Squaring is monotone on the nonnegatives

Depends on

• Ordered field
• Sign rules for products and monotonicity of multiplication
• Multiplication by zero: $0 \cdot a = 0$
• Basic properties of the absolute value

Used by

• Variance-aware quantum Monte Carlo Algorithm
• Constant-step BDF2 is second-order convergent with a consistent start and selected roots Corollary
• The induced length is a norm Corollary
• {q ∈ ℚ : q ≥ 0, q² < 2} is closed and bounded in ℚ and is not compact Counterexample
• A conjugate point at which there are many geodesics Counterexample
• A full-rank local Hessian does not capture both posterior modes Counterexample
• Adam diverges on a periodic bounded-gradient loss sequence Counterexample
• Boundary-scale derivative blowup despite bounded ball data Counterexample
• f(t) = (t², t³) on [0,1]: no ξ satisfies f(1)-f(0) = f'(ξ) Counterexample
• ℚ ∩ [0,2] is bounded and disconnected, so being an interval of ℚ is not enough Counterexample
• x ↦ √x is a uniformly continuous bijection of [0,∞) onto itself whose inverse x ↦ x² is not uniformly continuous Counterexample
• Left-invariant means on L^∞ of a locally compact group Definition
• √· on [0,∞) is uniformly continuous and exactly 1/2-Hölder, and is not Lipschitz Example
• √x is absolutely continuous but not Lipschitz on [0,1] Example
• A convergent sequence in ℝ³ and the integral ∫₀¹ (1, t, t²), computed componentwise Example
• A multimodal quartic receives an exact spatial branch-and-bound gap certificate Example
• An indefinite Coxeter form: infinite, but not of affine type Example
• An inverse-CDF singularity can be integrable but have infinite variation Example
• ARS(2,2,2) amplification on a two-cell reaction-diffusion grid Example
• Boundary-incompatible forcing reduces Strang order to five quarters in a Dirichlet modal model Example
• Cut locus on a flat rectangular torus from the Dirichlet cell Example
• Eight-bit Box–Muller has a finite radial cutoff Example
• Fixed points, local Newton behavior, and failure boundaries Example
• Gram determinants and principal minors of the non-crystallographic types H₃ and H₄ Example
• Hilbert spaces are uniformly convex Example
• Minkowski for two-entry vectors: equality and strict cases Example
• Path determinants dₖ=dₖ₋₁-cos²(π/m)dₖ₋₂ and the three-arm inequality Example
• Steinitz's confinement bound realised on an explicit list of six unit vectors in ℝ² summing to zero Example
• sup{q ∈ ℚ : q > 0, q² < 2} = √2 in ℝ, and no supremum in ℚ Example
• The comparison constants between ‖·‖₁, ‖·‖₂ and ‖·‖_∞ on ℝ², and vectors attaining each Example
• The cube [-M,M]ⁿ in ℝⁿ is totally bounded, with an explicit finite ε-net of grid points and no appeal to the integer part Example
• The hexagonal A₂ cell: Euclidean cell metric versus graph distance Example
• The map x ↦ (x + 2/x)/2 is a contraction of [1,2] with fixed point √2, and the a priori bound gives the error after n steps Example
• The metrics d₁, d₂ and d_∞ on ℝⁿ are metrics and are Lipschitz equivalent, with explicit constants Example
• The square roots √k diverge to +∞ Example
• The successive differences √k+1 - √k converge to 0 Example
• The two-term AM-GM inequality, its equality case, and x + x⁻¹ ≥ 2 Example
• Two squeeze computations: 1/((k+1)√k+1) → 0 and (k+(-1)ᵏ)/(k+1) → 1 Example
• Young's inequality at p = q = 2 and at p = 3, q = 3/2 Example
• FALSE: in every ordered field a closed bounded set is compact, so Heine-Borel needs no completeness False statement

…and 53 more results.

Dependency tree · two levels

7 results within two dependency steps 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)