Real powers for positive bases, with the zero-base positive-exponent convention

Depends on

• The natural logarithm as the inverse of the exponential function
• The real exponential function and the number $e$ by a power series

Used by

• Finite endpoint power substitution followed by Gauss–Jacobi quadrature Algorithm
• An α-narrow graph has a clique or stable set of size at least |V(G)|^1/(2α) Corollary
• Compatible extensions from the finite simple core Corollary
• Lᵖ is uniformly convex for 1<p<∞ Corollary
• Lyapunov central limit theorem Corollary
• Polynomial and exponential minimax estimation rates Corollary
• The edge-density form of Rödl's theorem: every nonempty H-free graph has a linearly large set of self-density at most ε or at least 1-ε Corollary
• The polynomial Rödl property implies the Erdős–Hajnal property Corollary
• Finite thresholds rule out an r-independent VC covering bound Counterexample
• Finite variance is not compact support Counterexample
• Knapp rules out extension below the Tomas exponent Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• Subcritical W^1,p is not closed under multiplication Counterexample
• A tau-critical graph Definition
• An α-narrow graph Definition
• Chebyshev weighted spectral Sobolev scale and projection Definition
• Covering, packing, bracketing numbers, and metric entropy Definition
• Euler's real Beta integral Definition
• Finite Rényi divergence Definition
• Fourier Bessel-potential and restriction-quotient Sobolev spaces Definition
• Hausdorff content at a prescribed scale Definition
• Legendre weighted spectral-Sobolev scale and projection Definition
• One-letter Gallager and expurgated exponent functions Definition
• The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class Definition
• The function space Lᵖ(μ) for 0 < p < ∞ Definition
• The polynomial Rödl property for a finite forbidden family Definition
• The real Gamma function by Euler's integral Definition
• The Sobolev conjugate exponent and the scaling identity Definition
• The viral property for a finite forbidden family Definition
• Weighted orthogonal-polynomial Galerkin spaces and spectral scale Definition
• A Caputo startup singularity and a graded-grid calculation Example
• A Dirac mass has precisely sufficiently negative Sobolev order Example
• A family containing K₁ is viral for vacuous reasons Example
• Boundary-incompatible forcing reduces Strang order to five quarters in a Dirichlet modal model Example
• Critical, subcritical and supercritical Gaussian regimes for Hardy's theorem Example
• Dilation determines the Riesz-potential target exponent Example
• Direct midpoint quadrature loses order on a half-order singularity Example
• Every hereditary graph class of bounded order has the Erdős–Hajnal property Example
• Exponential singular decay, logarithmic balls, and spectral gaps Example
• Exponential singular-value decay and an exponentially weighted truth class Example

…and 93 more results.

Dependency tree · two levels

9 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

• J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
• MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)