependent Choice, every nonempty compact metric space is a continuous image of Cantor space · revised
• Proposition The meagre subsets of a topological space form a sigma-ideal
• Proposition Open subspaces and residual subspaces of Baire spaces are Baire
• Lemma Every open subspace of a completely metrizable space is completely metrizable
• Corollary Open and closed subspaces of a completely metrizable space are completely metrizable, and under Dependent Choice so is every G_δ subspace
• Lemma The standard weighted metric on a countable product of bounded complete metric spaces is complete
• Theorem Every separable metrizable space embeds in the Hilbert cube [0,1]^ℕ

and 3 others on the page.

Examples & counterexamples →

Topology · 5 results added

• Lemma Complete metrizability: admitting a topologically equivalent complete metric is preserved by homeomorphism and by closed subspaces, and (0,∞) has it without being complete
• Remark Completeness belongs to the metric; complete metrizability is characterised on a later page
• False statement FALSE: d(fx, fy) < d(x,y) for all x ≠ y on a complete metric space forces a fixed point
• Example The map x ↦ (x + 2/x)/2 is a contraction of [1,2] with fixed point √2, and the a priori bound gives the error after n steps
• Counterexample x ↦ x + 1/x on [1,∞) strictly decreases every distance and has no fixed point

Examples & counterexamples →

Measure Theory · 1 result added

• Theorem Riesz-Thorin interpolation theorem

Linear Algebra · 1 result added

• Proposition Before convergence, every CG denominator pₖ^*Apₖ is positive

Real Analysis · 1 result added

• Theorem Heine-Cantor in ℝ: a continuous real function on a compact subset of ℝ is uniformly continuous

Topology · 1 result added

• Example For the lower-limit line, χ=d=L=c=ℵ₀ and w=2^ℵ₀ under choice

Computability Theory · 7 results added

• Lemma The Cook--Levin construction can be made parsimonious
• Definition MajoritySAT
• Theorem PP is strict sign testing for Gap-P
• Proposition NP is contained in P with a Sharp-P oracle
• False statement Every decision many-one reduction is parsimonious
• Example A two-branch computation and its parsimonious tableau formula
• Example Deciding satisfiability by a NumberSAT oracle query

Examples & counterexamples →

Topology · 5 results added

• Example Trivial coverings are products with a discrete fibre
• Example The quotient ℝ→ℝ/ℤ is a covering with integer translations as deck transformations
• Example The maps [x]↦[mx] on ℝ/ℤ are m-sheeted coverings for m≥1
• Example Pulling a covering back to an evenly covered open set gives a trivial covering
• Example The Hawaiian earring is locally path-connected but has no universal cover

Algebraic Topology · 8 results added

• Lemma A compact subspace of a CW complex meets only finitely many cells
• Corollary The image of a compact space lies in a finite CW subcomplex
• Theorem Relative homology of consecutive CW skeleta
• Lemma Homology of an infinite CW complex is the colimit of skeletal homology
• Theorem Cellular homology computes singular homology
• Proposition Cellular maps induce cellular chain maps
• Example Cellular homology of an infinite-dimensional projective space
• Counterexample The Hawaiian earring is not a CW complex with its punctured circles as cells

Examples & counterexamples →

Measure Theory · 12 results added

• Lemma Continuous compactly supported functions are translation-continuous in Lᵖ
• Lemma Borel representatives make the convolution integrand Borel measurable
• Lemma Convolution on L¹(ℝⁿ) is independent of the chosen Borel representatives
• Theorem If f,g ∈ L¹(ℝⁿ), then f*g exists almost everywhere, belongs to L¹, and ‖f*g‖₁ ≤ ‖f‖₁ ‖g‖₁
• Proposition Convolution on L¹(ℝⁿ) is bilinear, commutative, and associative
• Theorem The support of a convolution lies in the closure of the support sumset
• Lemma For 1 < p < ∞, the Lᵖ norm of a nonnegative function is the supremum of its pairings with L^q unit vectors
• Theorem Young's convolution inequality under Countable Choice

and 4 others on the page.

Examples & counterexamples →

Commutative Algebra · 58 results added

• Lemma Depth is infinite when the ideal acts surjectively
• Lemma A regular element exists by prime avoidance
• Corollary Depth zero and associated primes
• Theorem The local depth-zero associated-prime criterion
• Lemma A maximal regular sequence stops at an associated prime
• Lemma Maximal regular sequences have a common Ext length
• Corollary Depth as the first nonzero Ext degree
• Theorem Depth equals the maximal regular-sequence length

and 50 others on the page.

Examples & counterexamples →

Homological Algebra · 3 results added

• Theorem A bounded above complex of projectives is homotopically projective
• Proposition Homology of the derived tensor product is tor
• Example ex-derived-tensor-of-two-cyclic-abelian-groups.md

Examples & counterexamples →

Linear Algebra · 1 result added

• Definition Row swaps, arbitrary row scalings and row additions over a commutative ring, with reversible elementary cases distinguished

Computability Theory · 1 result added

• Theorem Halting on every input is not recognizable

Algebraic Geometry · 11 results added

• Lemma Function fields and dominant pullbacks on general varieties
• Corollary Dimension is birationally invariant
• Theorem A nontrivial principal section has pure codimension one
• Lemma r equations lower dimension by at most r
• Lemma Closed-point local dimension equals ambient irreducible dimension
• Lemma A point is locally cut out by dim X functions
• Lemma Zero-dimensional varieties are finite sets
• Lemma Dominant affine images contain a principal open

and 3 others on the page.

Number Theory · 3 results added

• Theorem The principal Dirichlet L-function factors through zeta
• Lemma A real nonprincipal Dirichlet L-function is nonzero at one
• Theorem Primes in one reduced residue class have Dirichlet density 1 over phi(q)

Functional Analysis · 19 results added

• Theorem The dual of a closed subspace is a dual quotient
• Corollary Distance to an annihilator is the restriction norm
• Lemma The transpose is bounded with the same norm
• Lemma Elementary kernel and range annihilator identities
• Corollary Dense range is equivalent to injectivity of the transpose
• Theorem The canonical bidual map is an isometry
• Corollary Distance to a closed subspace via unit annihilators
• Lemma The canonical map is natural

and 11 others on the page.

Examples & counterexamples →

Computability Theory · 3 results added

• Definition Constraint graph and labeling value
• Lemma Constraint expander overlay
• Definition Graph power and walk constraint

Scheme Theory · 2 results added

• Lemma Affineness is local on the target
• Lemma Base change and composition of affine morphisms

Set Theory · 2 results added

• Definition Ultrafilter
• False statement FALSE, once the ultrafilter lemma is available: every ultrafilter is principal

Combinatorics · 1 result added

• Corollary The symmetric Lovász Local Lemma under ep(d+1)≤1

Combinatorics · 1 result added

• Theorem A finite family of events is mutually independent exactly when its indicators are mutually independent

Real Analysis · 1 result added

• Lemma A C¹ map uniformly close to the identity derivative sandwiches a cube between contracted and expanded cubes

Topology · 2 results added

• Definition Locally compact metric space: every point has a compact neighbourhood
• Example Dini's theorem applied to a nondecreasing sequence of piecewise linear approximations on [0,1], and what fails when the limit is not continuous

Examples & counterexamples →

Group Theory · 6 results added

• Lemma Cobounded proper geodesic actions produce finite generating sets
• Theorem The Svarc-Milnor lemma
• Definition The homogeneous dimension of a finitely generated nilpotent group
• Remark Intermediate growth refutes universal polynomial growth (recorded)
• Example Free groups act geometrically on regular trees
• Example The discrete Heisenberg group has growth degree four

Examples & counterexamples →

Functional Analysis · 12 results added

• Theorem Separate a point from an open convex set
• Theorem Separation of disjoint convex sets when one is open
• Theorem Strong separation of a closed and a compact convex set
• Corollary A closed convex set is an intersection of closed half-spaces
• Theorem Geometric Hahn--Banach theorem for subspaces
• Corollary The annihilator detects the closure of a subspace
• Corollary Density is characterized by a zero annihilator
• Corollary Finite-dimensional subspaces are complemented

and 4 others on the page.

Examples & counterexamples →

Differential Topology · 4 results added

• Proposition Every Morse function admits a complete downward gradient-like field on a closed manifold
• Lemma Precompact trajectory tails have nonempty compact connected flow-invariant limit sets
• Lemma A negative-gradient trajectory on a compact Morse manifold has single critical alpha and omega limits
• Definition A Morse trajectory from one critical point to another

Homological Algebra · 4 results added

• Theorem Shapiro lemma for group cohomology
• Theorem Shapiro lemma for group homology
• False statement Shapiro lemma needs no induction/coinduction distinction
• Example Shapiro lemma for the trivial subgroup

Examples & counterexamples →

Lie Theory · 33 results added

• Proposition The Killing form is invariant and nondegenerate on a complex semisimple Lie algebra
• Theorem PBW gives an ordered monomial basis for the enveloping algebra
• Theorem Root-space decomposition relative to a Cartan subalgebra
• Proposition Brackets of root spaces add their roots
• Proposition Under Choice, the Killing form pairs only opposite root spaces
• Definition The Killing-dual vector attached to a root
• Proposition Opposite root spaces bracket to the Killing-dual line
• Theorem Triangular decomposition from a chosen positive root system

and 25 others on the page.

Examples & counterexamples →

Partial Differential Equations · 13 results added

• Definition Spherical averages and local ball means in Rn
• Lemma Sphere and ball measures scale in Rn
• Lemma Radial derivative of a spherical average
• Theorem Spherical mean-value property for harmonic functions
• Corollary Ball mean-value property for harmonic functions under Countable Choice
• Lemma Radial mollification fixes local mean-value functions
• Theorem Continuous ball-mean-value functions are harmonic
• Corollary A pointwise local ball mean property is enough

and 5 others on the page.

Examples & counterexamples →

Measure Theory · 1 result added

• False statement Hausdorff measure is countably additive on every subset

Number Theory · 1 result added

• Theorem Hasse-Minkowski theorem over Q

Algebraic Topology · 12 results added

• Theorem Singular homology satisfies homotopy exactness and excision
• Lemma Oriented simplex comparison for an ordinary homology theory
• Lemma Subdivision compatible continuous polyhedral homology comparison
• Lemma Finite cw pairs admit finite simplicial homotopy models
• Lemma Finite dimensional axiomatic homology has finite subcomplex support
• Theorem Eilenberg steenrod uniqueness on finite dimensional cw pairs
• Lemma Additivity and compact cell support control the infinite cw colimit
• Theorem Eilenberg steenrod uniqueness on all cw pairs

and 4 others on the page.

Examples & counterexamples →

Lie Theory · 9 results added

• Lemma Every Verma module contains a simple Verma submodule
• Proposition Simple-reflection embeddings of Verma modules
• Theorem Verma embedding for an arbitrary positive root
• Theorem The Jantzen sum formula for a Verma module
• Corollary Generic Verma modules are simple
• Corollary Antidominant regular Verma modules are simple
• Example The sl2 Verma embedding chain
• Example The A2 regular integral-dominant Verma embedding poset

and 1 other on the page.

Examples & counterexamples →

Topology · 1 result added

• Example A one-point space and ℝ are homotopy equivalent but not homeomorphic

Group Theory · 22 results added

• Theorem Slim triangles, the Gromov product, and the four-point condition are equivalent up to constants
• Theorem Morse stability of quasi-geodesics
• Theorem Hyperbolicity is a quasi-isometry invariant of geodesic spaces
• Theorem Hyperbolicity of a finitely generated group is independent of the finite generating set
• Proposition Free abelian groups of rank at least two are not hyperbolic
• Theorem Hyperbolic groups admit finite Dehn presentations
• Corollary Hyperbolic groups have solvable word problem
• Theorem Linear isoperimetric characterisation of hyperbolic groups

and 14 others on the page.

Examples & counterexamples →

Abstract Algebra · 4 results added

• Definition Virtual characters and the character ring R(G) of a finite group
• Theorem The values of a central character are algebraic integers
• Lemma An algebraic-integer average of roots of unity is either 0 or a common root of unity
• Example Inducing the trivial character of a subgroup of order two in S₃ gives 1 plus an irreducible degree-two character

Examples & counterexamples →

Probability · 2 results added

• Example Independent but non-identically distributed coordinates
• Example A canonical random walk from product increments

Differential Geometry · 15 results added

• Lemma Finite chart localization of compactly supported top forms, including boundary
• Proposition Linearity and additivity of the form integral
• Proposition Positivity of the oriented integral
• Theorem Change of variables on oriented manifolds
• Proposition Integration on an oriented embedded submanifold
• Definition Integral of a compactly supported smooth density
• Theorem Orientation-free density integration and its properties
• Proposition Orientation identifies top forms with signed densities

and 7 others on the page.

Examples & counterexamples →

Computability Theory · 2 results added

• Definition The graph-nonisomorphism interactive protocol
• Theorem Graph nonisomorphism is in IP

Group Theory · 1 result added

• Theorem Every finitely generated free group is subgroup separable

Computability Theory · 8 results added

• Definition Conditional Kolmogorov complexity
• Theorem Kraft inequality and effective prefix-code allocation
• Lemma Choice-free fair-coin content on open subsets of binary sequence space
• Definition Effectively open sets in Cantor space
• Definition Martin-Löf tests and random sequences
• Lemma A universal Martin-Löf test exists
• Theorem Martin-Löf randomness implies computable randomness
• Theorem Effective dimension is the liminf prefix-complexity rate

Computability Theory · 3 results added

• Definition Logspace many-one reduction
• Theorem Circuit Value is P-complete
• False statement NL equals coNL follows by swapping accepting and rejecting states

Partial Differential Equations · 13 results added

• Theorem Strong maximum principle for harmonic functions
• Theorem Harnack inequality on a ball
• Corollary Harnack inequality on compact subsets
• Theorem Harnack convergence principle
• Lemma Smooth sphere data have a harmonic replacement under Countable Choice
• Lemma Derivative estimate proof of one sided harmonic liouville
• Theorem Liouville theorem for bounded harmonic functions
• Corollary Positive entire harmonic functions are constant

and 5 others on the page.

Examples & counterexamples →

Differential Geometry · 14 results added

• Lemma Agreement of Borel overlap integrals
• Theorem Intrinsic density measure and its chart restriction
• Theorem Positive smooth densities give Radon volume
• Theorem Measurable integration extends smooth density integration
• Corollary Positive open-set and metric-ball volume
• False statement False: locally finite volume has finite total mass
• False statement False: density measures require an orientation
• Example Euclidean volume from chart gluing

and 6 others on the page.

Examples & counterexamples →

Measure Theory · 3 results added

• Proposition An indicator function is measurable exactly when its set is measurable
• Theorem Doob-Dynkin factorization through the sigma-algebra generated by a function
• False statement FALSE: if the absolute value is measurable, then the function is measurable

Recorded, Not Proved Here · 3 results added

• Remark Dominated convergence theorem
• Remark The Vitali set: a non-measurable subset of ℝ
• Remark Sierpiński 1938: no free ultrafilter on ℕ is measurable or has the Baire property

Topology · 3 results added

• Theorem Under choice, a space is metrizable if and only if it is regular, T₁, and has a σ-locally-finite basis
• Lemma A closure-controlled locally finite open cover transfers relative σ-locally-finite bases to the whole space
• Theorem Under choice, a space is metrizable if and only if it is paracompact, Hausdorff, and locally metrizable

Real Analysis · 1 result added

• Example A second-order Taylor polynomial computed from gradient and Hessian data

Probability · 3 results added

• Lemma The almost-sure convergence event is measurable
• Theorem Convergence in probability is metrized by d₀
• Theorem Dominated convergence in Lᵖ

Group Theory · 4 results added

• Definition The Jacobson radical of a finite-dimensional algebra is the intersection of its maximal left ideals
• Theorem For a finite-dimensional algebra, the Jacobson radical is nilpotent and the quotient by it is semisimple
• Theorem Finite-dimensional kG-modules decompose as finite direct sums of indecomposables uniquely up to order and isomorphism
• Definition A module is relatively H-projective when it is a direct summand of one induced from H

Category Theory · 1 result added

• Example On finite sets the ultrafilter monad is naturally isomorphic to the identity; assuming the ultrafilter lemma, its unit is not invertible on the natural numbers

Real Analysis · 1 result added

• Lemma An additive f : ℝ → ℝ satisfies f(0) = 0, f(-x) = -f(x) and f(qx) = q f(x) for every rational q and every real x; in particular f(q) = q f(1) at every rational q

Differential Topology · 12 results added

• Theorem Every smooth function admits arbitrarily fine strong-topology perturbations whose differential is transverse to zero, supported away from a closed set where transversality already holds
• Theorem In the strong C^∞ topology on C^∞(M,ℝ), the Morse functions form a residual subset
• Theorem On a compact smooth manifold, the Morse functions form an open dense subset in the C² and hence C^∞ topology
• Lemma For a compact Morse function, disjoint local bump perturbations can separate finitely many equal critical values without changing the Hessians
• Theorem On a compact smooth manifold, the excellent Morse functions form an open dense subset of the C² and hence C^∞ topology
• Theorem For a compact manifold embedded in Euclidean space, the restricted linear height is Morse for generic directions
• Theorem For a compact manifold embedded in Euclidean space, the squared-distance function from a generic center is Morse
• Corollary Every compact smooth manifold admits an excellent Morse function

and 4 others on the page.

Examples & counterexamples →

Functional Analysis · 2 results added

• Example c₀ is Banach for the supremum norm
• Counterexample An incomplete normed subspace need not be closed

Recorded, Not Proved Here · 4 results added

• Remark Is there a Dowker space of cardinality ℵ₁ in ZFC? (reported open in 2022)
• Remark The square of a Suslin line is not ccc (checked on audit)
• Remark Hahn-Banach and a Hamel basis for ℝ: a recorded question
• Remark Hahn-Banach and discontinuous additive maps: a recorded question

Computability Theory · 3 results added

• Lemma An everywhere-total functional has a computable use bound
• False statement Degree join is set union
• Example Changing representatives of a degree comparison

Examples & counterexamples →

Set Theory · 1 result added

• Definition Well-order and well-ordered set

Computability Theory · 1 result added

• Example Lexicographic certificate search uses polynomial space and exponential time

Number Theory · 12 results added

• Definition The Riemann zeta zero-counting function
• Theorem The Riemann--von Mangoldt zero count
• Corollary A unit-interval bound for zeta zeros
• Lemma A local formula for the logarithmic derivative of zeta
• Lemma A left-half-plane bound for the logarithmic derivative of zeta
• Lemma Residues in the von Mangoldt contour shift
• Theorem A smoothed von Mangoldt explicit formula
• Theorem The truncated von Mangoldt explicit formula

and 4 others on the page.

Examples & counterexamples →

Combinatorics · 3 results added

• Lemma A plane graph has finitely many faces and exactly one unbounded face
• Lemma Face frontiers are unions of whole edges; a cycle edge borders two faces and a bridge borders one
• Proposition Every plane forest has exactly one face

Number Theory · 1 result added

• Theorem Integral ideal factorisation in a number field, in ZF

Number Theory · 2 results added

• Theorem Analytic continuation of primitive Dirichlet L-functions
• Example Gauss sum for the character modulo 4

Examples & counterexamples →

Probability · 3 results added

• Theorem Finite probability spaces are exactly finite full-power-set probability spaces
• Theorem Probability laws correspond to distribution functions
• Lemma Variance and covariance identities for random variables

Measure Theory · 9 results added

• Theorem For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique
• Theorem Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
• Theorem Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
• Theorem On Borel subsets of Rᵐ⁺ⁿ, the product lambdaₘ times lambdaₙ agrees with lambdaₘ₊ₙ
• Theorem For 0 < p < infinity, the layer-cake formula computes the integral of |f|ᵖ from the distribution function
• Theorem Polar coordinates decompose Lebesgue measure into rⁿ⁻¹ dr d sigma
• Lemma A C¹ diffeomorphism maps Lebesgue null sets to Lebesgue null sets
• Theorem A C¹ diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions

and 1 other on the page.

Examples & counterexamples →

Algebraic Geometry · 5 results added

• Lemma The Zariski topology on an affine product is generally not the product topology
• Definition The Segre point map from two projective spaces
• Lemma The Veronese map is a well-defined closed immersion
• Theorem The Plucker image is a closed projective algebraic set
• Lemma A classical fibre is base change to a point

Algebraic Geometry · 2 results added

• Lemma projective regular function chart compatibility
• Definition morphism to projective space homogeneous coordinates

Combinatorics · 2 results revised · 14 results added

• Corollary For every forest H, graphs excluding H and H̄ have the Erdős-Hajnal property · revised
• Corollary Deleting a leaf and a co-leaf preserves the Erdős-Hajnal property of a finite forbidden family · revised
• Theorem Every H-free graph has a polynomial-size pure pair
• Definition Coherent graphs and support-regular blockades
• Lemma A long blockade has a wide support-invariant, support-uniform minor
• Lemma A wide concave support-regular blockade contains a rainbow rooted tree
• Theorem Every forest-free graph has a linear anticomplete pair or a linear-degree vertex
• Theorem For every forest H, graphs excluding H and H̄ have a linear pure pair

and 8 others on the page.

Examples & counterexamples →

Partial Differential Equations · 1 result added

• Remark Cauchy–Kovalevskaya for a noncharacteristic analytic Cauchy problem

Commutative Algebra · 63 results added

• Lemma embedding dimension is minimal maximal ideal generator number
• Theorem dimension at most embedding dimension
• Lemma regular system of parameters equivalent basis
• Lemma associated graded polynomial surjection
• Lemma regular local graded surjection has zero kernel
• Theorem associated graded ring of a regular local ring
• Corollary regular local hilbert samuel multiplicity one
• Lemma regular local domain induction

and 55 others on the page.

Examples & counterexamples →

Algebraic Topology · 5 results added

• Theorem Barycentric subdivision is a chain map
• Theorem Simplicial and singular homology agree
• Corollary Homology of spheres
• Example Relative homology of a disk and its boundary
• Example Mayer–Vietoris computation of first homology of the torus

Examples & counterexamples →

Computability Theory · 4 results added

• Definition Worst-case time and space complexity of a machine
• Lemma Any machine that fully reads every input of length n needs at least linear time
• Lemma An eventually positive time bound yields the same-order space bound
• Lemma A constructible time bound yields a uniformly clocked simulator

Differential Geometry · 1 result added

• Proposition First variation of volume for a normal variation

Real Analysis · 3 results added

• Remark Conventions of this page, the standing n ≥ 1 hypothesis, and what is taken up elsewhere in the reading order
• Counterexample ‖·‖₁ on ℝ² violates the parallelogram law, so no symmetric bilinear form induces it
• False statement FALSE: if a convergent series in ℝⁿ does not converge absolutely, then every point of ℝⁿ is the sum of some rearrangement of it

Examples & counterexamples →

Computability Theory · 2 results added

• Lemma The standard one-tape simulation of a fixed multitape machine has quadratic overhead
• Definition A universal Turing machine relative to a fixed machine encoding

Scheme Theory · 1 result added

• Lemma Finite type is affine-local on source and target

Representation Theory of Groups · 13 results added

• Lemma Base change for intertwiner spaces
• Lemma The character field is the stabilizer fixed field
• Theorem Absolute irreducibility via the endomorphism division algebra
• Definition The Schur index of an irreducible character
• Lemma The Schur index is independent of the splitting field
• Theorem Character formula over a nonsplitting field
• Corollary The Schur index divides the representation degree
• Theorem The Schur index equals the division-algebra index

and 5 others on the page.

Examples & counterexamples →

Group Theory · 26 results added

• Lemma Low-degree sequence of a free presentation
• Theorem Hopf formula for the Schur multiplier
• Corollary Hopf formula is presentation-independent
• Corollary Multiplier of a finitely presented group
• Proposition Multiplier of a free group
• Proposition Multiplier of a cyclic group
• Theorem Multiplier of an abelian group
• Theorem Existence criterion for universal central extensions

and 18 others on the page.

Examples & counterexamples →

Group Theory · 13 results added

• Theorem H² classifies extensions with fixed abelian kernel action
• Corollary The zero H² class is equivalent to splitting
• Theorem The Baer sum agrees with addition in H²
• Corollary Central extensions are classified by H² with trivial action
• Theorem Extension-theoretic interpretation of the standard five-term exact sequence
• Theorem Nonabelian extension obstruction and torsor under choice
• Remark Nonabelian extension obstruction in H³
• False statement FALSE: H² classifies extensions with arbitrary nonabelian kernel

and 5 others on the page.

Examples & counterexamples →

Topology · 1 result added

• Example The particular-point topology is T₀, it is not T₁ and not regular once the set has at least two points, and it is not normal once the set has at least three

Recorded, Not Proved Here · 2 results added

• Remark Martin's Axiom
• Remark The normal Moore space conjecture and its large-cardinal consistency strength

Algebraic Topology · 1 result added

• Proposition Zero-th simplicial homology is free on connected components

Real Analysis · 1 result added

• Lemma The sine and cosine power series converge absolutely for every real argument

Differential Geometry · 2 results added

• Proposition Smooth maps are continuous
• Proposition Identity maps and composites of smooth maps are smooth

Differential Geometry · 13 results added

• Lemma A locally finite positive smooth family normalizes to a partition of unity
• Lemma Every open cover of a manifold has a countable relatively compact coordinate-ball subcover
• Lemma A countable coordinate-ball cover has a countable locally finite shrinking
• Theorem Smooth partitions of unity exist on manifolds
• Corollary Smooth partitions subordinate to a countable coordinate cover
• Corollary Smooth functions separate points from closed sets
• Lemma Every manifold has a compact exhaustion
• Corollary Every closed subset of a manifold is the zero set of a smooth nonnegative function

and 5 others on the page.

Examples & counterexamples →

Topology · 1 result added

• Counterexample The hyperbola {(x,y) : xy = 1} is closed in ℝ² and its image under the first projection is ℝ ∖ {0}, which is not closed

Computability Theory · 4 results added

• Theorem Sigma₁⁰ sets are exactly the computably enumerable sets
• Theorem The totality set is Pi₂⁰-complete
• False statement False: a limit-computable function has a known stabilization stage
• Example A limit approximation computed from the halting oracle

Examples & counterexamples →

Topology · 1 result added

• Corollary Real and finite-dimensional Euclidean Ascoli–Arzelà criteria

Functional Analysis · 5 results added

• Lemma The closure of a bounded image contains a ball
• Lemma Successive approximation turns a closure-ball inclusion into an actual preimage
• Theorem Open mapping theorem
• Corollary Quantitative lifting form of the open mapping theorem
• Theorem Bounded inverse theorem

Real Analysis · 2 results added

• Lemma The binomial theorem over the complex field
• Lemma The complex exponential series converges absolutely for every complex argument

Real Analysis · 2 results added

• Theorem Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
• Example For a natural n ≥ 1, the derivative of x ↦ x^1/n on (0,∞) is 1/ι(n)x^1/n - 1, obtained from the inverse rule applied to x ↦ xⁿ; in particular (√x)' = 1/(ι(2)√x)

Examples & counterexamples →

Real Analysis · 1 result added

• Example The inverse-square field is divergence free, and its flux through the sphere bounding the translated unit ball vanishes

Measure Theory · 3 results added

• Theorem Every g∈ L^q(μ) defines a bounded linear functional on Lᵖ(μ)
• Proposition The functional Λ_g has norm ‖g‖_q; for q=∞ assume μ is semifinite
• Corollary The Lᵖ norm is the supremum of pairings against unit L^q functions

Differential Geometry · 1 result added

• Proposition Cartan commutator identities

Group Theory · 8 results added

• Lemma Quasisimple normal intersections and perfect central actions
• Lemma A normal quasisimple subgroup and a subnormal subgroup
• Lemma Distinct components commute
• Theorem The generalized Fitting subgroup contains its centralizer
• Remark What the library proves and does not prove about CFSG
• False statement The library proves CFSG
• Example A5 as the smallest nonabelian simple group
• Example PSL(2,7) and a low-rank family entry

Examples & counterexamples →

Real Analysis · 1 result added

• Definition The Henstock–Kurzweil integral on a compact interval

Complex Analysis · 3 results added

• Definition Noetherian commutative rings and modules
• Lemma Finite modules over Noetherian rings are Noetherian
• Lemma Noetherian domains are atomic

Measure Theory · 16 results added

• Theorem Every nonnegative measurable function is the increasing limit of simple measurable functions
• Proposition The nonnegative integral agrees with the simple integral on simple functions
• Theorem The indefinite integral of a nonnegative simple function is a measure
• Theorem Monotone convergence for the integral
• Corollary Additivity of the nonnegative Lebesgue integral
• Theorem The indefinite integral of a nonnegative measurable function is a measure
• Theorem Integrating against a density agrees with integrating the product
• Theorem Fatou's lemma

and 8 others on the page.

Measure Theory · 6 results added

• Definition The Radon-Nikodym derivative as an almost-everywhere equivalence class
• Theorem Integrating against a Radon-Nikodym derivative recovers integration against the measure
• Theorem The total variation of an absolutely continuous signed or complex measure has density the absolute value of the Radon-Nikodym derivative
• Proposition The Radon-Nikodym derivative is integrable exactly when the absolutely continuous part is finite
• Theorem For finite signed or complex measures, absolute continuity is equivalent to the epsilon-delta small-set condition
• Counterexample x⁻¹dλ on (0,1) shows finiteness is needed in the epsilon-delta criterion

Examples & counterexamples →

Real Analysis · 1 result added

• Theorem Lebesgue's criterion in ℝᵐ: a bounded function on a closed nondegenerate rectangle is Riemann integrable iff its discontinuity set is null

Algebraic Topology · 1 result added

• Theorem Serre-class transfer through a simply connected fibration

Linear Algebra · 2 results added

• Theorem Over a perfect field, every endomorphism has a unique commuting semisimple-plus-nilpotent decomposition, polynomial in the endomorphism
• Corollary The operator norm is submultiplicative and satisfies ||T^*T|| = ||T||²

Computability Theory · 9 results added

• Lemma The time-diagonal language respects its budget
• Theorem The deterministic time hierarchy theorem
• Corollary P is properly contained in EXP
• Theorem The nondeterministic time hierarchy theorem
• Lemma The read-only-workspace diagonal machine halts
• Theorem The read-only-workspace space hierarchy theorem
• Corollary L is properly contained in PSPACE
• Theorem Gap and union theorems for abstract complexity bounds

and 1 other on the page.

Topology · 2 results added

• Definition Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not
• Example On an infinite set the cofinite topology has every infinite subset dense and no two nonempty open sets disjoint

Examples & counterexamples →

Real Analysis · 2 results added

• Definition The classical Weierstrass function
• Example Tangent identifies a bounded incomplete interval with the unbounded complete real line

Examples & counterexamples →

Homological Algebra · 16 results added

• Lemma The homological universal-coefficient Tor obstruction map
• Theorem The universal coefficient theorem for homology over a PID
• Theorem The homology universal-coefficient sequence splits nonnaturally
• Lemma The cohomological universal-coefficient extension map
• Theorem The universal coefficient theorem for cohomology over a PID
• Corollary Cohomology over a field is dual to homology for finite-dimensional complexes
• Lemma The Kunneth Tor map
• Corollary Kunneth over a field

and 8 others on the page.

Examples & counterexamples →

Topology · 1 result added

• Example A continuous function on [0,1] ⊆ ℝ extended to all of ℝ, both by Tietze and by hand

Differential Geometry · 14 results added

• Theorem Derivations of smooth functions are exactly smooth vector fields
• Lemma A vector field along an embedded submanifold extends to a neighbourhood and globally when the submanifold is closed
• Proposition Coordinate formula for the Lie bracket
• Theorem Smooth vector fields form a Lie algebra under the Lie bracket
• Proposition Related vector fields have related Lie brackets
• Proposition Coordinate vector fields commute
• Theorem The fundamental theorem on flows
• Theorem Compactly supported smooth vector fields are complete

and 6 others on the page.

Examples & counterexamples →

Lie Theory · 8 results added

• Proposition Weights of a Verma module lie below lambda
• Lemma Preliminary factorization of a Shapovalov determinant
• Lemma The generic radical on a Shapovalov factor hyperplane
• Lemma The first transverse Shapovalov pairing is perfect
• Theorem The Shapovalov determinant formula
• Corollary The Verma irreducibility criterion from Shapovalov determinants
• Example Reducible and generic sl2 Verma modules
• Example The finite-dimensional sl2 quotient of a Verma module

Examples & counterexamples →

Differential Geometry · 23 results added

• Lemma A finite coordinate-bump map embeds a compact manifold in some Euclidean space
• Lemma A countable coordinate-bump map embeds a manifold in countable Euclidean data
• Lemma A smooth exhaustion separates the locally finite chart bands
• Lemma A generic linear projection preserves injectivity and immersion
• Lemma A generic projection can preserve properness
• Theorem The weak Whitney proper embedding theorem
• Corollary A closed Euclidean submanifold has a smooth neighborhood retraction
• Proposition Nearest-point projection is the tubular retraction after shrinking

and 15 others on the page.

Examples & counterexamples →

Functional Analysis · 25 results added · with a worked companion

Algebraic Topology · 30 results added · with a worked companion

Group Theory · 1 result added

• Proposition A recursive Dehn function yields a solution to the word problem

Homological Algebra · 1 result added

• Theorem Chain-homotopic maps induce the same map on homology

Number Theory · 18 results added

• Lemma Zeta logarithmic derivative zero bound
• Theorem Riemann zeta classical zero free region
• Lemma Zeta horizontal logarithmic derivative comparison
• Theorem Zeta bounds in classical zero free region
• Corollary Zeta zero count near the one line
• Lemma Zeta reciprocal zero sum bound
• Lemma Zeta explicit formula zero free error balance
• Theorem Chebyshev psi prime number theorem error

and 10 others on the page.

Examples & counterexamples →

Lie Theory · 6 results added

• Lemma Reflection basic invariants form a regular sequence
• Lemma Weyl coinvariant hilbert series has order w dimension
• Theorem Chevalley shephard todd for finite weyl groups
• Lemma Local Chevalley restriction for Kostant freeness
• Theorem Kostant harmonic decomposition of the symmetric algebra
• Example A2 coinvariant algebra and basic invariants

Examples & counterexamples →

Commutative Algebra · 4 results added

• Theorem A finite flat module over a local ring is free
• Corollary A finite flat module over a Noetherian ring is finite projective
• Example A residue-field basis lifts to a basis of a finite flat module over a local ring
• Example A residue-field basis lifts to a basis of a finite flat module over a local ring

Examples & counterexamples →

Differential Geometry · 3 results added

• Theorem The universal cover of a closed hyperbolic surface is the hyperbolic plane with geometric deck action
• Example Great circles as round-sphere geodesics
• Example Closed surface groups are hyperbolic

Examples & counterexamples →

Group Theory · 1 result added

• Remark Nonabelian extension obstructions live in H³ and realized classes form an H²-torsor

Commutative Algebra · 1 result added

• Lemma The parameter power-series map is injective by dimension

Algebraic Topology · 1 result added

• Theorem Relative cellular homology computes relative singular homology

Commutative Algebra · 1 result added

• Lemma Finite prime chains lift through module-finite domain extensions without Choice

Combinatorics · 1 result added

• Lemma Leaf-reducible families yield a large anticomplete pair or a deeper restricted induced subgraph

Commutative Algebra · 34 results added

• Lemma The symbolic-power step inside the principal ideal theorem
• Theorem Krull's principal ideal theorem
• Corollary A Noetherian local domain has dimension zero exactly when it is a field
• Corollary A minimal prime over a principal nonzerodivisor has height one
• Lemma Choose the first generator's minimal prime inside the target prime
• Lemma Quotienting by the first minimal prime reduces the remaining height count
• Theorem Krull's height theorem
• Corollary Height is bounded by the minimal number of local generators

and 26 others on the page.

Examples & counterexamples →

Fourier Analysis · 1 result added

• Corollary L-p convergence of a lacunary series is equivalent to ell-two coefficients

Measure Theory · 1 result added

• Corollary A bounded function on a closed bounded interval, or on a closed nondegenerate rectangle, is Riemann integrable exactly when its discontinuity set has Lebesgue measure zero

Category Theory · 1 result added

• Lemma Horizontally pasted commutative squares commute

Real Analysis · 1 result added

• Remark Conventions of this page, and which sharpenings of the integral are taken up later in the reading order

Commutative Algebra · 15 results added

• Theorem A finite graded module over a standard graded algebra has rational Hilbert series and eventual polynomial growth
• Theorem The Hilbert-Samuel function agrees eventually with a rational polynomial in binomial form
• Definition Hilbert-Samuel multiplicity as the factorial-scaled leading coefficient
• Theorem The degree of the Hilbert-Samuel polynomial equals the dimension of the support
• Theorem For a finite module, the dimension is the least size of an ideal of definition, and such tuples are systems of parameters
• Theorem Hilbert-Samuel leading coefficients are additive at the top polynomial degree
• Corollary For a nonzero finite module and an ideal of definition, Hilbert-Samuel multiplicity is a positive integer
• Example The polynomial ring and a homogeneous quotient have the expected Hilbert series and Hilbert polynomial

and 7 others on the page.

Examples & counterexamples →

Functional Analysis · 1 result added

• Corollary Separable reflexive space has separable dual

Differential Geometry · 23 results added

• Definition Null subsets of a smooth manifold
• Lemma C¹ local diffeomorphisms preserve null sets locally
• Proposition The null-set definition is independent of the smooth atlas
• Proposition A countable chart cover detects manifold null sets
• Proposition A null set has dense complement in a positive-dimensional manifold
• Proposition An equidimensional C¹ map sends null sets to null sets
• Proposition The image of a lower-dimensional C¹ manifold is null
• Corollary Positive-codimension immersed submanifolds are null

and 15 others on the page.

Examples & counterexamples →

Group Theory · 7 results added

• Theorem A word is trivial in a presented group exactly when it bounds a finite van Kampen diagram
• Theorem Minimal van Kampen area agrees with minimal algebraic relator area
• Theorem In a reduced C prime(1/6) null diagram, some face contributes more than half of its boundary to the outer boundary
• Corollary Finite C prime(1/6) presentations have solvable word problem
• Corollary Finite C prime(1/6) presentations satisfy a linear isoperimetric inequality
• Theorem In a C prime(1/6) group, every nontrivial torsion element is conjugate to a power of a relator root
• Corollary A C prime(1/6) presentation with no proper-power relators defines a torsion-free group

Combinatorics · 2 results added

• Corollary The bull graph has the Erdős-Hajnal property
• Corollary The two six-vertex prime H-graphs have the Erdős-Hajnal property

Differential Geometry · 7 results added

• Theorem The pullback fibre product is a smooth vector bundle
• Theorem Dual and Hom transition functions define smooth bundles
• Proposition Sections of Hom are the same as smooth fibrewise linear maps
• Theorem Every smooth vector bundle admits a smooth bundle metric
• Corollary Every vector subbundle has a smooth complement
• Corollary Every short exact sequence of smooth vector bundles splits
• False statement A short exact sequence of vector bundles has a canonical splitting

Group Theory · 4 results added

• Remark The O'Nan-Scott classification of finite primitive groups (recorded)
• Remark CFSG in the cited O'Nan-Scott proof and later refinements
• False statement FALSE: the cited LPS O'Nan-Scott proof uses no CFSG consequence
• Example A simple diagonal action

Examples & counterexamples →

Differential Topology · 1 result revised · 10 results added

• Theorem Morse--Smale metrics are residual for a fixed Morse function · revised
• Theorem Stable and unstable manifolds for a Morse gradient with an arbitrary smooth metric
• Lemma A half-line first-order operator with a hyperbolic limit has a right inverse
• Lemma Fredholm index and range of an asymptotically hyperbolic first-order operator
• Lemma Morse--Smale transversality and surjectivity of the linearized flow operator
• Lemma Countable proper local restrictions of a Fredholm map
• Lemma Critical images of proper local Fredholm restrictions are nowhere dense
• Theorem Sard--Smale residual regular values for Fredholm maps

and 3 others on the page.

Abstract Algebra · 1 result added

• Theorem Every proper subgroup of a finite nilpotent group is properly contained in its normalizer

Measure Theory · 7 results added

• Definition The space Lᵖ(μ) as the quotient by null functions
• Theorem Holder's inequality for integrals, including the endpoint cases
• Theorem Equality in Holder's inequality for 1 < p < ∞
• Theorem Lᵖ and L^∞ are vector spaces for p ≥ 1
• Corollary Assuming Countable Choice, Lᵖ-convergent sequences have almost-everywhere convergent subsequences
• Theorem The Lᵖ distance for 0 < p < 1 is a complete translation-invariant metric
• Theorem The p-power triangle inequality for nonnegative functions when 0 < p < 1

Real Analysis · 5 results added

• Definition The lower and upper Darboux integrals of a bounded f on [a,b] as sup_P L(f,P) and inf_P U(f,P), Darboux integrability as their equality, and the notation ∫ₐᵇ f
• Lemma A countable union of compact content-zero subsets of the line is null in ZF
• Theorem Lebesgue's criterion for Riemann integrability: a bounded f on [a,b] is Riemann integrable if and only if its set of discontinuities has measure zero
• Remark Choice ledger for the Riemann integral: the page results are proved in ZF
• False statement FALSE: a bounded function on [a,b] is Riemann integrable exactly when its set of discontinuities is nowhere dense

Commutative Algebra · 10 results added

• Theorem A nonzero module over a Noetherian ring has an associated prime
• Corollary A module over a Noetherian ring has no associated primes exactly when it is zero
• Theorem Minimal support primes of a finite module are associated
• Corollary The support is the union of the closures of the associated primes
• Theorem Finite modules over Noetherian rings have finitely many associated primes
• Lemma A zero divisor is contained in an associated prime
• Theorem Zero divisors on a module over a Noetherian ring are the union of its associated primes
• Lemma The radicals in a minimal primary decomposition are exactly the associated primes of the quotient

and 2 others on the page.

Functional Analysis · 1 result added

• Example Weak-star compactness of probability measures

Group Theory · 3 results added

• False statement FALSE: every finitely presented group has solvable word problem
• False statement FALSE: an unsolvable word problem means no individual word can be decided
• False statement FALSE: the Novikov-Boone theorem proves only the uniform problem is unsolvable

Measure Theory · 1 result added

• Lemma Complex translation, convolution, approximate identities, and mollification

Set Theory · 2 results added

• Theorem The recursion theorem
• Lemma Trichotomy of the order on ℕ

Fourier Analysis · 5 results added

• Definition Period-one Fourier coefficients, partial sums, and convolution on the torus
• Definition Dirichlet and Fejer kernels
• Lemma Fourier partial sums are Dirichlet convolutions
• Lemma Step functions on one period have vanishing Fourier coefficients
• Lemma Symmetric difference formula for Fourier partial sums

Fourier Analysis · 3 results added

• Definition Carleson maximal partial-sum operator
• Remark Reading the Kolmogorov example at the endpoint
• Counterexample The Carleson maximal operator is not strong type (1,1)

Examples & counterexamples →

Linear Algebra · 1 result added

• Example Rayleigh-quotient iteration on a 2×2 symmetric matrix exhibits fast local convergence

Partial Differential Equations · 1 result added

• Lemma Borel change of variables from the compact-support formula and Radon uniqueness

Linear Algebra · 2 results added

• Theorem The Hodge star exists uniquely and is given by the complementary-basis formula in an oriented orthonormal basis
• Definition Interior product on the exterior algebra

Fourier Analysis · 5 results added

• Definition Cesaro and Abel means of a Fourier series
• Lemma The Poisson kernel on the circle is a positive approximate identity
• Theorem Cesaro summability implies Abel summability
• Theorem Fejer means converge to midpoint values at jumps
• Example Fejer summation of the square wave

Examples & counterexamples →

Real Analysis · 1 result added

• Lemma Canonical naturals are positive and strictly increasing

Real Analysis · 1 result added

• Theorem t↦(cos t,sin t) is a bijection from [0,2π) onto the real unit circle

Probability · 1 result added

• Theorem Expectations factor over finite products of independent random variables

Commutative Algebra · 10 results added

• Lemma Local Koszul Acyclicity Inductive Converse
• Theorem Koszul Acyclicity Characterises Local Regular Sequences
• Corollary Local Koszul Acyclicity Iff Regular Sequence
• Lemma H One Regular Local Implies Koszul Regular
• Lemma Regular Sequence Permutation Adjacent Swap
• Corollary Regular Sequences Permutable Local
• Lemma Positive Powers Of A Regular Sequence Remain Regular
• Corollary Regularity Notions Coincide Local Finite

and 2 others on the page.

Examples & counterexamples →

Commutative Algebra · 1 result added

• Theorem Multiplicity under parameter reduction with annihilator correction

Measure Theory · 4 results added

• Theorem The Stieltjes interval set function is a premeasure
• Theorem Assuming countable choice, a nondecreasing right-continuous function defines a Borel measure on ℝ
• Theorem The interval data on (a,b] determines the Borel measure uniquely
• Theorem Assuming countable choice, finite-on-compacts Borel measures on ℝ correspond to nondecreasing right-continuous functions modulo constants

Algebraic Topology · 1 result added

• Lemma General Thom isomorphism from the relative Serre spectral sequence

Commutative Algebra · 3 results added

• Theorem Assuming the Axiom of Choice, Nakayama's lemma
• Corollary Assuming the Axiom of Choice, generators modulo an ideal in the Jacobson radical lift to generators
• Corollary Assuming the Axiom of Choice, minimal generators over a local ring are exactly residue-field bases

Differential Geometry · 5 results added

• Definition Smooth partition of unity on a manifold with boundary
• Lemma Relatively compact coordinate balls and half-balls form a boundary-manifold basis
• Lemma A countable boundary coordinate cover has a locally finite shrinking
• Theorem Smooth partitions of unity exist on manifolds with boundary
• Theorem Collar neighborhood theorem

Linear Algebra · 1 result added

• Proposition The reduced resolvent satisfies the standard projector and inverse identities on the complementary invariant subspace

Topology · 1 result added

• Definition The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement

Computability Theory · 1 result added

• Corollary DFAs and NFAs recognize the same languages

Measure Theory · 11 results added

• Theorem Positive functionals on C_c(X) are integration against a Radon measure
• Theorem Assuming Dependent Choice, uniqueness of the RMK representing measure among Radon measures
• Theorem Sigma-compact open sets make locally finite Borel measures regular
• Corollary Locally finite Borel measures on second-countable LCH spaces are regular
• Theorem C_c(X) is dense in Lᵖ(mu) for a Radon measure
• Lemma Positive C₀(X) functionals have finite regular representing measures
• Lemma A bounded real C₀(X) functional is a difference of positive functionals
• Theorem The bounded complex dual of C₀(X) is regular complex measures

and 3 others on the page.

Examples & counterexamples →

Set Theory · 1 result added

• Definition κ-trees and the tree property

Measure Theory · 1 result added

• Theorem The total variation of a signed or complex measure is a positive measure

Differential Geometry · 2 results added

• False statement Cohomologous symplectic forms on a noncompact manifold are always isotopic
• Counterexample A cohomology class obstructs a global symplectomorphism

Examples & counterexamples →

Measure Theory · 2 results added

• Lemma Euclidean balls have positive finite Lebesgue measure
• Theorem Marcinkiewicz interpolation from weak (1,1) and strong (∞,∞)

Probability · 1 result added

• Lemma The standard normal density has total mass one

Probability · 6 results added

• Corollary IID finite-variance weak law
• Theorem Kolmogorov maximal inequality
• Lemma Independent-copy symmetrization of random series
• Lemma Bounded centered convergent series have summable variances
• Theorem Truncation weak law for independent arrays
• Example A nonidentical Bernoulli weak law

Examples & counterexamples →

Functional Analysis · 40 results added · with a worked companion

Probability · 29 results added · with a worked companion

Probability · 28 results added · with a worked companion

Differential Geometry · 56 results added · with a worked companion

Algebraic Topology · 40 results added · with a worked companion

Set Theory · 46 results added · with a worked companion

Functional Analysis · 33 results added · with a worked companion

Functional Analysis · 33 results added · with a worked companion

Functional Analysis · 33 results added · with a worked companion

Functional Analysis · 53 results added · with a worked companion

Differential Geometry · 49 results added · with a worked companion

Functional Analysis · 34 results added · with a worked companion

Differential Geometry · 53 results added · with a worked companion

Set Theory · 34 results added · with a worked companion

Functional Analysis · 29 results added · with a worked companion

Differential Geometry · 62 results added · with a worked companion

Set Theory · 34 results added · with a worked companion

Functional Analysis · 29 results added · with a worked companion

Functional Analysis · 9 results added · with a worked companion

Probability · 27 results added · with a worked companion

Functional Analysis · 48 results added · with a worked companion