# Category:Mathematical fact

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## Pages in category "Mathematical fact"

The following 200 pages are in this category, out of 210 total.

(previous page) (next page)### C

- Cohen-Macaulay graded ring/Positive characteristic/Parameter/Tight closure/Degree criterion/Inclusion/Fact
- Cohen-Macaulay ring/Positive characteristic/Cohomological criterion for tight closure/Fact
- Compact interval/Real function/Riemann integrable on partition/Fact
- Complex Conjugation/Rules/Fact
- Complex numbers/Conjugation/Real part imaginary part/Fact
- Complex numbers/Field/Fact
- Complex numbers/Modulus/Rules/Fact
- Complex numbers/Real and imaginary part/Properties/Fact
- Continuous function/Closed bounded interval/Maximum is attained/Fact
- Continuous Function/Primitive function exists/Fact
- Continuous function/R/Intermediate value theorem/0-version/Fact
- Continuous function/R/Intermediate value theorem/Fact
- Convergent power series/R/Primitive function/Fact
- Convergent power series/R/Taylor series/Coincidence/Fact

### D

- Determinant/Field/Upper triangular matrix/Fact
- Determinant/Laplace expansion/Fact
- Determinant/Multiplication theorem/Fact
- Determinant/Recursively/Alternating/Swap property/Fact
- Determinant/Recursively/Multilinear/Fact
- Determinant/Transposed matrix/Universal property/Fact
- Determinant/Zero, linear dependent and rank property/Fact
- Differentiable function/D in R/Chain rule/Fact
- Differentiable function/D in R/Continuity in point/Fact
- Differentiable function/D in R/Inverse function/Fact
- Differentiable function/D in R/Linear approximation/Fact
- Differentiable function/D in R/Rules/Fact
- Differentiable function/D open in K/Chain rule/Fact
- Differentiable function/D open in K/Rules/Fact
- Differentiable function/Mean value theorem/Quotient version/Fact
- Differentiable functions/Mean value theorem/Fact

### E

- Endomorphism/Diagonalizable/Algebraic and geometric multiplicity/Fact
- Endomorphism/Eigenspaces are linear subspaces/Zero/Fact
- Endomorphism/Eigenvalue and characteristic polynomial/Fact
- Endomorphism/Eigenvectors/Linearly independent/Fact
- Endomorphism/Geometric and algebraic multiplicity/Fact
- Endomorphismus/Finite dimensional/Change of basis/Fact
- Exponential series/Real/Absolute convergence/Fact
- Exponential series/Real/Elementary properties/Fact
- Exponential series/Real/Functional equation/Fact

### F

- Field/Basic properties/Fact
- Field/Binomi/Fact
- Field/Uniqueness of negative and inverse element/Fact
- Finite field/Smooth projective curve/Vector bundel/Etale trivializable and Frobenius periodicity/Fact
- Finite field/Smooth projective curve/Vector bundle/Affineness and nonexistence of curves/Fact
- Finite field/Smooth projective curve/Vector bundle/Finite annihilation/Harder-Narasimhan-criterion/Fact
- Finite field/Smooth projective curve/Vector bundle/Strongly semistable degree 0/Trivializable/Fact
- Finite field/Smooth projective curve/Vector bundle/Strongly semistable degree nonnegative/Finite annihilation/Fact
- Forcing algebra/Primary ideal/Induced torsor/Fact
- Forcing algebra/Relation to tight closure/Local cohomology/Characterization/Fact
- Function/R/Limit/Epsilon/Characterizations/Fact
- Function/R/Limit/Epsilon/Rules/Fact
- Fundamental theorem of algebra/Nonconstant polynomial/Zero/Fact
- Fundamental theorem of calculus/Riemann/Fact

### H

- Hilbert-Kunz multiplicity/Two-dimensional graded/Brenner-Trivedi-formula/Fact
- Hilbert-Kunz multiplicity/Two-dimensional graded/Formula for maximal ideal in plane case/Fact
- Hilbert-Kunz/Invariant ring/Coinvariants/Fact
- Hilbert-Kunz/Invariant ring/Coinvariants/Polynomial ring/Fact
- Hilbert-Kunz/Relative two-dimensional situation/Limit and interpretation/Fact
- Hospital/Differentiable in inner interval/Fact
- Hyperbolic function/R/Monotonicity properties/Fact
- Hyperbolic functions/R/Elementary properties/Fact

### I

### L

- Linear inhomogeneous system/Elimination/Echelon form/Fact
- Linear inhomogeneous system/Strictly triangular/Solution/Fact
- Linear mapping/Bijective/Inverse mapping linear/Fact
- Linear mapping/Composition/Fact
- Linear mapping/Determination on basis/Fact
- Linear mapping/Diagonalizable/Characterizations/Fact
- Linear mapping/Different eigenvalues/Diagonalizable/Fact
- Linear mapping/Dimension formula/Fact
- Linear mapping/Eigenspace as kernel/Fact
- Linear mapping/Eigenspace for different eigenvalues/Zero/Fact
- Linear mapping/Eigenvalue zero/Characterization/Fact
- Linear mapping/Finite dimensional/Change of basis/Fact
- Linear mapping/Finite dimensional/Eigenvalues bounded by dimension/Fact
- Linear mapping/Kernel/Injectivity/Fact
- Linear mapping/Matrix and basis/Rank/Fact
- Linear mapping/Matrix to bases/Correspondence/Fact
- Linear mapping/Matrix to basis/Several properties/Fact
- Linear mapping/Matrix/Composition/Fact
- Linear mapping/Trigonalizable/Characterization with characteristic polynomial/Fact
- Linear system/Elimination lemma/Fact
- Linearly independent/Simple properties/Fact
- Logarithm/Base/Rules/Fact

### M

### N

### O

- One-dimensional ring/Hilbert-Kunz multiplicity is multiplicity/Fact
- Open subscheme/Affine and extended ideal/Finitely generated/Fact
- Ordered field/Bernoulli's inequality/Fact
- Ordered field/Convergent sequence/Bounded/Fact
- Ordered field/Convergent sequences/Rules/Fact
- Ordered field/Elementary properties for inverses/Fact
- Ordered field/Elementary properties/3/Fact

### P

- Polynomial function/Function behavior from differentiability/Fact
- Polynomial function/R/Continuous/Fact
- Polynomial ring/Field/One variable/Euclidean division/Fact
- Polynomial ring/Field/Zero/Linear factor/Fact
- Polynomial ring/Field/Zeroes/Number/Fact
- Polynomial/K/Interpolation/Fact
- Power function/Positive base/Real exponent/Fact
- Prime factorization/Existence/Fact
- Prime numbers/Infinity/Fact
- Primitive function/Inverse function/Fact
- Projective curve/Vector bundle/Cohomology class/Finite annihilation and torsor/Fact

### Q

### R

- Rational function/R/Continuous/Fact
- Real closed interval/Strictly increasing/Inverse function/Continuous/Fact
- Real cosine function/One zero between 0 and 2/Fact
- Real exponential function/Base/Derivative/Fact
- Real exponential function/Base/Properties/Fact
- Real exponential function/Continuity and image/Fact
- Real exponential function/Derivative/Fact
- Real function/Continuity in a point/Characterization/Fact
- Real function/Continuous/Addition, multiplication, inverting/Fact
- Real function/Continuous/Composition/Fact
- Real function/Derivative zero/Constant/Fact
- Real function/Derivative/Monotonicity/Fact
- Real function/Extrema/Higher derivatives/Fact
- Real function/Extrema/Second derivative/Fact
- Real function/Local extremum/Differentiable/Derivative zero/Fact
- Real function/Open interval/Local extrema/Differentiable/Derivative zero/Fact
- Real function/Taylor formula/(n+1)-times continuously differentiable/Estimate for error/Fact
- Real function/Taylor formula/(n+1)-times continuously differentiable/Lagrange/Fact
- Real function/Theorem of Rolle/Fact
- Real numbers/Bernoulli's inequality/Fact
- Real numbers/Bolzano Weierstraß/Fact
- Real numbers/Bounded monotonic increasing sequence/Cauchy sequence/Fact
- Real numbers/Bounded subset has supremum/Fact
- Real numbers/Convergent sequence/Bounded/Fact
- Real numbers/Convergent sequence/Cauchy sequence/Fact
- Real numbers/Convergent sequences/Compare/Fact
- Real numbers/Convergent sequences/Rules/Fact
- Real numbers/kth root of real number/Intermediate value theorem/Fact
- Real numbers/Modulus/Properties/Fact
- Real numbers/Nested intervals/Point/Fact
- Real numbers/Ordering axioms/Archimedean/Implications/Fact
- Real numbers/Sequence/Bounded monotone/Converges/Fact
- Real numbers/Sequence/Unique limit/Fact
- Real numbers/Sequences/Squeeze criterion/Fact
- Real numbers/Series converges/Summands/Null sequence/Fact
- Real numbers/Series/Cauchy-criterion/Fact
- Real numbers/Series/Rules/Fact
- Real positive number/Root/Unique existence/Fact
- Real power series/Convergence/Continuous function/Fact
- Real power series/Derivative by formal derivative/Fact
- Real power series/Infinitely often differentiable/Fact
- Real series/Absolute convergence and convergence/Fact
- Real series/Cauchy-product/Absolute convergence/Fact
- Real series/Direct comparison test/Fact
- Real series/Ratio test/Fact
- Real sine and cosine function/Derivative/Fact
- Regular ring/Characterication with Hilbert-Kunz multiplicity/Fact
- Regular ring/Dimension 2/Forcing algebra/Not inside/Affine/Fact
- Regular ring/Hilbert-Kunz multiplicity/Colength/Fact
- Regular ring/Positive characteristic/Forcing algebra/Not inside/Local cohomology/Fact
- Regular ring/Positive characteristic/Tight closure trivial/Fact
- Relative projective curve/Behavior of Hilbert-Kunz slopes/Fact
- Riemann integrability/Elementary properties/Fact
- Riemann integral/Step functions with equal limit/Integral/Fact

### S

- Scheme/Locally free sheaves and vector bundles/Equivalence/Fact
- Series/Real numbers/Leibniz criterion/Fact
- Sine and cosine/Monotonicity behavior/Fact
- Sine and cosine/R/Properties of periodicity/Fact
- Sine and cosine/Real/Properties/Fact
- Square matrix/C/Trigonalizable/Fact
- Square root/2/Irrational/Fact
- Standard space/K^n/Dimension n/Fact
- System of linear equations/Set of variables/Equivalent system/Manipulations/Fact
- System of linear equations/Set of variables/Solution space is vector space/Fact

### T

- Tangent and cotangent/Monotonicity behavior/Fact
- Tangent/Cotangent/Derivative/Fact
- Tight closure/Curve/Degree bound for inclusion/Strongly semistable/Fact
- Tight closure/Does not commute with localization/Fact
- Tight closure/Graded ring/Generic results/Fact
- Tight closure/Graded/Dimension 2/Gorenstein/Finite field/Test ideal exponent/Fact
- Tight closure/Localization/Geometric deformation over one-dimensional domain/Fact
- Tight closure/Monsky-quartic/Explicit non-inclusion/Fact
- Tight closure/Not plus closure/Dimension two/Standard-graded/Fact
- Tight closure/Plus closure/Two-dimensional/Standard-graded/Fact
- Tight closure/Relation to Hilbert-Kunz multiplicity/Fact
- Torsor over curve/Strong Harder-Narasimhan filtration/Affineness criterion/Fact
- Torsor over projective curve/Strongly semistable/Affineness criterion/Fact