Quarter-turn values and shifts by pi/2 and pi

Depends on

• Pi as twice the smallest positive zero of cosine
• Cosine has a smallest positive zero, lying strictly between zero and two
• The addition formulas for sine and cosine
• Parity and the Pythagorean identity for sine and cosine
• Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3

Used by

• Buffered certified rational phase approximation Algorithm
• Exact success amplification by attenuating the good branch Algorithm
• Known-amplitude amplification with integer rounding Algorithm
• Periodic Fourier collocation differentiation matrix Algorithm
• Szegedy-walk marked-set detection Algorithm
• exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0 Corollary
• Pi is the first positive zero of sine Corollary
• The normal component of the curl is the limiting circulation per unit area of shrinking discs Corollary
• x sin(1/x) extended by zero is continuous but not differentiable at zero Corollary
• A fixed Grover iteration count can fail when a is unknown Counterexample
• A map with two preimages but degree zero Counterexample
• A real-valued optimum is not a fractional Grover iteration Counterexample
• A surjective map need not be a fibration Counterexample
• Agreement accumulating only at the boundary does not force a holomorphic identity Counterexample
• An invertible derivative at one point does not give a local inverse without C¹ regularity Counterexample
• Antipodal points on a round sphere have many minimizing geodesics Counterexample
• f(x+iy)=eˣ(cos 2y+i sin 2y) is continuous, satisfies f(z+w)=f(z)f(w) and f(1)=e, but is not the standard complex exponential Counterexample
• Fidelity and trace-distance inequalities do not imply QRE data processing Counterexample
• Generator convention matters Counterexample
• High frequencies destroy uniform translation control Counterexample
• I₂(5) admits no crystallographic scaling and no reduced crystallographic root system with that base pairing Counterexample
• Principal arcsine has no finite derivative at -1 or 1 Counterexample
• Sharp frequency cutoffs have kernels that are not in L1 Counterexample
• sin(1/x) has no limit as x tends to zero Counterexample
• sin(3π/2)=-1 shows that the positive square root is not an unconditional half-angle formula Counterexample
• The continuous path γ(x)=(x,x sin(1/x)) on [0,1], with γ(0)=(0,0), is not rectifiable Counterexample
• The final-time face is not part of the parabolic boundary Counterexample
• The topologist's sine curve is connected but not path connected Counterexample
• u=v=π/4 shows that the tangent addition formula cannot omit its domain restrictions Counterexample
• Volterra's function is differentiable everywhere with bounded derivative, but its derivative is not Riemann integrable Counterexample
• Best uniform polynomial error and Chebyshev interpolation operators Definition
• Periodic Fourier collocation grid and even-grid Nyquist convention Definition
• Principal inverse sine and inverse cosine Definition
• The Coxeter nerve and its Moussong metric Definition
• The real Coxeter form, its radical, reflections, and form-preserving maps Definition
• A closed cylinder as a finitely patched oriented surface Example
• A continuous argument computed along a spiralling contour Example
• A cycle and an overlong arm: explicit non-positive witnesses Example
• A four-dimensional product exactly integrated on a sparse node union Example
• A moved-space intersection in A₃ that is not the meet Example

…and 155 more results.

Dependency tree · two levels

16 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

• NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)
• C. Schmeiser, Introduction to Analysis (standard reference, not scraped)