Multiplication by zero: $0\cdot a=0$

Depends on

• Field

Used by

• Bias-corrected Adam update Algorithm
• Bias-corrected AMSGrad update Algorithm
• Diagonal AdaGrad for online convex losses Algorithm
• RMSProp with an exponential second-moment estimate Algorithm
• Uncorrected Adam for online convex losses Algorithm
• Uncorrected AMSGrad with a coordinatewise historical maximum Algorithm
• ∑_k<n+1C(n, k) = 2ⁿ, and ∑_k<n+1(-1)ᵏiotaC(n, k) = 0 for n ≥ 1 Corollary
• AM-GM fails for negative entries: (-1,-4) has geometric mean 2 and arithmetic mean -5/2 Counterexample
• The union of the two coordinate axes of F² is closed under scalar multiplication and is not closed under addition, so neither closure condition implies the other Counterexample
• Three lines in F² that meet pairwise only in 0 and whose sum is F² with decompositions that are not unique, so pairwise trivial intersection does not give a direct sum Counterexample
• Integer powers aᵐ Definition
• (0,1) + (2,3) = (2,4), with supremum 4 = sup(0,1) + sup(2,3) Example
• A regularized logistic finite sum with an exact minimizer Example
• A signed Simpson-Smolyak weight makes a nonnegative chaos coefficient estimate negative Example
• Duplicate columns and explicit zeros count as stored entries Example
• Equispaced interpolation of Runge's function diverges Example
• F^ℕ is a vector space and the eventually zero families form a linear subspace of it that is the span of the standard unit families Example
• Five finite AM-GM proofs, with the same sharp equality case Example
• Five finite proofs of Bernoulli, with rational exponents and the quadratic refinement Example
• Geometric sequences: the complete classification for every real ratio Example
• In F³ the three coordinate lines are linear subspaces whose internal direct sum is F³, and F⁰ is the zero space Example
• Positive type on a discrete group: the identity mass, characters, and the regular GNS model Example
• sup{q ∈ ℚ : q > 0, q² < 2} = √2 in ℝ, and no supremum in ℚ Example
• The two-term Lagrange identity and Cauchy-Schwarz Example
• Two planes in F³ whose sum is F³ and whose intersection is a line, computed explicitly Example
• FALSE: ∑_k<n+1(-1)ᵏiotaC(n, k) = 0 for every n ∈ ℕ False statement
• FALSE: a^m/n := (a^1/n)ᵐ extends to negative bases False statement
• FALSE: every real number has a real square root False statement
• FALSE: the supremum of a set belongs to the set False statement
• FALSE: The union of two linear subspaces is a linear subspace False statement
• A cone generated by finitely many Euclidean vectors is closed Lemma
• A field has no zero divisors: ab = 0 ⟹ a = 0 or b = 0 Lemma
• A finite coefficient vector has a canonical polynomial representative Lemma
• A finite positive-spanning poll set has a positive uniform angle Lemma
• A real coefficient list of degree d has at most d real roots Lemma
• Bernoulli's inequality (1+x)ⁿ ≥ 1 + nx Lemma
• Cardinal interpolation functions are polynomials of the stated degrees Lemma
• Complex modulus restricts to real absolute value Lemma
• Finite group-relative updates preserve admissibility and have bounded oracle calls Lemma
• Finite-metric transport and normalized dual optima are attained Lemma

…and 21 more results.

Dependency tree · one level

1 result within one dependency step of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
• M. Spivak, Calculus, 4th ed., Ch. 1 (standard reference, not scraped)
• Janssen and Lindsey, Rings with Inquiry: Fields (standard reference, not scraped)