The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$

Depends on

• The exponential addition formula $\exp(x+y)=\exp(x)\exp(y)$
• The real exponential function and the number $e$ by a power series
• Squares of nonzero elements are positive

Used by

• Exp3 with importance-weighted loss feedback Algorithm
• Exp4 with unbiased estimated rewards Algorithm
• Exponential mechanism Algorithm
• Finite-class empirical-risk selection with exponential weights Algorithm
• Randomized weighted majority with full information Algorithm
• A sample-size consequence for compression schemes Corollary
• Dobinski's formula expresses the Bell numbers as Bₙ=e⁻¹∑_ℓ≥0ℓⁿ/ℓ! Corollary
• exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0 Corollary
• Lᵖ is uniformly convex for 1<p<∞ Corollary
• A boundary mode has a different concentration rate Counterexample
• A bounded continuous normalized function that is not of positive type Counterexample
• A discontinuous positive solution of F(x+y)=F(x)F(y) Counterexample
• A flat smooth real function has no holomorphic extension near zero Counterexample
• A generalized Bayes credible set can have zero coverage Counterexample
• A nonintegrable error envelope destroys the linear Hellinger rate Counterexample
• A point-mass Gaussian prior excluding the truth prevents contraction Counterexample
• A singular interior mode need not have Gaussian scaling Counterexample
• Boundary-scale derivative blowup despite bounded ball data Counterexample
• Finite variance does not imply a sub-Gaussian tail Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• Smooth data do not force an analytic solution Counterexample
• The final-time face is not part of the parabolic boundary Counterexample
• Carleson tiles wave packets and tile order Definition
• Chebyshev weighted spectral Sobolev scale and projection Definition
• Diagonal Dirichlet sine-mode heat model with constant forcing Definition
• Kullback–Leibler divergence Definition
• Legendre weighted spectral-Sobolev scale and projection Definition
• Matrix multiplicative-weights game and update Definition
• Periodized Gaussian gridding kernel Definition
• The heat kernel on ℝⁿ and its causal extension Definition
• The Jacobi theta function θ(t)=∑_n∈ℤ e^-π n² t for t>0 Definition
• The six hyperbolic functions and their natural domains Definition
• Weighted orthogonal-polynomial Galerkin spaces and spectral scale Definition
• A diagonal stiff diffusion step with linear reaction Example
• A family containing K₁ is viral for vacuous reasons Example
• A finite Euler adjoint gradient can differ from the exact flow gradient Example
• A finite identifiable model with posterior consistency Example
• A finite-mean polynomial-tail reward is not sub-Gaussian Example
• A Gaussian harmonic leapfrog resonance Example
• A globally Lipschitz Hamiltonian force Example

…and 135 more results.

Dependency tree · two levels

18 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

• MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
• J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
• J. K. Hunter, An Introduction to Real Analysis, Chapter 10 (standard reference, not scraped)
• S. G. Johnson, Exponential Functions (standard reference, not scraped)