Pages Needing Proofs
These pages have at least one proof that is empty or missing required knowledge links. A valid proof item needs content, a knowledge link in the statement, and a knowledge link in the proof body. Pages below 50% proof quality are hidden from directory pages until enough proof blocks are valid. Click a page title to show the specific proofs that need work.
Congruence 0% (0/36) page hidden
- congruence iff same remainder corollary statement needs a knowledge link; proof needs a knowledge link
- Congruence Commutes proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Division iff Congruent Mod 0 proposition statement needs a knowledge link
- Congruence is Transitive proposition statement needs a knowledge link; proof needs a knowledge link
- Sum of Two Congruent Numbers are Congruent proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Product of Two Congruent Numbers is Congruent proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Congruent Numbers raised to Congruent Powers may not be Congruent proposition statement needs a knowledge link; proof needs a knowledge link
- If two Two Numbers are Congruent, they are still Congruent mod a divisor of the Original Mod proposition statement needs a knowledge link; proof needs a knowledge link
- If Two Numbers are Congruent Using Two Different Moduli they are Congruent Using the LCM proposition statement needs a knowledge link; proof needs a knowledge link
- Switching Prime Powers exercise statement needs a knowledge link
- a p q exercise statement needs a knowledge link
- Division by a Common Factor Maintains Congruence proposition statement needs a knowledge link
- Division by a Common Factor Maintains Congruence if Mod is Divided by GCD proposition statement needs a knowledge link
- Multiplicative Cancellation if Mod Prime corollary statement needs a knowledge link; proof needs a knowledge link
- n Consecutive Numbers have Unique Remainders Mod n proposition statement needs a knowledge link
- When a Linear Modular Equation has a Solution proposition statement needs a knowledge link
- Relatively Prime Modular Equation has a Solution corollary statement needs a knowledge link; proof needs a knowledge link
- When a Rational Can be Converted to a Modular Inverse proposition statement needs a knowledge link
- 2p Choose p exercise statement needs a knowledge link
- Every Non Zero Number is Invertible Mod a Prime corollary statement needs a knowledge link; proof needs a knowledge link
- A Linear Modular Equation with a Solution has GCD many Solutions corollary statement needs a knowledge link
- 24 and 60 exercise statement needs a knowledge link
- Chinese Remainder theorem statement needs a knowledge link
- Seven Divides Ones and Threes exercise statement needs a knowledge link
- Alternating Congruence Again exercise statement needs a knowledge link; proof needs a knowledge link
- 43 Divides a Sum of Powers of 6 and 7 exercise statement needs a knowledge link; proof needs a knowledge link
- Remainder of a Sum of Powers exercise statement needs a knowledge link
- Remainder of a Sum of Factorials exercise statement needs a knowledge link; proof needs a knowledge link
- Modular Inverse iff GCD 1 proposition statement needs a knowledge link
- Not Relatively Prime iff No Modular Inverse corollary statement needs a knowledge link
- Equality is the Same as Congruent in a Complete System of Residues Modulo n corollary statement needs a knowledge link; proof needs a knowledge link
- Complete and Incomplete Systems Modulo n exercise statement needs a knowledge link; proof needs a knowledge link
- A Complete System of Residues Modulo n Hits all Possible Residues proposition statement needs a knowledge link; proof needs a knowledge link
- The Multiples of a Complete System of Residues Mod \( n \) is still a Complete System of Residues Mod \( n \) proposition statement needs a knowledge link
- Product of Non-Zero Residues Yields Factorial corollary statement needs a knowledge link; proof needs a knowledge link
- There are 22 Possibilities for the Last Two Digits of a Square proposition statement needs a knowledge link; proof needs a knowledge link
Quotient Rings 0% (0/32) page hidden
- The Natural Map is a Surjective Group Homomorphism corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Quotient Ring Mod a Polynomial proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Quotient Ring of Polynomial is Same as Its Remainder proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Integers Modulo N Equals the Quotient Ring of Z modulo n proposition statement needs a knowledge link; proof needs a knowledge link
- Modulo Integer Polynomial exercise proof needs a knowledge link
- Quotient Rings exercise statement needs a knowledge link; proof needs a knowledge link
- Crone Homomorphism induces an Isomorphism to its Image theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Isomorphic Rings exercise statement needs a knowledge link
- Polynomials and Prime Ideals exercise statement needs a knowledge link; proof needs a knowledge link
- Polynomials Mod a Polynomial is a Field iff it's irreducible proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Integers Modulo 2 Form a Field When Divided by a Certin Polynomial exercise statement needs a knowledge link; proof needs a knowledge link
- Crone Homomorphism Between Integers mod N exercise statement needs a knowledge link; proof needs a knowledge link
- Linear Polynomials exercise statement needs a knowledge link; proof needs a knowledge link
- Irreducible in C exercise statement needs a knowledge link; proof needs a knowledge link
- Polynomials Mod \( x + 1 \) exercise statement needs a knowledge link; proof needs a knowledge link
- Does it Contain All Roots? exercise statement needs a knowledge link; proof needs a knowledge link
- Quintic Formula Examples exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- Number of Irreducible Polynomials exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- Not Isomorphic Fields exercise statement needs a knowledge link; proof needs a knowledge link
- \( \pi \notin \mathbb{ Q } \left( \pi ^ 3 \right) \) lemma statement needs a knowledge link; proof needs a knowledge link
- Adjoin Pi exercise statement needs a knowledge link
- A Linear Factors Root Is an Element of the Field proposition statement needs a knowledge link; proof needs a knowledge link
- Splits Iff has all Roots corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Offset Splitting Fields exercise statement needs a knowledge link; proof needs a knowledge link
- Splitting Field in \( \mathbb{ Z } _ 3 ^ \% \) exercise statement needs a knowledge link; proof needs a knowledge link
- Irreducible and Monic with a Root has a Unique Smallest Degree lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Irreducible Polynomial Creates a Basis proposition statement needs a knowledge link
- Irreducible Polynomial of Degree \( d \) Creates a Field Extension of Degree \( d \) corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Counting Irreducible Polynomials exercise statement needs a knowledge link; proof needs a knowledge link
- Constructing a Reduced Polynomial proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Cubic Formula exercise statement needs a knowledge link
- Irreducible Polynomial of Degree Five exercise statement needs a knowledge link
Limits 0% (0/30) page hidden
- Convergent Sequence Doesn't Require Strict Inequality corollary statement needs a knowledge link; proof needs a knowledge link
- Eventually Increasing Index Mapping Maintains Limit proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Reciprocal Power Goes to Zero exercise statement needs a knowledge link; proof needs a knowledge link
- Limit of Sine Doesn't Exist at Infinity exercise statement needs a knowledge link
- Non-Zero Limit Implies Eventually Non-Zero Sequence proposition statement needs a knowledge link; proof needs a knowledge link
- Log Goes to Infinity exercise statement needs a knowledge link; proof needs a knowledge link
- Function Constant Inequality Has Solution if Function has Inverse proposition proof needs a knowledge link
- Two Limits that get Arbitrarily Close Have the Same Limit proposition statement needs a knowledge link; proof needs a knowledge link
- If a Sequence Converges it is Bounded proposition statement needs a knowledge link
- Squeeze Theorem theorem statement needs a knowledge link
- Integer Values Determine Monotone Function Limit at Infinity lemma statement needs a knowledge link
- limsup Exists iff the Sequence is Bounded Above proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Sequence Eventually gets Close to Limsups Limit proposition statement needs a knowledge link; proof needs a knowledge link
- Convergence iff Lim Sup and Inf are Equal proposition statement needs a knowledge link
- Sum Law proposition statement needs a knowledge link; proof needs a knowledge link
- Cauchy with Converging Subsequence Implies Entire Sequence Converges proposition proof needs a knowledge link
- Strictly Increasing Natural Number Function Surpasses All Values lemma statement needs a knowledge link; proof needs a knowledge link
- For all, There Exists Past a Point Yields Ordered Satisfaction lemma statement needs a knowledge link
- Limit to a Limit proposition statement needs a knowledge link
- Telescoping Triangle Inequality proposition statement needs a knowledge link; proof needs a knowledge link
- Series of Differences Converges Implies Cauchy proposition statement needs a knowledge link
- Cauchy Sequence Implies Subsequence with Convergent Differences proposition statement needs a knowledge link
- Zig-Zag Sequence has Nested Bound lemma statement needs a knowledge link; proof needs a knowledge link
- Zig-Zag Sequence is Cauchy iff Differences go to Zero proposition statement needs a knowledge link; proof needs a knowledge link
- Bounded Above and Monotone Increasing Converges to Supremum theorem statement needs a knowledge link
- Bounded Below and Monotone Decreasing Converges to Infimum theorem statement needs a knowledge link
- Nested Square Root Twos exercise statement needs a knowledge link; proof needs a knowledge link
- Inequality From Sequence to Limit proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Nested Intervals lemma statement needs a knowledge link; proof needs a knowledge link
- Bolzano-Weierstrass theorem statement needs a knowledge link; proof needs a knowledge link
Closed And Open Subsets 0% (0/27) page hidden
- The Set of One over N with Zero is Closed proposition statement needs a knowledge link; proof needs a knowledge link
- Finite Unions and Arbitrary Intersections are Closed proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Convergent Sequence With Finite Image Must Converge to a Point in Its Image lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Function on Naturals With Infinite Image Characterization lemma statement needs a knowledge link; proof needs a knowledge link
- The Image of a Sequence Union and its Limit is Closed proposition statement needs a knowledge link
- The Closure is Closed corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Closed iff Equals Closure proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Closure of a Set is the Smallest Closed Set Containing It proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Closure is the Set of Points with Intersecting Balls proposition statement needs a knowledge link; proof needs a knowledge link
- The Open Ball is Open corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Open iff Its Complement is Closed theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Arbitrary Union and Finite Intersection of Open sets is Open proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Compact Implies Closed and Bounded lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Closed Subset of a Compact Set is Compact lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Heine Borel theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Finite Union of Compact sets is Compact corollary statement needs a knowledge link; proof needs a knowledge link
- The Abitrary Intersection of Compact Sets is Compact corollary statement needs a knowledge link; proof needs a knowledge link
- Forced Convergence of a Summand lemma statement needs a knowledge link; proof needs a knowledge link
- The Sum of a Closed Subset and a Compact Subset is Closed proposition statement needs a knowledge link
- The Sum of two Closed sets Might Not be Closed proposition statement needs a knowledge link; proof needs a knowledge link
- If every Sub-sub-sequence Converges to the Same Point So does the Original proposition statement needs a knowledge link; proof needs a knowledge link
- Divergent Sequence in A Compact Set Must have Two Different Sub-limits proposition statement needs a knowledge link
- The Irrational Numbers are Dense in \( \mathbb{ R } \) exercise statement needs a knowledge link; proof needs a knowledge link
- The Rationals Have Empty Interior exercise statement needs a knowledge link; proof needs a knowledge link
- The Intersection of an Open and Dense Set is Dense proposition statement needs a knowledge link; proof needs a knowledge link
- Every Limit Point is Either a Cluster Point or Just An Element from the Set proposition statement needs a knowledge link; proof needs a knowledge link
- If a set Contains all Its Cluster Points Its Closed corollary statement needs a knowledge link; proof needs a knowledge link
Rings 0% (0/24) page hidden
- Two by Two Matrices with 0 In Bottom Left Form a Crone exercise empty proof; proof needs a knowledge link
- The Integers form a Ring exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- Every Field is a Ring corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Integers Modulo N are a Ring exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- Polynomials Form a Ring with Ring Coefficients proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Square Matrices form a Non-Commutative Ring exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- Zero Multiplication in a Ring proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Additive Inverse equals Multiplication by One's Additive Inverse proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Additive Inverse times Additive Inverse of One Yields Original proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- The set of One Element Forms a Ring proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Degree of Product equals Sum of Degrees proposition statement needs a knowledge link; proof needs a knowledge link
- Polynomials Formed from a Domain Form a Domain corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- \( \mathbb{ Z } \) is a Subring with Unity of \( \mathbb{ Q } \) proposition statement needs a knowledge link; proof needs a knowledge link
- Subring Criterion proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Domain Fractions Can be Extended to a Crone proposition statement needs a knowledge link; proof needs a knowledge link
- The Fraction Crone is a Field proposition statement needs a knowledge link; proof needs a knowledge link
- The complex numbers are a field extension of the reals exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- Field Extension Yields a Vector Space proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Degree of \( \mathbb{ C } \) over \( \mathbb{ R } \) exercise statement needs a knowledge link; proof needs a knowledge link
- The Kernel of a Ring Homomorphism is a Proper Ideal proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Ring Homomorphism is an Injection iff It's Kernel is Zero proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Field to Ring Homomorphism Implies Injective proposition statement needs a knowledge link; proof needs a knowledge link
- Field to Crone Homomorphism Implies the Image of the Field Is a Subfield Isomorphic to the Original Field proposition statement needs a knowledge link; proof needs a knowledge link
- GCD of two Polynomials exercise statement needs a knowledge link; proof needs a knowledge link
Absolute Value 0% (0/23) page hidden
- A Negative Number is Smaller than a Positive One corollary statement needs a knowledge link
- Multiplying by -1 Inverts the Sign proposition statement needs a knowledge link; proof needs a knowledge link
- Negative Zero is Zero proposition statement needs a knowledge link; proof needs a knowledge link
- A Positive Number is Greater than the Negative Version of Itself corollary statement needs a knowledge link
- Multiplying by -1 Changes the Inequality Direction proposition statement needs a knowledge link
- Absolute Value Differs By Sign proposition statement needs a knowledge link; proof needs a knowledge link
- Absolute Value Equals Max of Itself and it's Negation proposition statement needs a knowledge link
- Multiplication in Absolute Value theorem statement needs a knowledge link; proof needs a knowledge link
- Value Between Positive and Negative Equivalence proposition statement needs a knowledge link
- Maximum is Always Bigger than It's arguments proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Value Between Implies It's Absolute Value is Less than the Max of the Others proposition statement needs a knowledge link
- Absolute Value Bounds Distance corollary statement needs a knowledge link
- absolute value greater inequality proposition statement needs a knowledge link; proof needs a knowledge link
- absolute value equality proposition statement needs a knowledge link; proof needs a knowledge link
- not equal range proposition statement needs a knowledge link; proof needs a knowledge link
- changing sign doesn't change absolute value corollary statement needs a knowledge link; proof needs a knowledge link
- absolute value is greater than itself corollary statement needs a knowledge link; proof needs a knowledge link
- absolute value is greater than the negative version of itself corollary statement needs a knowledge link; proof needs a knowledge link
- Inverse Absolute Value corollary statement needs a knowledge link; proof needs a knowledge link
- Inverse Triangle Inequality corollary statement needs a knowledge link; proof needs a knowledge link
- Triangle Inequality theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Generalized Triangle Inequality proposition statement needs a knowledge link; proof needs a knowledge link
- Isolate Absolute value Inequality proposition statement needs a knowledge link; proof needs a knowledge link
Everything 0% (0/20) page hidden
- If a Cauchy Sequence has a Convergent Subsequence, then the Original Converges There theorem proof needs a knowledge link
- Unitary Operator Equivalences proposition proof needs a knowledge link
- The Unilateral Shift Is an Isometry proposition statement needs a knowledge link; proof needs a knowledge link
- The Unilateral Shift Is Not Unitary proposition statement needs a knowledge link; proof needs a knowledge link
- The Adjoint of the Unilateral Shift Is the Bilateral Shift proposition statement needs a knowledge link; proof needs a knowledge link
- The Bilateral Shift Is Unitary proposition statement needs a knowledge link; proof needs a knowledge link
- Closed Graph Theorem theorem statement needs a knowledge link; proof needs a knowledge link
- A Convex Subset of a Hilbert Space Has a Unique Closest Element to a Point in the Hilbert Space lemma statement needs a knowledge link
- Compact Operators Are Limits of Finite Rank Operators proposition statement needs a knowledge link; proof needs a knowledge link
- Spectral Theorem for Compact Operators theorem statement needs a knowledge link; proof needs a knowledge link
- Real Spectrum Iff Self Adjoint proposition statement needs a knowledge link; proof needs a knowledge link
- If T Is Self Adjoint Then T Is Bounded theorem statement needs a knowledge link; proof needs a knowledge link
- The Dirichlet Kernel Is Real Valued and Integrates to 1 proposition statement needs a knowledge link; proof needs a knowledge link
- Bounded Linear Functional Integral lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Embryionic Spectral Theorem theorem statement needs a knowledge link; proof needs a knowledge link
- Open Mapping Theorem theorem proof needs a knowledge link
- Fredholm Operators Plus Finite Rank Operators Are Fredholm Operators proposition statement needs a knowledge link; proof needs a knowledge link
- The Product of Linear Operators With Index Still Has Index proposition statement needs a knowledge link; proof needs a knowledge link
- A Bounded Linear Operator Has Non-empty Spectrum lemma statement needs a knowledge link; proof needs a knowledge link
- An Eigenvalue for a Normal Linear Operator Is an Eigenvalue for the Adjoint When Taking the Conjugate lemma statement needs a knowledge link; proof needs a knowledge link
Quotient And Remainder 0% (0/19) page hidden
- Quotient Remainder theorem statement needs a knowledge link
- Quotient Remainder using Quotient and Remainder Function corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Specific Value of the Divisor Function corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Specific Value of the Remainder Function corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Quotient Remainder on Positive Integer has Positive Divisor corollary statement needs a knowledge link; proof needs a knowledge link
- Divides iff Zero Remainder proposition statement needs a knowledge link; proof needs a knowledge link
- Every integer is either Odd or Even proposition proof needs a knowledge link
- Exactly one of any two Consecutive Numbers is Even proposition statement needs a knowledge link; proof needs a knowledge link
- When the Remainder Function does Nothing proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Remainder of Sum Less Than First Summand When Sum Exceeds Modulus proposition statement needs a knowledge link
- Repeated Application of the Remainder Function does Nothing proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Quotient Remainder and Addition proposition statement needs a knowledge link; proof needs a knowledge link
- If a Number Divides a Sum and Already Divides one of the Summands it Must Divide the Other corollary statement needs a knowledge link; proof needs a knowledge link
- Adding Multiples of the Divisor Never changes the Remainder proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- n Factorial Plus or Minus One Restricts Factors proposition statement needs a knowledge link
- Every Perfect Square has Remainder Zero or One Upon Division by Four corollary statement needs a knowledge link
- Every Odd Perfect Square has Remainder 1 mod 8 proposition statement needs a knowledge link
- Quotient Rule For Exponents proposition statement needs a knowledge link; proof needs a knowledge link
- The Divisor Counting Function Is Multiplicative proposition statement needs a knowledge link; proof needs a knowledge link
Spectral 0% (0/18) page hidden
- The Entries of the Power of an Adjacency Matrix Represent Walks proposition statement needs a knowledge link; proof needs a knowledge link
- The Laplacian Is the Sum of Edge Laplacians corollary statement needs a knowledge link; proof needs a knowledge link
- Quadratic Form of the Edge Laplacian Is a Squared Difference proposition statement needs a knowledge link; proof needs a knowledge link
- The Laplacian Is Positive-semidefinite proposition statement needs a knowledge link; proof needs a knowledge link
- Every Eigenvalue of the Laplacian Is Non-negative proposition statement needs a knowledge link; proof needs a knowledge link
- 0 Is an Eigenvalue for the Laplacian proposition statement needs a knowledge link; proof needs a knowledge link
- There Is Only One Zero Eigen Value for the Laplacian of a Connected Graph proposition statement needs a knowledge link; proof needs a knowledge link
- The Geometric Multiplicity of the Eigenvalue Zero of the Laplacian Is the Number of Connected Components corollary statement needs a knowledge link; proof needs a knowledge link
- The Eigenvalues of the Laplacian of a Connected Graph proposition statement needs a knowledge link
- Laplacian of the Cycle Graph corollary statement needs a knowledge link; proof needs a knowledge link
- An Eigenvector of the Laplacian of the Cycle Graph Yields More Eigenvectors by Its Cyclic Permutations corollary statement needs a knowledge link; proof needs a knowledge link
- Laplacian of the Cycle Graph Has Trigonometric Eigenvalues proposition statement needs a knowledge link
- The Rayleigh Quotient of an Eigenvector Is Its Eigenvalue corollary statement needs a knowledge link; proof needs a knowledge link
- Quadradic Form as a Summation of Eigenvalues lemma statement needs a knowledge link; proof needs a knowledge link
- Courant Fischer theorem statement needs a knowledge link
- Courant Fischer Specific corollary statement needs a knowledge link; proof needs a knowledge link
- The Largest Eigenvalue of the Laplacian of a Graph Is at Least as Large as the Degree of Any Vertex theorem statement needs a knowledge link
- The Second Smallest Eigenvalue of the Laplacian of the Path Graph on N Vertices Is in the Big O of One Over N Squared theorem statement needs a knowledge link; proof needs a knowledge link
The Galois Group 0% (0/17) page hidden
- Automorphism Fixed Pointwise Takes Roots to Roots proposition statement needs a knowledge link; proof needs a knowledge link
- Degree 1 iff Same as Fields proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Prime Degree Extension Restricts Subfields proposition statement needs a knowledge link
- Irreducible Polynomial Restricts Size of Galois Group proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Polynomial to Galois exercise statement needs a knowledge link; proof needs a knowledge link
- The Irreducible Polynomial of 1 + i exercise statement needs a knowledge link; proof needs a knowledge link
- Degree of Extension exercise statement needs a knowledge link; proof needs a knowledge link
- Quartic Roots exercise statement needs a knowledge link; proof needs a knowledge link
- Square Root Coefficients exercise statement needs a knowledge link; proof needs a knowledge link
- Polynomials as Matrices exercise statement needs a knowledge link; proof needs a knowledge link
- Big Adjoin exercise statement needs a knowledge link; proof needs a knowledge link
- Automorphism Takes Roots of a Factor to the Root of the Same Factor corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Galois Group of Factored Polynomial exercise statement needs a knowledge link
- A Familiar Group exercise statement needs a knowledge link; proof needs a knowledge link
- Only One exercise statement needs a knowledge link; proof needs a knowledge link
- Big Polynomial exercise statement needs a knowledge link; proof needs a knowledge link
- Cos 20 Is not a Surd exercise statement needs a knowledge link; proof needs a knowledge link
Ideals 0% (0/15) page hidden
- Trivial Ideal corollary empty proof; proof needs a knowledge link
- Every Crone is an Ideal corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Kernel of a Crone Homomorphism is a Proper Ideal proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Crone Homomorphism is an Injection iff \( \operatorname{ ker } \left( \phi \right) = \left\{ 0 _ R \right\} \) proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- An Ideal is a Normal Subgroup proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Intersection of Ideals is an Ideal proposition statement needs a knowledge link; proof needs a knowledge link
- Ideal Generated by a Set is an Ideal corollary statement needs a knowledge link
- Ideal Generated by a Set is the Smallest Ideal Containing the Set proposition statement needs a knowledge link
- Left Multiplication Yields an Ideal proposition statement needs a knowledge link; proof needs a knowledge link
- Principal Ideal Equals Generated Ideal proposition statement needs a knowledge link; proof needs a knowledge link
- Ideal Generated by a Finite Set is their Linear Combinations proposition statement needs a knowledge link; proof needs a knowledge link
- Product of Ideals is Contained in their Intersection proposition statement needs a knowledge link; proof needs a knowledge link
- Can't get to All Polynomials From a Generated Ideal exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- An Ideal of Continuous Functions exercise statement needs a knowledge link; proof needs a knowledge link
- Quotient Remainder For Polynomials proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Compact Operators 0% (0/15) page hidden
- Bounded Operator in Terms of Sequences corollary statement needs a knowledge link
- A Compact Operator Is Bounded theorem statement needs a knowledge link; proof needs a knowledge link
- The Compact Operators Are a Linear Subspace of the Bounded Operators proposition statement needs a knowledge link; proof needs a knowledge link
- The Composition of Linear Operators Is Compact If at Least One Is proposition statement needs a knowledge link; proof needs a knowledge link
- Riesz' lemma statement needs a knowledge link; proof needs a knowledge link
- The Unit Disk nor the Unit Circle Is Compact in an Infinite Dimensional Space theorem statement needs a knowledge link
- The Identity Operator on an Infinite Dimensional Space Is Not Compact theorem statement needs a knowledge link; proof needs a knowledge link
- A Compact Operator on a Infinite Dimensional Space Is Not Invertible corollary statement needs a knowledge link
- Compact Symmetric Operators That Commute Can Be Simultaneously Diagonalized proposition statement needs a knowledge link; proof needs a knowledge link
- Complex Form of a Linear Operator proposition statement needs a knowledge link; proof needs a knowledge link
- If a Bounded Linear Operator Has Finite Rank Then It Is Compact proposition statement needs a knowledge link; proof needs a knowledge link
- Every Compact Operator Is Bounded proposition statement needs a knowledge link
- Every Compact Operator Is Continuous corollary statement needs a knowledge link
- A Bounded Finite Rank Operator Is Compact proposition statement needs a knowledge link
- If a Linear Operator Has a Finite Dimensional Domain Then It Is Compact proposition statement needs a knowledge link
Exponential Function 0% (0/14) page hidden
- exponentiation of a positive number is positive proposition statement needs a knowledge link; proof needs a knowledge link
- product of two reals of the same sign is positive proposition statement needs a knowledge link; proof needs a knowledge link
- Powers Of X Are Positive proposition statement needs a knowledge link; proof needs a knowledge link
- binomial theorem statement needs a knowledge link; proof needs a knowledge link
- TODO theorem statement needs a knowledge link; proof needs a knowledge link
- Derivative theorem statement needs a knowledge link; proof needs a knowledge link
- exponential sum product equality theorem statement needs a knowledge link; proof needs a knowledge link
- reciprocal of exponential corollary statement needs a knowledge link; proof needs a knowledge link
- TODO proposition statement needs a knowledge link; proof needs a knowledge link
- Ln proposition statement needs a knowledge link; proof needs a knowledge link
- exponential is always positive theorem statement needs a knowledge link; proof needs a knowledge link
- exponentiation inverse proposition statement needs a knowledge link; proof needs a knowledge link
- Exponential Bound proposition statement needs a knowledge link; proof needs a knowledge link
- Exponential Inequality proposition statement needs a knowledge link; proof needs a knowledge link
Floor And Ceiling 0% (0/13) page hidden
- Fractional Part of Floor Bound corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Floor of an Integer is Itself corollary statement needs a knowledge link; proof needs a knowledge link
- Ceiling of an Integer is Itself corollary statement needs a knowledge link; proof needs a knowledge link
- Floor of a Non Integer is Smaller proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Ceiling of a Non Integer is Greater proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- The floor of a Sum of a Real and an Integer proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Number is a Perfect Square if its Square Root is an Integer lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Counting Squares with Floor corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- When Flooring Tells us Two Numbers Divide Eachother proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Counting Multiples with Floor proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Largest Prime Power Dividing the Factorial proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Prime Factorization of the Factorial corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Double Sum of Divisors Equation proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Union And Intersection 0% (0/12) page hidden
- partition of the integers exercise statement needs a knowledge link
- a set intersected with a superset is itself proposition statement needs a knowledge link
- A Set Union a Subset is Itself proposition statement needs a knowledge link
- Subset of an Intersection proposition statement needs a knowledge link; proof needs a knowledge link
- intersection factors from union proposition statement needs a knowledge link; proof needs a knowledge link
- intersection factors from intersection proposition statement needs a knowledge link; proof needs a knowledge link
- Union of Subsets is Still a Subset proposition statement needs a knowledge link; proof needs a knowledge link
- Intersection of Subsets is Still a Subset proposition statement needs a knowledge link; proof needs a knowledge link
- An Intersection of Supersets is still a Superset proposition statement needs a knowledge link; proof needs a knowledge link
- Intersection Decreases as It Intersects More Things proposition statement needs a knowledge link; proof needs a knowledge link
- The Intersection of a Collection of Sets Is a Subset of Any Set Part of the Intersection corollary statement needs a knowledge link
- A set Covered in Subsets is a Union proposition statement needs a knowledge link
Legendre Symbol 0% (0/12) page hidden
- Eulers Criterion lemma empty proof; proof needs a knowledge link
- Quadratic Residue iff Legendre Symbol is One corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Quadratic Residues mod p Come in Pairs proposition statement needs a knowledge link
- Half of them are Quadratic Resides and the Rest Aren't proposition statement needs a knowledge link; proof needs a knowledge link
- Quadratic Residue with a Composite Modulus proposition statement needs a knowledge link
- When a Quadratic Congruence has a Solution mod p Squared proposition statement needs a knowledge link
- Obtaining a Solution to a Quadratic Congrugence mod p Squared from a Solution mod p proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Quadratic Equation mod p Squared has 2 Solutions if the Legendre Symbol is One and None if Minus One proposition statement needs a knowledge link
- Solutions to 2783 exercise statement needs a knowledge link; proof needs a knowledge link
- A Number is a Solution of a Quadratic Equation mod a Power of 2 iff a Power of 2 Minus the Number is as well lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Only Numbers Congruent to 1 mod 8 Have Quadratic Residues mod a Power of 2 proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Quadratic Congruence mod a Power of 2 has 4 Solutions proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Prime Ideals And Maximal Ideals 0% (0/11) page hidden
- Degree One Polynomials are Irreducible corollary statement needs a knowledge link; proof needs a knowledge link
- Polynomial of Degree 2 or 3 is Irreducible Iff It has no Roots proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Irreducible Either has GCD 1 or Divides any other Non-Constant Polynomial proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Irreducibility Criterion with Prime Ideals theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Proper Ideal is Prime iff the Quotient Ring is a Domain proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Maximal iff Quotient Ring is a Field proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Maximal Implies Prime corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- In a PID every Non-Zero Prime Ideal is Maximal proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Field Polynomials Mod an Irreducible Yields a Root corollary statement needs a knowledge link
- Splits iff Roots corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Unique Roots in a Field of Characteristic p proposition statement needs a knowledge link; proof needs a knowledge link
Primitive Roots 0% (0/11) page hidden
- Order Exists iff There is a Power that Yields 1 proposition statement needs a knowledge link; proof needs a knowledge link
- Order Exists iff Coprime proposition statement needs a knowledge link
- Primitive Roots are a Subset of Coprimes corollary statement needs a knowledge link; proof needs a knowledge link
- Every Prime number has a Primitive Root theorem statement needs a knowledge link; proof needs a knowledge link
- Connection Between Primitive Roots of \( p ^ k \) and 2 \( p ^ k \) proposition statement needs a knowledge link
- Order of a Primitive Root mod the next Power proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- When a Primitive Remains a Primitive Root mod the next Power corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Every Power of an odd Prime has a Primitive Root proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Classification of the Existence of Primitive Roots theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Number to the Phi n over Two is Congruent to 1 When it is not a Primitive Root proposition statement needs a knowledge link; proof needs a knowledge link
- Distinct Powers up to the Order lemma statement needs a knowledge link; proof needs a knowledge link
Algebra Of The Complex Plane 0% (0/10) page hidden
- conjugate cancels proposition statement needs a knowledge link; proof needs a knowledge link
- modulus ignores conjugates corollary statement needs a knowledge link; proof needs a knowledge link
- complex number times it's conjugate equals the modulus squared proposition statement needs a knowledge link; proof needs a knowledge link
- value of the inverse theorem statement needs a knowledge link; proof needs a knowledge link
- Normalize Exercise exercise statement needs a knowledge link; proof needs a knowledge link
- extracting the real part of a complex number proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- modulus is greater than it's components proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- imaginary part distributes proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- modulus plus one of it's components is positive proposition statement needs a knowledge link; proof needs a knowledge link
- creating squares exercise statement needs a knowledge link; proof needs a knowledge link
Complex Numbers 0% (0/10) page hidden
- conjugate Cancels proposition statement needs a knowledge link; proof needs a knowledge link
- Modulus Ignores Conjugates corollary statement needs a knowledge link; proof needs a knowledge link
- complex number times it's conjugate equals the modulus squared proposition statement needs a knowledge link; proof needs a knowledge link
- value of the inverse theorem statement needs a knowledge link; proof needs a knowledge link
- Normalize Exercise exercise statement needs a knowledge link; proof needs a knowledge link
- extracting the real part of a complex number proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- modulus is greater than it's components proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- imaginary part distributes proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- modulus plus one of it's components is positive proposition statement needs a knowledge link; proof needs a knowledge link
- creating squares exercise statement needs a knowledge link; proof needs a knowledge link
Logic 0% (0/10) page hidden
- Logical And True iff both Arguments are True exercise statement needs a knowledge link; proof needs a knowledge link
- Addition Mod 2 is Associative lemma statement needs a knowledge link; proof needs a knowledge link
- Exclusive Or is Associative proposition statement needs a knowledge link; proof needs a knowledge link
- Sum of Ones Mod 2 becomes 0 iff Even proposition statement needs a knowledge link; proof needs a knowledge link
- Exclusive Or Evaluates to True iff an odd number of arguments are True corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Contrapositive proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Forced Consequence proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Empty And is True corollary proof needs a knowledge link
- Arbitrary And Indexed Notation with Reversed Integers is True corollary statement needs a knowledge link
- For Every, There Exists is a Function proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Basics 0% (0/8) page hidden
- You can add to get to any Number proposition statement needs a knowledge link; proof needs a knowledge link
- Numbers from Zero to K proposition statement needs a knowledge link; proof needs a knowledge link
- The Number of Integers between Two Integers Inclusive proposition statement needs a knowledge link
- Pigeon Hole Principle proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Formula For the \( n \)-th Catalan Number proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Bijection Between Valid Bracket Strings and Lattice Paths proposition statement needs a knowledge link; proof needs a knowledge link
- Number of Valid Bracketings is Equal to the \( n \)-th Catalan Number proposition statement needs a knowledge link; proof needs a knowledge link
- Counting Stars exercise statement needs a knowledge link; proof needs a knowledge link
A Star 0% (0/8) page hidden
- A* Execution Table exercise statement needs a knowledge link; proof needs a knowledge link
- A* is Complete theorem statement needs a knowledge link; proof needs a knowledge link
- A* is Not Cost Optimal if The Heuristic is Not Admissible proposition statement needs a knowledge link; proof needs a knowledge link
- A* Never Expands Large f-values lemma statement needs a knowledge link; proof needs a knowledge link
- A* is Optimal theorem statement needs a knowledge link; proof needs a knowledge link
- Monontonicity implies Admissibility theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Later Expansions Imply Larger f-values Under Monotonic Heuristic proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- First Expansion is Minimum Cost Uner Monotonic Heuristic proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Complexity 0% (0/8) page hidden
- Adding to Reduce exercise statement needs a knowledge link; proof needs a knowledge link
- Complexity exercise statement needs a knowledge link; proof needs a knowledge link
- Subset Sum is in NP corollary statement needs a knowledge link; proof needs a knowledge link
- Subset Sum is NP-Complete proposition statement needs a knowledge link; proof needs a knowledge link
- Set Partition is in NP proposition statement needs a knowledge link; proof needs a knowledge link
- Set Partition is NP-Complete proposition statement needs a knowledge link; proof needs a knowledge link
- Squared Sum Partition is NP-Complete proposition statement needs a knowledge link; proof needs a knowledge link
- Max-2-Sat is NP-Complete proposition statement needs a knowledge link; proof needs a knowledge link
Tuples 0% (0/8) page hidden
- When an Integer Indexed Tuple is Empty proposition statement needs a knowledge link
- Length of a Zero Indexed Tuple proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Converting from Regular to Zero Indexing proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Reverse Cancels proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- sorted_asc Holds iff It is Sorted in Ascending Order corollary empty proof; proof needs a knowledge link
- Ascending Tuple exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- Descending Tuple exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- The reverse of Ascending is Descending proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Banach Algebras 0% (0/8) page hidden
- Continuous and Bounded Functions With the Sup-norm Form a Unital C* Algebra exercise statement needs a knowledge link; proof needs a knowledge link
- Polynomials and Spectrum Commute proposition statement needs a knowledge link; proof needs a knowledge link
- Newmann Series proposition statement needs a knowledge link; proof needs a knowledge link
- The Spectrum of an Element From a Unital Banach Algebra Is Non-Empty theorem statement needs a knowledge link; proof needs a knowledge link
- Gelfand theorem statement needs a knowledge link; proof needs a knowledge link
- Abstract Wiener's theorem statement needs a knowledge link; proof needs a knowledge link
- Spectrum Is the Gelfand Image of Multiplicative Functionals proposition statement needs a knowledge link; proof needs a knowledge link
- Spectral Sub Additivity and Multiplicativity proposition statement needs a knowledge link
Domains And Fields 0% (0/7) page hidden
- Product Zero Implies Factor Zero Domain Equivalence corollary proof needs a knowledge link
- Domain iff Cancellation proposition statement needs a knowledge link
- Nilpotent Implies Zero or Zero Divisor proposition statement needs a knowledge link; proof needs a knowledge link
- Constant times Nilpotent is Nilpotent proposition statement needs a knowledge link; proof needs a knowledge link
- One plus Nilpotent is Nilpotent proposition proof needs a knowledge link
- The Sum of a Unit and a Nilpotent Element is a Unit corollary statement needs a knowledge link
- Field is a Ring with Units proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
The Fundamental Theorem Of Calculus 0% (0/7) page hidden
- Fundamental Theorem of Calculus I theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Fundamental Theorem of Calculus II theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- MVT for Integrals proposition statement needs a knowledge link
- Sign Functions Integral exercise statement needs a knowledge link; proof needs a knowledge link
- One over Epsilon Integral exercise statement needs a knowledge link; proof needs a knowledge link
- Sup Norm Sandwich Inequality lemma statement needs a knowledge link
- Twice Differentiable Triangle exercise statement needs a knowledge link
Hilbert Space Theory 0% (0/7) page hidden
- Cauchy-Schwarz Inequality theorem empty proof; proof needs a knowledge link
- Parallelogram Identity theorem empty proof; proof needs a knowledge link
- Closest Point Projection onto Closed Convex Set theorem empty proof; proof needs a knowledge link
- Orthogonal Decomposition theorem empty proof; proof needs a knowledge link
- Bessel's Inequality theorem empty proof; proof needs a knowledge link
- Parseval's Identity theorem empty proof; proof needs a knowledge link
- Riesz Representation Theorem for Hilbert Spaces theorem empty proof; proof needs a knowledge link
Metric And Banach Spaces 0% (0/7) page hidden
- Banach Fixed Point Theorem theorem empty proof; proof needs a knowledge link
- Linear Map Continuous iff Bounded theorem empty proof; proof needs a knowledge link
- Bounded Linear Maps Form a Banach Space theorem empty proof; proof needs a knowledge link
- Baire Category Theorem theorem empty proof; proof needs a knowledge link
- Open Mapping Theorem theorem empty proof; proof needs a knowledge link
- Closed Graph Theorem theorem empty proof; proof needs a knowledge link
- Uniform Boundedness Principle theorem empty proof; proof needs a knowledge link
Random Walk 0% (0/7) page hidden
- Spatial Homogeneity lemma proof needs a knowledge link
- Temporal Homogeneity lemma proof needs a knowledge link
- Markov Property lemma proof needs a knowledge link
- Position Distribution proposition proof needs a knowledge link
- Reflection Principle theorem proof needs a knowledge link
- Gamblers Ruin Probability theorem proof needs a knowledge link
- Symmetric Gamblers Ruin Bankruptcy Probability corollary proof needs a knowledge link
Convergence And Completeness 0% (0/6) page hidden
- Convergence iff Limit gets arbitrarily Close to a Point corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Sequence in Rn Converges to a Point iff Each Component Converges proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Less Than or Equal to Iff Less Than Or Equal to With Epsilon lemma statement needs a knowledge link; proof needs a knowledge link
- Closed Balls are Complete exercise statement needs a knowledge link
- \( \mathbb{ R } ^ n \) is Complete theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- If a Sequence in \( \mathbb{ R } ^ n \) Converges, Then their Norms Converge proposition statement needs a knowledge link; proof needs a knowledge link
Integration On Chains 0% (0/6) page hidden
- Tensor Product Basis theorem empty proof; proof needs a knowledge link
- Wedge Product Properties theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Alternating Tensor Basis theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Exterior Derivative Properties theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Boundary Squared is Zero theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Stokes' Theorem for Chains theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
Intermediate Value Theorem 0% (0/6) page hidden
- Intermediate Value theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Intermediate Value for Paths corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- The circle into R is not One to One exercise statement needs a knowledge link; proof needs a knowledge link
- We can make the Absolute value of a Polynomial as Small as we Want lemma statement needs a knowledge link; proof needs a knowledge link
- We can make the Absolute value of a Polynomial as small as we Want Inverse Edition corollary statement needs a knowledge link; proof needs a knowledge link
- A Polynomial of Odd Degree has at Least One Root exercise statement needs a knowledge link; proof needs a knowledge link
Mean Value Theorem 0% (0/6) page hidden
- Sign of the Derivative Tells us about Monotonicity corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Matching Derivatives Implies Differing by a Constant exercise statement needs a knowledge link; proof needs a knowledge link
- Non-Zero Derivative Implies Locally One-to-One exercise statement needs a knowledge link
- Rolles and Zeros corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Nth Derivative has a Zero Condition corollary statement needs a knowledge link; proof needs a knowledge link
- Forced Limit of the Derivative exercise statement needs a knowledge link
Riemann Integration 0% (0/6) page hidden
- Riemann Sum Inequalities under Refinement proposition empty proof; proof needs a knowledge link
- Any Lower Riemann Sum is Below Any Upper Riemann Sum proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Lower Riemann Integral is Below Upper Riemann Integral proposition empty proof; proof needs a knowledge link
- Continuous Functions on Closed Bounded Intervals are Riemann Integrable theorem empty proof; proof needs a knowledge link
- Bounds on a Riemann Integral proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Interchanging Riemann Integral and Pointwise Limit theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
Symmetric Difference 0% (0/6) page hidden
- Set Equality through the Characteristic corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Symmetric Difference Characterization proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Characteristic Version of Symmetric Difference proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Symmetric Difference is Associative proposition statement needs a knowledge link; proof needs a knowledge link
- Characteristic Version of Symmetric Difference Generalized proposition statement needs a knowledge link; proof needs a knowledge link
- Element is in the Fold of Symmetric Differences iff it is an an odd Number of Sets corollary statement needs a knowledge link
Linear Functionals 0% (0/6) page hidden
- The Dual Space Is a Normed Vector Space corollary statement needs a knowledge link; proof needs a knowledge link
- The Dual Space Is a Banach Space proposition statement needs a knowledge link; proof needs a knowledge link
- The Dual Space of Rn Is Rn proposition statement needs a knowledge link; proof needs a knowledge link
- The Dual of Lp Is Lq proposition statement needs a knowledge link; proof needs a knowledge link
- The Dual Space of l1 Is l Infinity proposition statement needs a knowledge link; proof needs a knowledge link
- Hahn Banach theorem statement needs a knowledge link; proof needs a knowledge link
Quadratic Congruences 0% (0/6) page hidden
- x Squared is Congruent to 1 iff x is Congruent to Plus or Minus One Mod p proposition statement needs a knowledge link
- Quadratic's Little corollary statement needs a knowledge link; proof needs a knowledge link
- Number of Solutions to x Squared Congruent to Minus One Mod p proposition statement needs a knowledge link
- Incongruent mod 37 exercise statement needs a knowledge link; proof needs a knowledge link
- a Squared b Squared exercise statement needs a knowledge link
- Number of Solutions to a Quadratic Congruence proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Product Topology 0% (0/6) page hidden
- Cartesian Product of open sets is a basis for the Cartesian Product proposition statement needs a knowledge link; proof needs a knowledge link
- Product Topology by Universal Properties theorem statement needs a knowledge link; proof needs a knowledge link
- The Basis for the product Topology of two Topologies generated by Bases are all the Cartesian Products of all the Basis Elements theorem statement needs a knowledge link; proof needs a knowledge link
- The Dictionary Order Topology on Rxr Is the Same as the Product Topology of Rdxr exercise statement needs a knowledge link
- inverse of a projection corollary statement needs a knowledge link; proof needs a knowledge link
- subbasis for the product topology theorem statement needs a knowledge link; proof needs a knowledge link
Polynomial Rings Over Fields 0% (0/5) page hidden
- GCD for Polynomials Is Unique corollary statement needs a knowledge link; proof needs a knowledge link
- GCD as a Linear Combination proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Every Polynomial Ideal is Principal proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Show that \( \mathbb{ Z } \) is a Principal Ideal Domain exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- If \( F \) is a Field then \( F \left[ x \right] \) is a Principal Ideal Domain theorem statement needs a knowledge link; proof needs a knowledge link
Roots 0% (0/5) page hidden
- Extending to a Field with Four Elements exercise statement needs a knowledge link; proof needs a knowledge link
- A Field with Four elements is not Isomorphic to a subfield of a field with 8 proposition statement needs a knowledge link; proof needs a knowledge link
- Roots exercise statement needs a knowledge link; proof needs a knowledge link
- Roots exercise statement needs a knowledge link; proof needs a knowledge link
- Double Kronecker exercise statement needs a knowledge link; proof needs a knowledge link
Compactness And Subsets Of C(K) 0% (0/5) page hidden
- Compact Subsets of Continuous Functions with Compact Domain is Equicontinuous lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- An Equicontinuous Family of Functions whose Domain is Compact is also Uniformly Equicontinuous proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Bounded subset of Rm is Totally Bounded proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Arzela-Ascoli theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Functions Bounded by 1 are Not Compact exercise statement needs a knowledge link; proof needs a knowledge link
Mean Value Theorem 0% (0/5) page hidden
- Limit Exists iff the Left and Right limits Exist and Agree theorem statement needs a knowledge link; proof needs a knowledge link
- Function goes to Infinity at a Point From the Right Alternate Characterization proposition proof needs a knowledge link
- local extreme value theorem statement needs a knowledge link; proof needs a knowledge link
- TODO corollary statement needs a knowledge link; proof needs a knowledge link
- mean value theorem statement needs a knowledge link; proof needs a knowledge link
Uniform Cost Search 0% (0/5) page hidden
- UCS Example Graph exercise statement needs a knowledge link; proof needs a knowledge link
- Later Expansion means Higher Cost lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Strictly Less Cost Implies Earlier Expansion Time lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- UCS Produces Minimal Cost Paths lemma statement needs a knowledge link; proof needs a knowledge link
- UCS Runtime theorem statement needs a knowledge link; proof needs a knowledge link
Linear 0% (0/5) page hidden
- Dual & Simplex exercise statement needs a knowledge link; proof needs a knowledge link
- Largest Disk in a Polygon exercise statement needs a knowledge link; proof needs a knowledge link
- Simple Scheduling with Prerequisites exercise statement needs a knowledge link; proof needs a knowledge link
- Logical Operators exercise statement needs a knowledge link; proof needs a knowledge link
- Cargo Plane exercise statement needs a knowledge link; proof needs a knowledge link
Eulers Totient Function 0% (0/5) page hidden
- Bijection between Cops and G's proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- The GCD Sets form a Partition proposition empty proof; proof needs a knowledge link
- Euler's theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Sum over Coprimes Totient Identity proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Sum over Divisors of the Totient function is the Identity lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
Number Bases 0% (0/5) page hidden
- Every Number has a Base \( b \) Representation theorem statement needs a knowledge link; proof needs a knowledge link
- Every Base \( b \) Representation is Unique theorem statement needs a knowledge link; proof needs a knowledge link
- Number of Digits Required in a Base 2 Representation proposition statement needs a knowledge link; proof needs a knowledge link
- Base Plus Minus One To a Power is Congruent to Plus Minus One lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Division by 9 Rule proposition statement needs a knowledge link; proof needs a knowledge link
Square Free Integers 0% (0/5) page hidden
- Square Free Division Characterization corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Divisor of Square Free Product is Divisor of Other proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- If a Square Free Number Divides a Square then It divides the Square Root proposition statement needs a knowledge link
- If a Square Free Number Divides the Nth Power then it Divides the Nth Square Root proposition statement needs a knowledge link; proof needs a knowledge link
- Power of a Square free Number Divides the Power of a Number Implies the Square Free Number Divides the Other proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Symmetry Group 0% (0/4) page hidden
- The Symmetry Group is a Group proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Symmetries of the Square are Isomorphic to \( D _ 4 \) proposition statement needs a knowledge link; proof needs a knowledge link
- Rectangle Symmetries are Isomorphic to the 4-Group proposition statement needs a knowledge link; proof needs a knowledge link
- Quadrilaterals that have Low Symmetry exercise statement needs a knowledge link; proof needs a knowledge link
Limits And Continuity 0% (0/4) page hidden
- The Characteristic Function is Discontinuous on the Boundary exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- Every Lipschitz Function is Continuous proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Base and Exponent in the Range 0 to 1 Inequality lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- x to the Alpha is in Lip Alpha exercise statement needs a knowledge link
Approximations 0% (0/4) page hidden
- Set Cover exercise statement needs a knowledge link; proof needs a knowledge link
- Metric TSP exercise statement needs a knowledge link; proof needs a knowledge link
- Randomized Max Cut exercise statement needs a knowledge link; proof needs a knowledge link
- Randomized Max-k-Sat exercise statement needs a knowledge link; proof needs a knowledge link
Recursion 0% (0/4) page hidden
- List Operation Tail exercise statement needs a knowledge link; proof needs a knowledge link
- foldr Expression proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- foldl Expression proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- foldr foldl Equivalent When Operation is Commutative and Associative corollary statement needs a knowledge link; proof needs a knowledge link
Fourier Analysis 0% (0/4) page hidden
- Riemann-Lebesgue Lemma theorem empty proof; proof needs a knowledge link
- Fejer's Theorem theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Fourier Series Converges in \( L ^ 2 \) theorem empty proof; proof needs a knowledge link
- Plancherel Identity for Fourier Series theorem empty proof; proof needs a knowledge link
Operators On Hilbert Spaces 0% (0/4) page hidden
- Adjoint Exists and is Unique theorem empty proof; proof needs a knowledge link
- Spectrum of a Bounded Operator is Compact theorem empty proof; proof needs a knowledge link
- Spectral Radius Formula theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Spectral Theorem for Compact Self-Adjoint Operators theorem empty proof; proof needs a knowledge link
Structures And Languages 0% (0/4) page hidden
- numbers in the number theory structure exercise proof needs a knowledge link
- For All Makes a Variable not Free corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Free Variable For All Easy exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- Free Variable For All Medium exercise proof needs a knowledge link
Dirichlet Theorem 0% (0/4) page hidden
- Dirichlet Theorem1 theorem statement needs a knowledge link; proof needs a knowledge link
- Dirichlet Charater1 proposition statement needs a knowledge link; proof needs a knowledge link
- Dirichlet Charater2 proposition statement needs a knowledge link; proof needs a knowledge link
- Euler product of L-function theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
Euclidean Algorithm 0% (0/4) page hidden
- euclidean algorithm theorem statement needs a knowledge link; proof needs a knowledge link
- bounded sum implies one less then half lemma statement needs a knowledge link; proof needs a knowledge link
- euclidean algorithm has exponentially decreasing remainders proposition statement needs a knowledge link; proof needs a knowledge link
- upper bound on euclidean algorithm iterations corollary statement needs a knowledge link; proof needs a knowledge link
Irrational Numbers 0% (0/4) page hidden
- irrational to the power of irrational can be rational exercise statement needs a knowledge link; proof needs a knowledge link
- Square Root of a Non Perfect Square is Irrational proposition statement needs a knowledge link; proof needs a knowledge link
- There is a Rational Between any Two Reals proposition statement needs a knowledge link; proof needs a knowledge link
- There is an Irrational Number between any two Real Numbers proposition statement needs a knowledge link; proof needs a knowledge link
Probability Measures 0% (0/4) page hidden
- Monotone Convergence Theorem for Expectations theorem empty proof; proof needs a knowledge link
- Dominated Convergence Theorem for Expectations theorem empty proof; proof needs a knowledge link
- Weak Law of Large Numbers theorem empty proof; proof needs a knowledge link
- Strong Law of Large Numbers theorem empty proof; proof needs a knowledge link
Bayesian Networks 0% (0/4) page hidden
- Bayes Burritos exercise statement needs a knowledge link; proof needs a knowledge link
- Bayesian Network Independence exercise statement needs a knowledge link; proof needs a knowledge link
- Bayesian Construction exercise statement needs a knowledge link; proof needs a knowledge link
- Bayesian Network Computation exercise statement needs a knowledge link; proof needs a knowledge link
Probability Measures 0% (0/4) page hidden
- Finite Pairwise Disjoint Probability proposition statement needs a knowledge link; proof needs a knowledge link
- probability of a single point with equally likely outcomes is zero proposition statement needs a knowledge link; proof needs a knowledge link
- Union Superset yields Sum Inequality proposition statement needs a knowledge link; proof needs a knowledge link
- Probability of a Random Variable being an Element of a Set using Inverse Image corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
Isomorphisms 0% (0/3) page hidden
- Identities are Carried proposition proof needs a knowledge link
- Powers Pass Through a Homomorphism proposition statement needs a knowledge link; proof needs a knowledge link
- Powers pass through an Isomorphism corollary statement needs a knowledge link
Normal Subgroups 0% (0/3) page hidden
- Normal Subgroup Test proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Solvable exercise statement needs a knowledge link; proof needs a knowledge link
- The Symmetric Group for n Greater Than or Equal to 5 Is not Solvable proposition statement needs a knowledge link; proof needs a knowledge link
Rubiks Cube 0% (0/3) page hidden
- The Rubiks Cube is a Group theorem statement needs a knowledge link; proof needs a knowledge link
- If a Sequence of Moves Yields the Identity then applying those Moves to the Physical Cube Means it is Solved proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Rubiks Cube corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
Systems Of Equations 0% (0/3) page hidden
- Encoding a system of Equations into Matrix Vector Multiplication proposition statement needs a knowledge link; proof needs a knowledge link
- The solution space of a homogeneous system of equations forms a subspace proposition statement needs a knowledge link; proof needs a knowledge link
- Basis for a Solution Space exercise statement needs a knowledge link; proof needs a knowledge link
Inverse Functions 0% (0/3) page hidden
- Invertible Iff Bijective theorem statement needs a knowledge link; proof needs a knowledge link
- Strictly Monotone Implies Invertible proposition statement needs a knowledge link; proof needs a knowledge link
- Strictly Monotone Implies Continuous Inverse proposition statement needs a knowledge link; proof needs a knowledge link
Trigonometric Functions 0% (0/3) page hidden
- Trigonometric Angle Addition Formulas proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Sine is Odd and Cosine is Even proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Sine is Zero on Zero to Pi Only at the Endpoints proposition statement needs a knowledge link; proof needs a knowledge link
Weierstrass Approximation 0% (0/3) page hidden
- Weierstrass Approximation theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Integrals Equal Zero Implies Function is Zero exercise statement needs a knowledge link
- Continuous Extension on Compact Subset exercise statement needs a knowledge link; proof needs a knowledge link
Counting Inversions 0% (0/3) page hidden
- Inversion Lookahead in Sorted Concatenation lemma statement needs a knowledge link; proof needs a knowledge link
- No Inversion Lookahead in Sorted Concatenation lemma statement needs a knowledge link; proof needs a knowledge link
- \( \texttt{counting\_inversions} \) is Correct proposition empty proof; proof needs a knowledge link
Dynamic Programming 0% (0/3) page hidden
- Programming Team exercise statement needs a knowledge link; proof needs a knowledge link
- Romeo and Juliet exercise statement needs a knowledge link; proof needs a knowledge link
- Mario and Bowser exercise statement needs a knowledge link; proof needs a knowledge link
Ford Fulkerson 0% (0/3) page hidden
- Flow out of the Source is the Same as Flow Entering the Sink corollary statement needs a knowledge link; proof needs a knowledge link
- Max-Flow Min-Cut theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Sum of Capacities Runtime proposition statement needs a knowledge link; proof needs a knowledge link
Interval Partitioning 0% (0/3) page hidden
- Inversion Lookahead in Sorted Concatenation lemma statement needs a knowledge link; proof needs a knowledge link
- No Inversion Lookahead in Sorted Concatenation lemma statement needs a knowledge link; proof needs a knowledge link
- \( \texttt{counting\_inversions} \) is Correct proposition empty proof; proof needs a knowledge link
Runtime 0% (0/3) page hidden
- Squared Dominates Linear exercise empty proof; proof needs a knowledge link
- Lower Degree Polynomial plus Constant is Eventually Dominated by Higher Degree exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- Basic Polynomial exercise statement needs a knowledge link; proof needs a knowledge link
Summations 0% (0/3) page hidden
- Constant can be Factored proposition statement needs a knowledge link; proof needs a knowledge link
- Sum of Added or Subtracted Sequence is the Addition or Subtraction of Sums proposition statement needs a knowledge link; proof needs a knowledge link
- Sum of Consecutive Integers theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
Graphs 0% (0/3) page hidden
- Sum of Degrees proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Graph with One Vertex is Connected corollary statement needs a knowledge link; proof needs a knowledge link
- A Disconnected Graph is the Union of Many Disjoint Connected Subgraphs proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Diophantine Equations 0% (0/3) page hidden
- A Linear Diophantine Equation Has Solutions if the GCD of the coefficients divides the Constant proposition statement needs a knowledge link
- One Solution Yields Infinitely Many proposition statement needs a knowledge link; proof needs a knowledge link
- The Set of Linear Combinations Are All Integral Multiples of Their GCD proposition statement needs a knowledge link
Generalized Gcd And Lcm 0% (0/3) page hidden
- GCD of GCD's Is the Same as Their Union proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Generalized Bezout Identity proposition statement needs a knowledge link
- Divisible by Relatively Prime Numbers Means Divisible by Their Product proposition statement needs a knowledge link; proof needs a knowledge link
Homomorphisms 0% (0/2) page hidden
- Crone Homomorphism Between Two by Two Matices with 0 in Bottom Left and Integers exercise statement needs a knowledge link; proof needs a knowledge link
- Homomorphisms carry Zero to Zero proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Dihedral Group 0% (0/2) page hidden
- Elements of a Dihedral Group proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Caley Table for D _ 4 proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Integers Modulo N 0% (0/2) page hidden
- choice of representative doesn't matter for addition proposition statement needs a knowledge link; proof needs a knowledge link
- representitive doens't matter for multiplication proposition statement needs a knowledge link; proof needs a knowledge link
Subgroups 0% (0/2) page hidden
- The Identity in a Subgroup is the Same corollary proof needs a knowledge link
- The Subgroup generated by S is the Smallest Subgroup containing S corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
Span And Linear Independence 0% (0/2) page hidden
- Linearly Independent iff Unique Representation proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Linearly Depedent iff Zero has a Non-Trivial Representation corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
Compactness And Extreme Values 0% (0/2) page hidden
- The Continuous Image of a Compact set is Compact theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Extreme Value theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
Power Series 0% (0/2) page hidden
- Term by Term Operations on Series proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Hadamard theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
Uniform Convergence And Continuity 0% (0/2) page hidden
- Uniform Convergence Equivalence In Terms of Lim Sup theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Dini's theorem statement needs a knowledge link; proof needs a knowledge link
Uniform Convergence And Integration 0% (0/2) page hidden
- Integral Convergence theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Derivative Convergence proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Category 0% (0/2) page hidden
- The opposite category is a category proposition statement needs a knowledge link; proof needs a knowledge link
- The Fundamental Group Is a Functor proposition statement needs a knowledge link
Clock Deltas 0% (0/2) page hidden
- Time Conversion Function corollary statement needs a knowledge link; proof needs a knowledge link
- Isolating for Travel Time and Clock Delta From Packets proposition statement needs a knowledge link; proof needs a knowledge link
Merge Sort 0% (0/2) page hidden
- \( \texttt{merge} \) is Correct lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- \( \texttt{merge\_sort} \) is Correct theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
Missing Number 0% (0/2) page hidden
- A Sorted Tuple Missing an Element Has a Change Point proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Equal Number of Ones and Zeros proposition empty proof; proof needs a knowledge link
Depth First Search 0% (0/2) page hidden
- DFS is not Complete theorem statement needs a knowledge link; proof needs a knowledge link
- DFS Example exercise statement needs a knowledge link; proof needs a knowledge link
Interval Scheduling 0% (0/2) page hidden
- Earliest Finish Time Adding Criterion lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Earliest Finish Time is Optimal proposition statement needs a knowledge link; proof needs a knowledge link
Racket 0% (0/2) page hidden
- Let in Racket exercise statement needs a knowledge link; proof needs a knowledge link
- Let Error exercise statement needs a knowledge link; proof needs a knowledge link
Syntax And Grammar 0% (0/2) page hidden
- Syntactically Valid Arithmetic exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- Ambiguous If Then Grammar exercise statement needs a knowledge link; proof needs a knowledge link
Vision 0% (0/2) page hidden
- Every Convolution Is a Linear Shift Invariant Filter proposition statement needs a knowledge link; proof needs a knowledge link
- Dirac Comb Convolved With a Function proposition statement needs a knowledge link; proof needs a knowledge link
Induction 0% (0/2) page hidden
- induction theorem statement needs a knowledge link; proof needs a knowledge link
- linear recursive sum proposition statement needs a knowledge link; proof needs a knowledge link
Cartesian Product 0% (0/2) page hidden
- intersection and cartesian product commute proposition statement needs a knowledge link
- cartesian product distributes over intersection proposition statement needs a knowledge link
Hilbert Spaces 0% (0/2) page hidden
- Uniform Boundedness Theorem theorem empty proof; proof needs a knowledge link
- There Is a Unique Hilbert Adjoint Operator proposition statement needs a knowledge link; proof needs a knowledge link
Mobius Inversion 0% (0/2) page hidden
- The Dirichlet Inverse for 1 is the Inverse proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- If an Arithmetic Function Convolved with 1 is Multiplicative so is the Original corollary statement needs a knowledge link
Limits Of Sets 0% (0/2) page hidden
- Half Closed Subset of Open for Smaller Value lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Open Sets as an Increasing Sequence of Sets proposition statement needs a knowledge link; proof needs a knowledge link
Infinite Product Topologies 0% (0/2) page hidden
- Sequences Converge Iff Projections Converge proposition statement needs a knowledge link
- Closure of R Infinity in the Box and Product Topologies proposition statement needs a knowledge link
Order Topology 0% (0/2) page hidden
- The Order Basis Is a Basis corollary statement needs a knowledge link; proof needs a knowledge link
- The Order Basis for the Dictionary Order on RxR proposition statement needs a knowledge link; proof needs a knowledge link
Splitting Fields 0% (0/1) page hidden
- Field Extension must Adjoin a Square Root proposition statement needs a knowledge link; proof needs a knowledge link
External Direct Products 0% (0/1) page hidden
- The External Direct Product With Componentwise Multiplication Is a Group proposition statement needs a knowledge link; proof needs a knowledge link
Properties Of Continuous Functions 0% (0/1) page hidden
- Sequential and Open Set Equivalences for Continuity theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
Continuity 0% (0/1) page hidden
- continuity equivalence exercise statement needs a knowledge link; proof needs a knowledge link
Extreme Value Theorem 0% (0/1) page hidden
- Continuous Implies Bounded proposition statement needs a knowledge link; proof needs a knowledge link
Monotone Sequences 0% (0/1) page hidden
- Reciprocal of the Reciprocal exercise statement needs a knowledge link
Cardinal Numbers 0% (0/1) page hidden
- The Countable Union of Countable Sets is Countable theorem statement needs a knowledge link; proof needs a knowledge link
Combinatorics 0% (0/1) page hidden
- painting exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
Searching 0% (0/1) page hidden
- Sudoku Representation exercise statement needs a knowledge link; proof needs a knowledge link
Lambda Calculus 0% (0/1) page hidden
- Lambda Evaluation of Euclidean Distance exercise statement needs a knowledge link; proof needs a knowledge link
Review 0% (0/1) page hidden
- Nested Triangles with Recursion exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
Master Theorem 0% (0/1) page hidden
- Master theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
Electrical Components 0% (0/1) page hidden
- Ohm's Law for Capacitors theorem statement needs a knowledge link; proof needs a knowledge link
Electricity 0% (0/1) page hidden
- Metals are Conductors proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Cardinality 0% (0/1) page hidden
- product rule proposition statement needs a knowledge link; proof needs a knowledge link
Geometry 0% (0/1) page hidden
- equilateral triangle construction theorem statement needs a knowledge link; proof needs a knowledge link
Decimal Expansion 0% (0/1) page hidden
- Finite Decimal Expansion Implies Restriction on Denominator proposition statement needs a knowledge link
Fibonacci 0% (0/1) page hidden
- Sum Equals Jump 2 minus One proposition statement needs a knowledge link; proof needs a knowledge link
Misc 0% (0/1) page hidden
- Max Length Sequence proposition statement needs a knowledge link; proof needs a knowledge link
Natural Numbers 0% (0/1) page hidden
- Subset of Naturals with a Max Element has Bounded Difference proposition statement needs a knowledge link; proof needs a knowledge link
Partial Summation 0% (0/1) page hidden
- Partial Summation (or Abel's Summation) theorem statement needs a knowledge link; proof needs a knowledge link
Riemann Zeta Function 0% (0/1) page hidden
- Zeta Function1 theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
Rsa Theory 0% (0/1) page hidden
- Why Raising the Ciphertext to the Power of The Private Key Yields the Plaintext proposition statement needs a knowledge link; proof needs a knowledge link
Introduction 0% (0/1) page hidden
- Union proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Gcd And Lcm 3% (1/30) page hidden
- GCD's Characterization corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- GCD is a Linear Combination proposition statement needs a knowledge link
- Relatively Prime implies there is a Linear Combination of One corollary statement needs a knowledge link
- Factoring GCD proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Forced Division through GCD proposition statement needs a knowledge link; proof needs a knowledge link
- If you Can find a Linear Combination that Yields One, then Their gcd is one proposition statement needs a knowledge link
- Swapping Factors exercise statement needs a knowledge link
- Coefficients have GCD one corollary statement needs a knowledge link
- Dividing out by GCD removes all Common Factors corollary statement needs a knowledge link; proof needs a knowledge link
- Non Trivial Divisors Don't Change with the Sign Changes corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Non Trivial Divisors is a Subset of 2 up to The Number corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Non Trivial Divisors of n Factorial Plus or Minus One corollary statement needs a knowledge link
- Common Divisors as the Intersection of the Non Trivial Divisors corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Common Divisor Set is Bounded Above corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Add a multiple of the other Yields Same Common Divisors proposition statement needs a knowledge link
- No Common Non Trivial Divisors Implies Relatively Prime proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Superset Divisors Implies Greater GCD proposition statement needs a knowledge link; proof needs a knowledge link
- Same Divisors Implies same GCD corollary statement needs a knowledge link; proof needs a knowledge link
- Add a multiple of the other Yields same GCD corollary statement needs a knowledge link
- Two Consecutive Integers Are Coprime proposition statement needs a knowledge link
- Division by Individuals Implies Division by Product when GCD is one proposition statement needs a knowledge link; proof needs a knowledge link
- GCD of a Multiple is Greater proposition statement needs a knowledge link
- A Divisor of a Coprime number Is also Coprime proposition statement needs a knowledge link; proof needs a knowledge link
- Relatively Prime Factors Implies Relatively Prime Product proposition statement needs a knowledge link
- GCD of a Factorial exercise statement needs a knowledge link
- Coprime GCD Characterization corollary statement needs a knowledge link; proof needs a knowledge link
- Least Common Multiple Characterization corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- GCD and LCM Connection theorem statement needs a knowledge link; proof needs a knowledge link
- Divisor Function is a Bijection if Relatively Prime exercise statement needs a knowledge link
Primes 4% (2/52) page hidden
- If a Prime Divides Another they are Equal corollary statement needs a knowledge link
- Primes of the Form 4k Plus Something proposition statement needs a knowledge link
- Composite iff Not Prime corollary statement needs a knowledge link; proof needs a knowledge link
- A Prime is Relatively Prime to Anything Else That is Not a Multiple of It lemma proof needs a knowledge link
- Every Number is a Product of Primes proposition statement needs a knowledge link
- There are Infinitely Many Primes theorem statement needs a knowledge link
- There is a Prime Between n and n Factorial proposition statement needs a knowledge link
- n Minus 1 Factorial is Divisible by n proposition statement needs a knowledge link; proof needs a knowledge link
- There are Gaps of Arbitrary Size between Consecutive Primes proposition statement needs a knowledge link
- Fundamental Theorem of Algebra theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Products as the Addition of Power Sequences corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Max of the Prime Power Sequence is 0 iff They share no Prime Factors lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Relatively Prime In terms of Prime Factorization proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Coprime and 5 exercise statement needs a knowledge link
- Subsequence of the sum of Two Disjoint Power Sequences Is Still a Disjoint Sum lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Number Divides a Coprime Product iff It is the product of Individual Divisors proposition statement needs a knowledge link; proof needs a knowledge link
- Euclid lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- If a Prime Divides a Product it Divides at Least One of the Factors corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Prime Divisors of a Factorial are Bounded by the Argument proposition statement needs a knowledge link
- A Divisor of a Product of Primes is a Product of Primes with Smaller Powers proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- GCD and LCM as Prime Powers proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Perfect Square iff Every Power in the Prime Factorization is Even proposition statement needs a knowledge link; proof needs a knowledge link
- Division and Prime Power Equivalence proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Every Composite Three Digit Number must have a Prime Factor Less than or Equal to 31 exercise statement needs a knowledge link; proof needs a knowledge link
- Lower Bound on Primes in Factorization Limits the Number of Prime Factors exercise statement needs a knowledge link; proof needs a knowledge link
- Goldbach Equivalence exercise statement needs a knowledge link; proof needs a knowledge link
- p Minus one Factorial is Congruent to Minus one Mod p proposition statement needs a knowledge link
- p Minus two Factorial is Congruent to One Mod p corollary statement needs a knowledge link; proof needs a knowledge link
- Factorial Fun exercise statement needs a knowledge link; proof needs a knowledge link
- 1 2 3 n go exercise statement needs a knowledge link
- Generalized Wilson's Theorem theorem statement needs a knowledge link
- Wilson Folded in Half corollary statement needs a knowledge link; proof needs a knowledge link
- Fermats Helper lemma statement needs a knowledge link
- Fermats Helper Mod Exponent corollary statement needs a knowledge link; proof needs a knowledge link
- Fermats Little theorem statement needs a knowledge link; proof needs a knowledge link
- Remainder of Power Plus One exercise statement needs a knowledge link; proof needs a knowledge link
- 2's and 5's exercise statement needs a knowledge link
- Prime Power Sum exercise statement needs a knowledge link; proof needs a knowledge link
- If a Doesn't Divide p Then Fermats Little Theorem and Fermats Little Helper are Equivalent corollary statement needs a knowledge link
- Almost Prime exercise statement needs a knowledge link
- 561 is Carmichael exercise statement needs a knowledge link
- Korselt's Criterion corollary empty proof; proof needs a knowledge link
- 1729 is Carmichael exercise statement needs a knowledge link; proof needs a knowledge link
- Fermats Compositeness Test proposition statement needs a knowledge link
- \( 341 \) is a Pseudoprime to base \( 2 \) exercise statement needs a knowledge link; proof needs a knowledge link
- Every Carmichael Number is a Probable Prime corollary statement needs a knowledge link
- At Least Half of the Tested Elements will be Witnesses to a Composite Non-Carmichael Number proposition statement needs a knowledge link; proof needs a knowledge link
- Every Odd Number is a Difference of Squares proposition statement needs a knowledge link; proof needs a knowledge link
- The Sum is less than or Equal to the Product lemma statement needs a knowledge link; proof needs a knowledge link
- Fermats Factorization proposition statement needs a knowledge link
Banach Spaces 4% (1/23) page hidden
- Norm Is Zero Iff Input Is Zero corollary proof needs a knowledge link
- Non-negativity of the Norm corollary proof needs a knowledge link
- Norm Induced Metric proposition proof needs a knowledge link
- A Compact Subset of a Metric Space Is Closed and Bounded lemma statement needs a knowledge link; proof needs a knowledge link
- Linear Independence Coefficient Inequality lemma statement needs a knowledge link; proof needs a knowledge link
- In a Finite Dimensional Normed Space Compactness Is Equivalent to Closed and Bounded proposition statement needs a knowledge link; proof needs a knowledge link
- Riesz's lemma statement needs a knowledge link; proof needs a knowledge link
- Every Finite Dimensional Subspace of a Normed Space Is Complete proposition statement needs a knowledge link; proof needs a knowledge link
- A Subspace of a Complete Metric Space Is Complete Iff It Is Closed proposition statement needs a knowledge link; proof needs a knowledge link
- Every Finite Dimensional Subspace of a Normed Space Is Closed proposition statement needs a knowledge link
- If the Closed Unit Ball Is Compact Then the Space Is Finite Dimensional theorem statement needs a knowledge link
- Any Finite Dimensional Vector Space With a Norm Is Complete corollary statement needs a knowledge link; proof needs a knowledge link
- Operator Norm Restricted to the Circle corollary statement needs a knowledge link; proof needs a knowledge link
- The Operator Norm of a Bounded Operator Is The Smallest Number Satisfying an Inequality proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Image of a Linear Opertor Is a Vector Space proposition statement needs a knowledge link; proof needs a knowledge link
- If a Linear Operator Has a Finite Dimensional Domain Then the Domain of Its Image Is Less Than It proposition statement needs a knowledge link; proof needs a knowledge link
- The Nullspace of a Linear Operator Is a Vector Space proposition statement needs a knowledge link; proof needs a knowledge link
- Continuity of a Linear Operator corollary proof needs a knowledge link
- Continuity Boundedness Equivalence of Linear Operators proposition proof needs a knowledge link
- The Adjoint Is a Unique Linear Operator corollary statement needs a knowledge link; proof needs a knowledge link
- The Adjoint of a Linear Operator Exists When the Vector Space Is Finite Dimensional theorem statement needs a knowledge link; proof needs a knowledge link
- A Linear Operator With a Finite Dimensional Domain Is Bounded proposition statement needs a knowledge link; proof needs a knowledge link
Division 5% (1/22) page hidden
- Divisible iff Integer Quotient corollary statement needs a knowledge link; proof needs a knowledge link
- n Factorial is Divisible by Its Factors proposition statement needs a knowledge link
- Not Divisible iff Non-Integer Quotient corollary statement needs a knowledge link; proof needs a knowledge link
- 1 Divides Everything proposition statement needs a knowledge link
- If something Divides One then it is One or Negative One proposition statement needs a knowledge link; proof needs a knowledge link
- If Zero Divides a Number, then that Number is Zero proposition statement needs a knowledge link; proof needs a knowledge link
- 0 is Divisible by Everything proposition statement needs a knowledge link; proof needs a knowledge link
- Everything Divides Itself proposition statement needs a knowledge link; proof needs a knowledge link
- Division is Transitive proposition statement needs a knowledge link; proof needs a knowledge link
- If you Divide a Number, then You Divide their Multiples proposition statement needs a knowledge link; proof needs a knowledge link
- If it Divides it's Less than or Equal with Absolute Values proposition proof needs a knowledge link
- Larger Absolute value Implies no Division corollary statement needs a knowledge link
- If two Numbers Divide each other then Their Absolute Values are Equal proposition statement needs a knowledge link
- If a Number Divides another Then Given a Common Factor Their Quotients Divide Each Other proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Linearity of Division proposition statement needs a knowledge link; proof needs a knowledge link
- A power of an Even Number is Even proposition statement needs a knowledge link; proof needs a knowledge link
- A power of an Even Number is Odd proposition statement needs a knowledge link; proof needs a knowledge link
- Product of Two Equal Factors means They are Square Roots lemma statement needs a knowledge link; proof needs a knowledge link
- Product Implies Square Root bound on Factors proposition statement needs a knowledge link
- Number To the Power m Minus 1 Divides Number To the Power n if m Divides n proposition statement needs a knowledge link; proof needs a knowledge link
- Forced Division Of Sum proposition statement needs a knowledge link; proof needs a knowledge link
Binary Operations And Groups 5% (1/21) page hidden
- Bracket Placement Doesn't Matter for Associative Binary Operations theorem proof needs a knowledge link
- Associativity Implies Folds are Equal corollary statement needs a knowledge link
- The Integers are a Group proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- identity element is unique in a group proposition statement needs a knowledge link; proof needs a knowledge link
- inverse element is unique proposition statement needs a knowledge link; proof needs a knowledge link
- two inverses cancel proposition statement needs a knowledge link; proof needs a knowledge link
- inverse of ab proposition statement needs a knowledge link
- Cancellation proposition statement needs a knowledge link; proof needs a knowledge link
- The Integers with Addition are Abelian proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Cartesian Product of Abelian Groups is Abelian proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Power Properties proposition statement needs a knowledge link; proof needs a knowledge link
- Finite Order Implies Powers Have Same Remainder proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- If the power of an element with finite order yields the identity then the order divides the power corollary statement needs a knowledge link; proof needs a knowledge link
- Binary Operations And Groups proposition statement needs a knowledge link; proof needs a knowledge link
- TODO exercise statement needs a knowledge link; proof needs a knowledge link
- TODO exercise statement needs a knowledge link; proof needs a knowledge link
- cyclic implies abelian proposition statement needs a knowledge link; proof needs a knowledge link
- power equivalence proposition statement needs a knowledge link; proof needs a knowledge link
- gcd generates the same theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- TODO corollary statement needs a knowledge link; proof needs a knowledge link
Summations And Series 5% (1/20) page hidden
- Closed form of a Geometric Summation proposition statement needs a knowledge link; proof needs a knowledge link
- Geometric Summation with Offset corollary statement needs a knowledge link; proof needs a knowledge link
- Harmonic Series Diverges proposition statement needs a knowledge link; proof needs a knowledge link
- Geometric Power Series proposition statement needs a knowledge link; proof needs a knowledge link
- Geometric Power Series with Offset corollary statement needs a knowledge link
- Series of Halves is Bounded corollary statement needs a knowledge link; proof needs a knowledge link
- geometric sum with incrementing factor proposition statement needs a knowledge link; proof needs a knowledge link
- geometric series with incrementing factor proposition statement needs a knowledge link; proof needs a knowledge link
- Convergent Series Implies Sequence goes to Zero proposition statement needs a knowledge link; proof needs a knowledge link
- Cauchy Criterion for Series theorem statement needs a knowledge link; proof needs a knowledge link
- If the Absolute Valued Series Converges then so does the Original corollary statement needs a knowledge link
- Non-Negative Series Converges iff Partial Sums are Bounded Above proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Monotone Increasing Sequence with a Subsequence which is Bounded Above Implies Entire Sequence is Bounded Above lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Cauchy Condensation Test theorem statement needs a knowledge link
- One over N to the p Convergence iff p is greater than One proposition statement needs a knowledge link
- Root Test theorem statement needs a knowledge link
- Ratio Test proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Leibniz Alternating Series Test proposition empty proof; proof needs a knowledge link
- Every Rearrangment of an Absolutely Convergent Series converges to the same Limit theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
Discrete Distributions 5% (1/20) page hidden
- Probability Mass Function Properties lemma proof needs a knowledge link
- Discrete Marginal Mass Functions lemma proof needs a knowledge link
- Law of Total Expectation theorem proof needs a knowledge link
- Discrete Convolution Formula theorem proof needs a knowledge link
- Expectation of Indicator proposition proof needs a knowledge link
- Expected Value of a Bernoulli Random Variable corollary proof needs a knowledge link
- Expected Value of the Binomial Distribution corollary empty proof; proof needs a knowledge link
- Probability Mass Function of the Binomial Distribution proposition empty proof; proof needs a knowledge link
- Coin Counting exercise statement needs a knowledge link; proof needs a knowledge link
- Poisson as a Limit of Binomial theorem statement needs a knowledge link; proof needs a knowledge link
- \( T _ k = n \) Equivalence proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Probability Mass Function of the Negative Binomial Distribution proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Large Number of Tossed Coins exercise statement needs a knowledge link; proof needs a knowledge link
- Poisson Process exercise statement needs a knowledge link; proof needs a knowledge link
- Fundamental Properties of Covariance proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Covariance Equations exercise statement needs a knowledge link; proof needs a knowledge link
- Sum of Bernoulli exercise statement needs a knowledge link; proof needs a knowledge link
- Geometric Expectation exercise statement needs a knowledge link; proof needs a knowledge link
- Poisson Quotient exercise statement needs a knowledge link; proof needs a knowledge link
Algebraic Topology 5% (1/19) page hidden
- Path Homotopy Is Invariant Modulo Continuous Function Composition proposition statement needs a knowledge link
- Path Homotopy Is an Equivalence Relation proposition statement needs a knowledge link; proof needs a knowledge link
- Path Concatentation Yields a Path proposition statement needs a knowledge link; proof needs a knowledge link
- Path Homotopy Leads to Concatenation Homotopy proposition statement needs a knowledge link; proof needs a knowledge link
- The Fundamental Group Is a Group With Concatenation proposition statement needs a knowledge link; proof needs a knowledge link
- Function Homotopy Is an Equivalence Relation proposition statement needs a knowledge link
- Function Homotopy Leads to Composition Homotopy proposition statement needs a knowledge link; proof needs a knowledge link
- Rn Is Simply Connected corollary statement needs a knowledge link; proof needs a knowledge link
- Simply Connected Iff Any Two Paths With Shared Ends Are Homotopic proposition statement needs a knowledge link; proof needs a knowledge link
- Circle Representative and Simply Connected Equivalences proposition statement needs a knowledge link; proof needs a knowledge link
- A Continous Function From S1 to S2 Is Homotopic to the Same Which Is Not Surjective proposition statement needs a knowledge link; proof needs a knowledge link
- S2 Is Simply Connected proposition statement needs a knowledge link; proof needs a knowledge link
- Path Lifting lemma statement needs a knowledge link; proof needs a knowledge link
- Covering Alternate Characterization proposition proof needs a knowledge link
- Connected Covering Yields a Lift proposition statement needs a knowledge link; proof needs a knowledge link
- Lifting Criterion proposition statement needs a knowledge link
- The Fundamental Group of a Product Is Isomorphic to a Product of Fundamental Groups proposition statement needs a knowledge link
- The Fundamental Group of the Figure 8 Is in Bijection to the Free Group of Two Elements exercise statement needs a knowledge link; proof needs a knowledge link
Set Difference 6% (1/17) page hidden
- set difference is empty iff one is a superset of the other proposition statement needs a knowledge link; proof needs a knowledge link
- Difference of a Superset is Empty proposition statement needs a knowledge link; proof needs a knowledge link
- Empty Intersection implies Superset Difference proposition statement needs a knowledge link; proof needs a knowledge link
- Set Difference with Itself is Empty proposition statement needs a knowledge link; proof needs a knowledge link
- Set Difference with the Empty set is Itself proposition statement needs a knowledge link; proof needs a knowledge link
- Redundant Intersection in Difference proposition statement needs a knowledge link; proof needs a knowledge link
- Intersection Distributes through Set Difference proposition statement needs a knowledge link; proof needs a knowledge link
- Intersection with an Ambient Difference proposition statement needs a knowledge link; proof needs a knowledge link
- Intersection of Ambient Differences proposition statement needs a knowledge link; proof needs a knowledge link
- Union Intersection Inversion through set Difference proposition statement needs a knowledge link; proof needs a knowledge link
- Double Difference creates Intersection proposition statement needs a knowledge link; proof needs a knowledge link
- Triple Difference Equality proposition statement needs a knowledge link; proof needs a knowledge link
- Double Set Difference Yields Intersection proposition statement needs a knowledge link; proof needs a knowledge link
- Double Set Difference Cancels for Subsets corollary statement needs a knowledge link; proof needs a knowledge link
- DeMorgan's Laws theorem statement needs a knowledge link; proof needs a knowledge link
Boundedness 7% (1/14) page hidden
- Supremum of Suprema Is the Same as Their Union proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Supremum is Unique corollary statement needs a knowledge link
- Greater than Infimum Implies Not a Lower Bound corollary statement needs a knowledge link; proof needs a knowledge link
- less than sup implies not an upper bound corollary statement needs a knowledge link; proof needs a knowledge link
- Infimum or Supremum of a Set may not be part of Itself proposition statement needs a knowledge link; proof needs a knowledge link
- If an Upper Bound is Contained in the Set it is the Supremum lemma statement needs a knowledge link; proof needs a knowledge link
- Maximum Value iff Contains Sup proposition statement needs a knowledge link
- A Number is Greater than the Other iff the Difference is Positive proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Multiplying an Inequality by Minus One Flips it corollary statement needs a knowledge link
- Negative of Infimum is Supremum of Negatives proposition statement needs a knowledge link
- If a Set Equals its Negative Then its Absolute Value is Its Non-Negative Half proposition statement needs a knowledge link; proof needs a knowledge link
- If a Set Equals its Negative the Sup of the Absolute value and the Original Set are the Same proposition statement needs a knowledge link
- Half Length Interval lemma statement needs a knowledge link; proof needs a knowledge link
Arithmetic Functions 7% (1/14) page hidden
- Equation for the Divisor Counting Function proposition empty proof; proof needs a knowledge link
- Equation for the Divisor Sum Function proposition empty proof; proof needs a knowledge link
- Multiplicative Functions that map 1 to 0 Are the Constant 0 Function proposition proof needs a knowledge link
- The Base Function is Totally Multiplicative corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- When the Constant Function is Multiplicative proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Product of Two Multiplicative Functions is Multiplicative theorem empty proof; proof needs a knowledge link
- The Quotient of Two Multiplicative Functions is Multiplicative theorem empty proof; proof needs a knowledge link
- Two Multiplicative Functions are Equal if they Agree on Prime Powers theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Sum of Reciprocals of Divisors Equation exercise statement needs a knowledge link
- The Sum of a Multiplicative Function over Divisors of a Number is Multiplicative theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Dirichlet Convolution of two Multiplicative Functions is Multiplicative theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Dirichlet Identity is Completely Multiplicative corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Dirichlet Identity is an Identity corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
Metric Topology 8% (1/13) page hidden
- The Set of Open Balls Is a Basis proposition proof needs a knowledge link
- Balls in a Metric Are Open proposition statement needs a knowledge link; proof needs a knowledge link
- All Metric Topologies Are T2 corollary proof needs a knowledge link
- Zero Is Part of the Closure of Positive Sequences in the Box Topology, but Not in the Sequential Closure proposition statement needs a knowledge link
- The Sequential Closure Is a Subset of the Closure lemma statement needs a knowledge link
- The Max of Two Metrics Is a Metric lemma statement needs a knowledge link; proof needs a knowledge link
- The Product of Two Metrizable Spaces Is Metrizable proposition statement needs a knowledge link
- \( \mathbb{ R } \times \mathbb{ R } \) in the Dictionary Order Topology Is Metrizable corollary statement needs a knowledge link
- The Metric Is Continuous in Its Metric Space proposition statement needs a knowledge link; proof needs a knowledge link
- The Metric Topology Is the Coarsest Topology Such That the Metric Is Continuous proposition statement needs a knowledge link; proof needs a knowledge link
- Closure of \( \mathbb{ R } ^ \infty \) in \( \mathbb{ R } ^ \mathbb{ N } \) in the Uniform Topology proposition statement needs a knowledge link
- Infinite Product of Intervals in the Uniform Topology exercise statement needs a knowledge link; proof needs a knowledge link
Functions On Euclidean Space 8% (1/12) page hidden
- Euclidean Norm Properties theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Euclidean Inner Product Properties theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Heine-Borel for Closed Intervals theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Tube Lemma for a Compact Factor theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Product of Compact Sets is Compact corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Finite Product of Compact Sets is Compact corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Closed Bounded Subset of Euclidean Space is Compact corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Continuity Open Set Characterization theorem empty proof; proof needs a knowledge link
- Continuous Image of Compact Set is Compact theorem empty proof; proof needs a knowledge link
- Continuity Oscillation Characterization theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Large Oscillation Set is Closed theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
Hausdorff Spaces 9% (1/11) page hidden
- Limit Point Iff Every Neighborhood Contains Infinitely Many Points proposition statement needs a knowledge link
- Convergence Iff All but Finitely Many corollary statement needs a knowledge link; proof needs a knowledge link
- A Subspace of a T2 Space Remains T2 corollary statement needs a knowledge link; proof needs a knowledge link
- Unique Limits in a Hausdorff Space proposition statement needs a knowledge link
- Every Finite Set is Closed in a Hausdorff Space corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Topological Space Is Hausdorff Iff the Diagonal Is Closed proposition statement needs a knowledge link; proof needs a knowledge link
- Pairwise Disjoint Sets for a Collection of Points in a Hausdorff Space proposition proof needs a knowledge link
- A Subspace of a T2 Space Is T2 corollary statement needs a knowledge link; proof needs a knowledge link
- Product of T2 Spaces Is T2 proposition statement needs a knowledge link; proof needs a knowledge link
- Every Ordered Set With the Order Topology Is Hausdorff theorem proof needs a knowledge link
Vectors 10% (3/31) page hidden
- Dot Product is Symmetric proposition statement needs a knowledge link
- Dot Product Distributes into Vector Addition from the Right proposition statement needs a knowledge link
- Dot Product Distributes into Vector Addition from the Left corollary statement needs a knowledge link
- Dot Product is Compatible with Scalar Multiplication proposition statement needs a knowledge link
- Dot Product is Positive Definite proposition statement needs a knowledge link
- Norm Squared is the Dot Product corollary statement needs a knowledge link
- Nonzero Vector has Positive Length corollary statement needs a knowledge link
- Dot Product Expansion of Norm Squared proposition statement needs a knowledge link
- Euclidean Law of Cosines proposition statement needs a knowledge link; proof needs a knowledge link
- Norm Equals Absolute Dot Product with a Parallel Unit Vector proposition proof needs a knowledge link
- Dot Product of a Vector Parallel to a Unit Vector proposition proof needs a knowledge link
- Cross Product Vector Component Formula proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Cross Product is Orthogonal to its Factors proposition statement needs a knowledge link
- Cross Product Distributes Over Vector Addition in the Second Input proposition statement needs a knowledge link
- Cross Product Scalar Multiplication in the Second Input proposition statement needs a knowledge link
- Cross Product is Anti-Commutative proposition statement needs a knowledge link
- Cross Product Distributes Over Vector Addition in the First Input corollary statement needs a knowledge link
- Cross Product Scalar Multiplication in the First Input corollary statement needs a knowledge link
- Cross Product Expands Over Two Vector Sums corollary statement needs a knowledge link
- Span Plane has Cross Product Normal proposition statement needs a knowledge link
- Coordinate Equation of a Plane is Normal Form proposition statement needs a knowledge link
- Translated Span Plane Equals Translated Normal Plane proposition statement needs a knowledge link
- Equation of a Line as a 3d Dot Product corollary statement needs a knowledge link; proof needs a knowledge link
- Equation of a Line Given by the Cross Product proposition statement needs a knowledge link
- Parallelogram Law proposition statement needs a knowledge link
- The Norm of The Half of The Sum of Two Unit Vectors is One iff They are Equal exercise statement needs a knowledge link
- Vector Cosine Law proposition statement needs a knowledge link
- Dot Product Is a Sum of Norms corollary statement needs a knowledge link
Homomorphisms 10% (1/10) page hidden
- The order of The Mapped Element Divides the Original proposition statement needs a knowledge link
- Associative Implies Left Equals Right Iteration proposition statement needs a knowledge link; proof needs a knowledge link
- Left Tuples as Iteration proposition statement needs a knowledge link; proof needs a knowledge link
- Right Tuples as Iteration proposition statement needs a knowledge link; proof needs a knowledge link
- Iterated Homomorphism theorem statement needs a knowledge link; proof needs a knowledge link; empty proof
- Homomorphism Between a Group and it's Cartesian Product exercise empty proof; statement needs a knowledge link; proof needs a knowledge link
- Homomorphisms proposition statement needs a knowledge link
- Finding Group Homomorphisms exercise statement needs a knowledge link; proof needs a knowledge link
Breadth First Search 11% (1/9) page hidden
- BFS Example exercise statement needs a knowledge link; proof needs a knowledge link
- Breadth First Search is Correct theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- BFS Searches Less Deep Vertices First proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- BFS Expands by Depth Layer corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- BFS Expands All Enqueued Vertices proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- BFS Time Complexity theorem statement needs a knowledge link; proof needs a knowledge link
- BFS Space Complexity theorem statement needs a knowledge link; proof needs a knowledge link
- BFS Cost Optimal? exercise statement needs a knowledge link; proof needs a knowledge link
Expected Value 11% (1/9) page hidden
- Squared Expectation exercise statement needs a knowledge link; proof needs a knowledge link
- Expectation is Linear proposition empty proof; proof needs a knowledge link
- Variance as Squares of Expectation corollary empty proof; proof needs a knowledge link
- Variance is Not Linear proposition empty proof; proof needs a knowledge link
- Covariance as Expectation Product proposition proof needs a knowledge link
- Independent Random Variables Factor Expectation lemma proof needs a knowledge link
- Cauchy Schwarz Inequality for Random Variables theorem proof needs a knowledge link
- Correlation Coefficient is Bounded corollary proof needs a knowledge link
Riemann Integration 13% (3/24) page hidden
- A Delta Sum Telescopes corollary statement needs a knowledge link
- Upper Sum Decreases over Refinements proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Lower Sum Decreases over Refinements proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Riemann Integrable Characterizations theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Function Difference Set is the Same as its Negative lemma statement needs a knowledge link; proof needs a knowledge link
- Supremum of the Function Difference Set is the Same with Absolute Values corollary statement needs a knowledge link
- Supremum of the Sum of Two Functions is Less than or Equal to the Sum of the Supremums proposition statement needs a knowledge link; proof needs a knowledge link
- Supremum of the Sum of two Functions is the Same as the Sum of the Supremums if the Variables are Independent proposition statement needs a knowledge link; proof needs a knowledge link
- Bound on the Upper minus Lower Sum for The Same Partition proposition statement needs a knowledge link
- Riemann Integrable on Two Parts of the Interval Means Riemann Integrable on the Whole Interval lemma statement needs a knowledge link; proof needs a knowledge link
- Multivariable Refinement Improves Darboux Sums lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Multivariable Lower Sums are Below Upper Sums corollary empty proof; statement needs a knowledge link; proof needs a knowledge link
- Multivariable Riemann Integrability Criterion theorem empty proof; proof needs a knowledge link
- Countable Union of Measure Zero Sets has Measure Zero theorem empty proof; proof needs a knowledge link
- Compact Measure Zero Implies Content Zero theorem empty proof; proof needs a knowledge link
- Lebesgue Criterion for Multivariable Riemann Integrability theorem empty proof; proof needs a knowledge link
- Jordan Measurable iff Boundary has Measure Zero theorem empty proof; proof needs a knowledge link
- Removing a Measure-Zero Set Does Not Change a Riemann Integral proposition proof needs a knowledge link
- Fubini for Riemann Integrals on Rectangles theorem empty proof; proof needs a knowledge link
- Polar Coordinate Change of Variables corollary statement needs a knowledge link
- Sard's Theorem for the Equal-Dimension \( C ^ 1 \) Case theorem empty proof; proof needs a knowledge link
Connectedness And Compactness 13% (3/23) page hidden
- Connected Iff Every Clopen Is Trivial corollary statement needs a knowledge link
- A Union of Connected Sets That Intersect Is Connected lemma statement needs a knowledge link; proof needs a knowledge link
- The Continuous Image of a Connected Set Is Connected proposition statement needs a knowledge link; proof needs a knowledge link
- Intermediate Value theorem proof needs a knowledge link
- Circle Function Has a Meet Point exercise statement needs a knowledge link; proof needs a knowledge link
- Path Connected Implies Connected proposition statement needs a knowledge link; proof needs a knowledge link
- The Topologists Sine Curve Is Connected but Not Path Connected proposition statement needs a knowledge link; proof needs a knowledge link
- The Topologists Sine Curve Is Connected but Not Path Connected proposition statement needs a knowledge link
- Balls in Rn Are Path Connected proposition statement needs a knowledge link; proof needs a knowledge link
- Any Open Connected Subset of Rn Is Path-connected exercise statement needs a knowledge link
- The Finite Complement Topology on an Uncountable Space Is Compact proposition statement needs a knowledge link; proof needs a knowledge link
- The Countable Complement Toplogy on an Uncountable Set May Not Be Compact exercise proof needs a knowledge link
- Every Compact Subspace of a Hausdorff Space Is Closed proposition statement needs a knowledge link; proof needs a knowledge link
- Every Closed Subspace of a Compact Space Is Compact proposition statement needs a knowledge link; proof needs a knowledge link
- The Image of a Compact Set Under a Continuous Function Is Compact proposition statement needs a knowledge link; proof needs a knowledge link
- Hausdorff Spaces Separate Compact Subspaces and Points lemma statement needs a knowledge link; proof needs a knowledge link
- Disjoint Compact Subspaces of a Hausdorff Space Can Be Separated proposition statement needs a knowledge link
- A Continuous Function From a Compact Space to a Hausdorff Space Is Closed proposition statement needs a knowledge link; proof needs a knowledge link
- There Is Always Space for an Open Rectangle Between a Cartesian Product of Compact Spaces and an Open Set Containing Them proposition statement needs a knowledge link; proof needs a knowledge link
- Properties of the Infimum Metric and Compact Spaces exercise statement needs a knowledge link; proof needs a knowledge link
Limit Points 14% (1/7) page hidden
- Limit Point Iff Every Basis Element Containing Intersects at a Different Point corollary proof needs a knowledge link
- Limit Point iff Element of Closure minus a Point corollary statement needs a knowledge link
- Limit Points are a Subset of Closure proposition statement needs a knowledge link
- Closure Equals Limit Points Union Itself proposition statement needs a knowledge link
- The Closure Is the Union of a and the Limits Points of a That Are Not in A corollary statement needs a knowledge link; proof needs a knowledge link
- A Subset of a Topological Space is Closed if and only if it contains all it's Limit Points corollary statement needs a knowledge link
Induction 15% (3/20) page hidden
- Well Ordering Principle proposition statement needs a knowledge link; proof needs a knowledge link
- Empty Negation Set means True for Everything proposition proof needs a knowledge link
- Triangular Numbers with the Well Ordering Principle exercise statement needs a knowledge link
- There is Not an Infinitely Strictly Decreasing Sequence of Natural Numbers exercise proof needs a knowledge link
- Archimedean Property proposition statement needs a knowledge link; proof needs a knowledge link
- There is a Source and a Sink exercise statement needs a knowledge link
- Formula for the Triangular Numbers corollary statement needs a knowledge link; proof needs a knowledge link
- Telescoping Sum proposition proof needs a knowledge link
- Formula for n-term arithmetic Progression proposition statement needs a knowledge link; proof needs a knowledge link
- Summation of Consecutive Odd Numbers corollary statement needs a knowledge link; proof needs a knowledge link
- Sum of the Squares of Consecutive Integers proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Multiplying by a non Negative Constant Retains Inequality lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- The Square of a Real Number is Non-Negative proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Bernouilli's Inequality proposition statement needs a knowledge link
- The Well Ordering Principle, Induction and Strong Induction are Equivalent corollary statement needs a knowledge link; proof needs a knowledge link
- Finite Induction proposition empty proof; proof needs a knowledge link
- Upward Downward Induction theorem statement needs a knowledge link
Basis 15% (2/13) page hidden
- The Size of a Basis is Unique proposition statement needs a knowledge link; proof needs a knowledge link
- Every Vector Space has a Basis proposition proof needs a knowledge link
- Dimension of Rn corollary statement needs a knowledge link
- Coordinate Vector in a Nonstandard Basis exercise statement needs a knowledge link; proof needs a knowledge link
- generating a unit column matrix corollary statement needs a knowledge link; proof needs a knowledge link
- basis implies unique representation theorem statement needs a knowledge link; proof needs a knowledge link
- something theorem statement needs a knowledge link; proof needs a knowledge link
- matrix representation of a transformation theorem statement needs a knowledge link; proof needs a knowledge link
- Change of Basis Matrix Columns theorem statement needs a knowledge link
- Linear Idependent Set Expansion To Basis proposition statement needs a knowledge link; proof needs a knowledge link
- Extending a Basis exercise statement needs a knowledge link; proof needs a knowledge link
Lebesgue Measure 15% (2/13) page hidden
- Outer Measure is Additive when One Set is Open theorem statement needs a knowledge link
- Outer Measure is Additive when One Set is Closed theorem statement needs a knowledge link
- Borel Sets Can be Approximated from Below by Closed Sets theorem statement needs a knowledge link
- Outer Measure is Additive when One Set is Borel theorem statement needs a knowledge link
- There is a Non-Borel Set of Finite Outer Measure theorem statement needs a knowledge link
- Equivalent Characterizations of Lebesgue Measurable Sets theorem statement needs a knowledge link
- Base Three Description of the Cantor Set theorem statement needs a knowledge link; proof needs a knowledge link
- Basic Properties of the Cantor Set theorem statement needs a knowledge link
- Cantor Function theorem statement needs a knowledge link; proof needs a knowledge link
- Cantor Set is Uncountable corollary statement needs a knowledge link
- Continuous Image of a Lebesgue Measurable Set Can be Nonmeasurable theorem statement needs a knowledge link
Linked Lists 17% (1/6) page hidden
- All Input Arrays have a Meet Value proposition statement needs a knowledge link; proof needs a knowledge link
- All Index Array Graphs have A Cycle Continuing Forever corollary statement needs a knowledge link
- Termination of the Seond while loop and the Value of \(\mathtt{slow1}\) is the Meet Value lemma statement needs a knowledge link
- The Meet Value is the Duplicated Value lemma statement needs a knowledge link; proof needs a knowledge link
- \( \mathtt{find\_duplicate} \) is Correct theorem statement needs a knowledge link
Functions 20% (5/25) page hidden
- An Element Belongs to a set Implies its Image belongs to the Image of the Set corollary statement needs a knowledge link
- Image Maintains Subsets proposition statement needs a knowledge link
- Union Factors through Image proposition statement needs a knowledge link
- Image of the Intersection is a Subset of the Intersection of the Images proposition statement needs a knowledge link
- The Difference of Images is a Subset of the Image of the Difference proposition statement needs a knowledge link; proof needs a knowledge link
- Inverse Image Maintains Subsets proposition statement needs a knowledge link
- Inverse Image of The Range is The Domain proposition statement needs a knowledge link
- Inverse Image of a Composition proposition statement needs a knowledge link; proof needs a knowledge link
- Union Factors through Inverse Image proposition statement needs a knowledge link
- Set Difference Factors through Inverse Image proposition statement needs a knowledge link; proof needs a knowledge link
- Intersection Factors through Inverse Image proposition statement needs a knowledge link
- Image of the Inverse Image proposition statement needs a knowledge link; proof needs a knowledge link
- Inverse Image of the Image proposition statement needs a knowledge link; proof needs a knowledge link
- The Inverse of the Identity Function Is Itself corollary proof needs a knowledge link
- Invertible iff Bijective proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Inverse Image of the Inverse Function is the Image exercise statement needs a knowledge link
- Image of Inverse Image Equality proposition statement needs a knowledge link
- composition of two bijective functions is bijective proposition proof needs a knowledge link
- The Inverse of an Order Preserving Function is Order Preserving proposition statement needs a knowledge link; proof needs a knowledge link
- When a Function Defined in Terms of a Function is a Function proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Outer Measure 22% (2/9) page hidden
- Outer Measure Preserves Order proposition statement needs a knowledge link
- Outer Measure is Translation Invariant proposition statement needs a knowledge link
- Countable Subadditivity of Outer Measure proposition statement needs a knowledge link
- Outer Measure of a Closed Interval theorem statement needs a knowledge link
- Nontrivial Intervals are Uncountable corollary statement needs a knowledge link
- Outer Measure is Not Additive theorem statement needs a knowledge link
- No Countably Additive Translation Invariant Extension of Length to All Subsets theorem statement needs a knowledge link
Topological Spaces 22% (2/9) page hidden
- The Finite Complement Topology proposition statement needs a knowledge link; proof needs a knowledge link
- The Countable Complement Topology proposition proof needs a knowledge link
- The Intersection of Topologies Is a Topology proposition statement needs a knowledge link
- Given a Family of Topologies There Is a Unique Smallest Topology Containing All of Them proposition statement needs a knowledge link
- Given a Family of Topologies There Is a Unique Largest Topology Contained in Every Topology in the Family proposition statement needs a knowledge link
- Example of the Largest and Smallest Topology exercise statement needs a knowledge link; proof needs a knowledge link
- The Power Set is a Topology proposition statement needs a knowledge link
Unbounded Linear Operators 25% (1/4) page hidden
- Adjoint Operators Respect the Subset Relation proposition statement needs a knowledge link; proof needs a knowledge link
- Double Adjoint Is a Superset proposition statement needs a knowledge link; proof needs a knowledge link
- If a Linear Operator Is Injective and Has Dense Image Then the Adjoint Has an Inverse proposition statement needs a knowledge link; proof needs a knowledge link
Quotient Topology 25% (1/4) page hidden
- Quotient Topology by Universal Properties proposition statement needs a knowledge link; proof needs a knowledge link
- The Direction of the Reals exercise statement needs a knowledge link
- Direction of R2 exercise statement needs a knowledge link; proof needs a knowledge link
Dedekind Cuts 26% (7/27) page hidden
- Natural Numbers are Well-Ordered theorem empty proof; proof needs a knowledge link
- Integer Pair Relation is an Equivalence Relation proposition empty proof; proof needs a knowledge link
- Integer Addition and Multiplication are Well-Defined proposition empty proof; proof needs a knowledge link
- Order on Integers is Well-Defined proposition empty proof; proof needs a knowledge link
- Integers Form an Ordered Integral Domain theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Rational Number Pair Relation is an Equivalence Relation proposition empty proof; proof needs a knowledge link
- Rational Addition and Multiplication are Well-Defined proposition empty proof; proof needs a knowledge link
- Order on Rationals is Well-Defined proposition empty proof; proof needs a knowledge link
- Rationals Form an Ordered Field theorem empty proof; proof needs a knowledge link
- Rational Upper Rays are Dedekind Cuts lemma empty proof; proof needs a knowledge link
- Complement of a Dedekind Cut lemma empty proof; proof needs a knowledge link
- Dedekind Cuts are Linearly Ordered by Containment lemma empty proof; proof needs a knowledge link
- Union of Dedekind Cuts is a Dedekind Cut lemma empty proof; proof needs a knowledge link
- Dedekind Cut Operation Closure lemma empty proof; proof needs a knowledge link
- Dedekind Cut Approximation lemma empty proof; proof needs a knowledge link
- Order Comparison with Rational Cuts lemma empty proof; statement needs a knowledge link; proof needs a knowledge link
- Dedekind Reals Form an Ordered Field theorem empty proof; proof needs a knowledge link
- Greatest Lower Bound Property for Dedekind Reals theorem empty proof; proof needs a knowledge link
- Least Upper Bound Property for Dedekind Reals theorem empty proof; proof needs a knowledge link
- Rationals Embed into Dedekind Reals theorem empty proof; proof needs a knowledge link
Matrices 27% (4/15) page hidden
- Two by Two Block Matrix Multiplication theorem proof needs a knowledge link
- Reflection Matrix About a Line Through the Origin proposition statement needs a knowledge link
- Composition of Two Reflections is a Rotation proposition statement needs a knowledge link
- Composition of Two Plane Reflections is a Rotation proposition statement needs a knowledge link
- The Packing and Unpacking Functions are Inverses proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- One by one Matrices are Isomorphic to R proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Row and Column Matrices are Isomorphic to \( \mathbb{ R } ^ n \) proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- Nested Matrix Multiplication Unpacking lemma empty proof; proof needs a knowledge link
- Matrix Multiplication with Rows and Columns proposition empty proof; proof needs a knowledge link
- column matrix multiplication corollary statement needs a knowledge link; proof needs a knowledge link
- column extraction corollary statement needs a knowledge link; proof needs a knowledge link
Measurable Spaces And Functions 31% (4/13) page hidden
- Algebra of Inverse Images proposition statement needs a knowledge link
- Inverse Image of a Composition proposition statement needs a knowledge link
- Ray Test for Measurability proposition statement needs a knowledge link
- Continuous Functions are Borel Measurable theorem statement needs a knowledge link
- Composition of Measurable Functions proposition statement needs a knowledge link
- Algebraic Operations with Measurable Functions proposition statement needs a knowledge link
- Pointwise Limit of Measurable Functions is Measurable theorem statement needs a knowledge link
- Extended Ray Test for Measurability proposition statement needs a knowledge link
- Infimum and Supremum of Measurable Functions proposition statement needs a knowledge link
Basis 33% (4/12) page hidden
- Intervals With Rational Endpoints Are a Basis for the Real Numbers exercise statement needs a knowledge link
- Half Open Intervals With Rational Endpoints Are a Basis for the Real Numbers exercise statement needs a knowledge link; proof needs a knowledge link
- basis elements are open in a generated topology corollary statement needs a knowledge link
- basis finer equivalence lemma proof needs a knowledge link
- topology equality with basis corollary statement needs a knowledge link; proof needs a knowledge link
- The Real Intervals With Rational Endpoints Generate Generate the Standard Topology proposition statement needs a knowledge link
- The Topology Generated by the Half Open Intervals With Rational Endpoints Is Strictly Corser Lower Limit Topology exercise statement needs a knowledge link
- intersections of subbasis elements forms a basis proposition proof needs a knowledge link
Fields 33% (2/6) page hidden
- The Complex Numbers Form a Field proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
- A Field is a Crone corollary empty proof; proof needs a knowledge link
- A Crone with Multiplicative Inverses is a Field corollary empty proof; proof needs a knowledge link
- Multiplicative Inverse in a Finite Extension of the Rationals proposition statement needs a knowledge link; proof needs a knowledge link
Subspace Topology 33% (2/6) page hidden
- Basis for the Subspace Topology proposition statement needs a knowledge link
- open in a subspace implies open lemma statement needs a knowledge link; proof needs a knowledge link
- TODO theorem statement needs a knowledge link; proof needs a knowledge link
- Lines Intersected With Boxes exercise statement needs a knowledge link; proof needs a knowledge link
Linear Transformations 33% (1/3) page hidden
- Schwarz Inequality proposition empty proof; proof needs a knowledge link
- The Norm Satisfies the Triangle Inequality proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Real Numbers 33% (1/3) page hidden
- A Convergent Rational Sequence is Cauchy proposition statement needs a knowledge link
- The Cauchy Subtraction Relation is an Equivalence Relation proposition proof needs a knowledge link
Monte Carlo 35% (7/20) page hidden
- Joint Density Factors as Marginal Times Conditional Density proposition empty proof; proof needs a knowledge link
- Uniform Monte Carlo Estimator is Unbiased corollary statement needs a knowledge link
- Discrete Inversion Method Has the Target Probabilities proposition proof needs a knowledge link
- Inversion Method Samples from the Target Distribution theorem proof needs a knowledge link
- Inversion Sampling for a Linear Function Density proposition proof needs a knowledge link
- Change of Variables for Density Functions theorem statement needs a knowledge link; proof needs a knowledge link
- One Dimensional Density Transformation corollary statement needs a knowledge link
- Polar Coordinate Density Transformation corollary statement needs a knowledge link
- Spherical Coordinate Density Transformation corollary statement needs a knowledge link
- Stratified Sampling Does Not Increase Variance proposition statement needs a knowledge link
- Ideal Importance Sampling Has Zero Variance for Nonnegative Integrands proposition proof needs a knowledge link
- Balance Heuristic Weights Sum to One proposition proof needs a knowledge link
- Russian Roulette with a Fallback Value Preserves Expectation proposition proof needs a knowledge link
Continuous Functions 35% (7/20) page hidden
- The Set of Points Where a Function Is Larger Than the Other Is Open proposition proof needs a knowledge link
- The Min Function of Two Functions Is Continuous corollary statement needs a knowledge link; proof needs a knowledge link
- Two Topologies Are Equal If the Identity Is a Homeomorphism corollary proof needs a knowledge link
- Closed Inverse Image Implies Continuous lemma statement needs a knowledge link
- Continuous iff Every Neighborhood Contains the Image of another Neighborhood proposition statement needs a knowledge link
- Continuous, Image Closure Expansion, Closed Inverse Image, Nested Neighborhoods Equivalence theorem statement needs a knowledge link
- Constant Functions are Continuous proposition proof needs a knowledge link
- Inclusion is Continuous proposition proof needs a knowledge link
- Compositions of Continuous Functions are Continuous proposition statement needs a knowledge link; proof needs a knowledge link
- Domain Restriction is Continuous proposition statement needs a knowledge link
- Restricting the Range maintains Continuity proposition statement needs a knowledge link; proof needs a knowledge link
- Pasting lemma statement needs a knowledge link; proof needs a knowledge link
- Given a Convergent Sequence Then the Continuous Image Also Converges proposition statement needs a knowledge link; proof needs a knowledge link
Interior And Closure 36% (5/14) page hidden
- Closed Topology theorem statement needs a knowledge link; proof needs a knowledge link
- Transitivity of Closure proposition statement needs a knowledge link
- The Closure is Closed corollary statement needs a knowledge link
- The Interior is Open corollary statement needs a knowledge link; proof needs a knowledge link
- Is the Interior of the Closure of an Open Set Itself? exercise statement needs a knowledge link; proof needs a knowledge link
- Closed Supersets are Supersets of the Closure proposition statement needs a knowledge link; proof needs a knowledge link
- Open Subsets are Subsets of the Interior proposition statement needs a knowledge link; proof needs a knowledge link
- Interior is Smaller, Closure is Bigger proposition statement needs a knowledge link
- Open Sets Equal their Interior proposition statement needs a knowledge link
Relations 40% (4/10) page hidden
- The Subset Relation Is Transitive proposition statement needs a knowledge link; proof needs a knowledge link
- Congruences induce a Equivalence Relation exercise proof needs a knowledge link
- Every Partial Order Admits a Strict Partial Order corollary statement needs a knowledge link; proof needs a knowledge link
- well ordering theorem proof needs a knowledge link
- The Smallest Element of a Set is Unique proposition proof needs a knowledge link
- Intersection of Intervals theorem statement needs a knowledge link; proof needs a knowledge link
Quaternions 41% (15/37) page hidden
- to_quat from \( \mathbb{R}^3 \) Produces a Pure Quaternion corollary proof needs a knowledge link
- Vector Part of to_quat from \( \mathbb{R}^3 \) is the Identity corollary statement needs a knowledge link; proof needs a knowledge link
- to_quat from \( \mathbb{R} \) Has Zero Vector Part corollary statement needs a knowledge link; proof needs a knowledge link
- Quaternion Decomposes as Sum of to_quats corollary statement needs a knowledge link
- Equality by Vector and Real Equality corollary statement needs a knowledge link; proof needs a knowledge link
- Quaternion Multiplication Distributes Over Addition proposition statement needs a knowledge link
- Real Part of Quaternion Product corollary statement needs a knowledge link
- Vector Part of Quaternion Product corollary statement needs a knowledge link
- Quaternion Multiplication is Associative proposition statement needs a knowledge link; proof needs a knowledge link
- Product with a Real Quaternion proposition statement needs a knowledge link
- Conjugation is Additive corollary statement needs a knowledge link; proof needs a knowledge link
- Vector Part of the Conjugate is Minus 1 Times the Original corollary statement needs a knowledge link; proof needs a knowledge link
- Real Part of the Conjugate Doesn't Change corollary statement needs a knowledge link; proof needs a knowledge link
- Conjugate Fixes Real Quaternions corollary statement needs a knowledge link; proof needs a knowledge link
- Conjugate Only Applies to Vector Part corollary statement needs a knowledge link
- Product Commutes in the Real Part corollary statement needs a knowledge link
- Quaternion Times its Conjugate proposition statement needs a knowledge link; proof needs a knowledge link
- Conjugation is a Homomorphism in the Real Part lemma statement needs a knowledge link
- Conjugation Swaps Order in the Vector Part lemma statement needs a knowledge link
- Conjugation Distributes by Swapping proposition statement needs a knowledge link
- Conjugation Preserves Norm lemma statement needs a knowledge link; proof needs a knowledge link
- Quaternion Product is Norm-Multiplicative lemma statement needs a knowledge link
Differentiation 43% (10/23) page hidden
- Partial Derivative becomes frozen function derivative proposition statement needs a knowledge link; proof needs a knowledge link
- Todo exercise statement needs a knowledge link; proof needs a knowledge link
- Todo exercise statement needs a knowledge link; proof needs a knowledge link
- Todo exercise statement needs a knowledge link; proof needs a knowledge link
- continuous partials implies differentiability theorem statement needs a knowledge link; proof needs a knowledge link
- Derivative is Unique theorem empty proof; proof needs a knowledge link
- Differentiability Implies Continuity theorem empty proof; proof needs a knowledge link
- Chain Rule for Total Derivatives theorem empty proof; proof needs a knowledge link
- Basic Derivative Rules in Euclidean Space theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Mixed Partials Commute Under Continuity theorem empty proof; statement needs a knowledge link; proof needs a knowledge link
- Interior Extremum has Zero Partials theorem empty proof; proof needs a knowledge link
- Mean Value Inequality in Euclidean Space theorem empty proof; proof needs a knowledge link
- Derivative of the Vector Scalar Division Map proposition statement needs a knowledge link
Integration On Manifolds 47% (7/15) page hidden
- Coordinate Characterization of Manifolds theorem statement needs a knowledge link
- Boundary of a Manifold with Boundary is a Manifold theorem statement needs a knowledge link
- Exterior Derivative on Manifolds theorem statement needs a knowledge link
- Stokes' Theorem on Manifolds theorem statement needs a knowledge link
- Surface Area Element in \( \mathbb{ R } ^ 3 \) theorem statement needs a knowledge link
- Green's Theorem theorem statement needs a knowledge link
- Divergence Theorem theorem statement needs a knowledge link
- Classical Stokes' Theorem theorem statement needs a knowledge link
Continuous Distributions 47% (8/17) page hidden
- Probability of an Interval in the Uniform Distribution corollary empty proof; proof needs a knowledge link
- Density Determines Distribution Function proposition proof needs a knowledge link
- Density Determines Interval Probabilities lemma proof needs a knowledge link
- Continuous Expectation is Linear for Finite Sums proposition statement needs a knowledge link
- Continuous Marginal Density Functions lemma empty proof; proof needs a knowledge link
- Continuous Convolution Formula theorem proof needs a knowledge link
- Change of Variables for Joint Densities theorem empty proof; proof needs a knowledge link
- Density Gives Zero Probability to Points corollary proof needs a knowledge link
- Probability of an Interval in the Exponential Distribution corollary proof needs a knowledge link
Uniform Continuity 50% (1/2)
- Every Lipschitz function is Uniformly Continuous proposition empty proof; statement needs a knowledge link; proof needs a knowledge link
Basics 50% (1/2)
- Empty Integer Set corollary statement needs a knowledge link
Probability Models 53% (8/15)
- Intersection as Conditional Probability corollary empty proof; proof needs a knowledge link
- Product Rule theorem proof needs a knowledge link
- Symmetric Conditional Equation corollary statement needs a knowledge link
- Probability of the Impossible Event proposition statement needs a knowledge link
- Probability of an Event through a Partition lemma proof needs a knowledge link
- Law of Total Probability theorem empty proof; proof needs a knowledge link
- Independence is Symmetric corollary proof needs a knowledge link
Random Variables 57% (4/7)
- Independence as a Product corollary empty proof; proof needs a knowledge link
- Independence is Reflexive corollary statement needs a knowledge link; proof needs a knowledge link
- Independence Practice exercise statement needs a knowledge link
Real And Complex Measures 75% (3/4)
- Radon-Nikodym Theorem theorem proof needs a knowledge link
Radiometry 83% (20/24)
- Incident Radiance from Raycast proposition proof needs a knowledge link
- Derivative of the Normalization Map proposition statement needs a knowledge link
- Irradiance from a Unit-Radius Disk exercise proof needs a knowledge link
- Irradiance from a Unit Square exercise proof needs a knowledge link