- [z1 , z2 , z3 , z4 ] (cross ratio), 4.5
- cosσ, 3.7.1
- δjk, 1.1
- ∞ (point at infinity), 8.1
- Λ(g), 1.2
- ⋋ (perpendicularity), 7.4
- AB (directed interval), 7.2
- SL2(ℝ), 6.1, 6.3, 6.4, 6.8
- SL2(ℝ) group
- Hardy representation, 1.1
- representation, 3.2, 3.2
- [2], 2.1
- ℍn (Heisenberg group), 2.1
- Jn (jet space), 1.3
- ⟨ ·,·
⟩, 1.1
- σ-cosine, 3.7.1, 5.4, 7.1
- σ-sine, 3.7.1, 7.1
- σ-tangent, 3.7.1
- sinσ, 3.7.1
- ⊣ (f-orthogonality), 6.6
- tanσ, 3.7.1
- tr, 1.1
- ℝσ (point space), 3.4
- A subgroup, 1.1, 3.2
- A-orbit, 1.1, 3.5, 10.4.2
- A′ subgroup, 3.2, 3.7, 9.1, 10.2
- A″ subgroup, 3.7
- -orbit, 3.7
- ax+b group, 1, 1, 2.1, 3.5, 4.2, 4.4, 4.4, 5.1, 5.1, 5.2, 6.3, 6.4, 6.5, 6.6
- Haar measure, 2.4
- invariant measure, 6.3
- left regular representation, 6.3
- representation, 1.1, 4.2
- co-adjoint, 6.3
- quasi-regular, 6.3
- G σc (ghost cycle), 6.3
- ℝe (complex numbers), 3.4
- ℝp (dual numbers), 3.4
- ⅅσ (unit disk), 10.1
- ε (parabolic unit), 1.1
- є (infinitesimal), 7.5
- F subgroup, 3.1
- F′ subgroup, 3.1
- F′ subgroup, 3.1
- G(), 2.1
- ℍ1 (Heisenberg group), 2.1
- h1 (Lie algebra of the Heisenberg group), 2.2
- i (imaginary unit), 1.1
- ι (hypercomplex unit), 1.1, 3.3.4, B.1
- ιc (hypercomplex unit in cycle space), 1.2, 4.2
- є (hyperbolic unit), 1.1
- K subgroup, 1.1, 1.7, 3.2, 9.1
- K-orbit, 1.1, 3.5, 3.5, 3.5, 3.6, 3.7, 5.1, 8.2, 8.2, 10.4.2
- k-normalised cycle, 1.6, 5.2, 5.4, 5.4, 7.1, 7.1, 8.1, C.4
- lc(AB) (length from centre), 7.2
- lf(AB) (length from focus), 7.2
- M map, 4.2, 8.1
- N subgroup, 1.1, 3.2
- N-orbit, 1.1, 3.5, 10.4.2
- N′ subgroup, 3.2, 3.7, 9.1
- N′-orbit, 3.7
- ℝh (double numbers), 3.4
- ℂ′ (two-fold cover of double numbers), 8.2
- ℂ′+ (hyperbolic upper half-plane), 8.2
- ω (symplectic form), 2.1
- P (projection from the projective space), 9.5
- ℙ1(ℝσ) (projective space), 4.5
- ℙ3 (projective space), 4.2
- p map, 2.2.2, 8.3.3, 1.4
- Q map, 4.2, 6.2, 8.1
- Rσcs (cycle representing the real line), 6.5
- ℝ2, 2.2.1
- $ℝ^_$ (projective real line), 3.1, 3.1
- ℝ* (hyperreals), 7.6
- r map, 2.2.2, 8.3.3, 1.4
- S (section to the projective space), 4.5
- SL2(ℝ) group, 1, 2.1
- SU(1,1) group, 10.1
- s map, 2.2.2, 8.3.3, 1.4
- σ (σ:=ι2), 1.1, 3.4
- σc (σc:=ιc2), 4.2
- σ1, 1.2
- σr-focus, 5.1, 9.4
- sl2 (Lie algebra), 2.3.3
- Tσ (unit cycle), 10.1
- χ (Heaviside function), 6.3
- Z (the centre of the Heisenberg group), 2.1
- Zσcs(y) (σc-zero-radius cycle), 5.4
- Z∞ (zero-radius cycle at infinity), 8.1
- Mouse, see CAS exercise
- Abel summation, 6.2
- Apollonian gasket, 4.2
- Archimedes, 7.5, B.2
- Asymptote, C, C.3.2
- Atiyah–Singer index theorem, 1.1
- abelian
- group, see commutative group
- about hearing a drum’s shape, 1.3
- absolute
- abstract group, 2.1, 2.2
- acceleration, 8.4
- action
- adjoint
- adjoint representation, 1.1
- admissible
- admissible vector, 2.2
- admissible wavelet, 4, 4.2, 4.4, 6.4, 6.5
- affine
- affine group, see ax+b group
- algebra
- associative, B.1
- Clifford, 1.6, 4.3, 4.4, 6.4, 4.2, 4, A.2, A.4, B.5
- cycle matrix, B.5
- Möbius map, B.5
- matrix similarity, B.5
- commutative, B.1
- homomorphism, 1.1
- Lie, 2.3, 2.3.3, 4.4.2
- Weyl, 2, 2.2
- algebraically-closed, B.1
- analysis
- complex, 1.7
- functional, 4.2
- harmonic, 1.7
- multiresolution, 4.2
- non-Archimedean, see non-standard
- non-standard, 7.5, 7.6
- analytic
- contravariant calculus, 1.2
- function on discrete sets, A.2
- annihilation operator, see ladder operator
- annulus, 5.5
- approximation
- arithmetic of infinity, 3.1
- arrow
- associative
- associativity, 2.1
- astronomy
- asymptote
- atom, 4.4, 6.5
- atomic
- automatic theorem proving, C.3.3
- automorphism
- Baker–Campbell–Hausdorff formula, 2.2
- Benz
- Bergman
- Birkhoff orthogonality, see perpendicular
- Blaschke
- birational
- boundary effect on the upper half-plane, 1.3, 5.1, 5.4, 6.1, 6.3, 6.6
- bracket
- CAS, ??, 4.3, 7.5, B.5, C, C, C.2, C.5
- Calderón–Zygmund
- Campbell–Hausdorff formula, see Baker–Campbell–Hausdorff formula
- Carleson
- Cartan
- Casimir operator, 5.3
- Cauchy
- integral, 1.7, 4.2, 5.3, 5.4, 5.4, 5.5, 6.2, 6.4, 6.8, 1.2, A.2
- Cauchy integral formula, 2.1
- Cauchy–Riemann
- Cauchy–Riemann operator, 5.1, 6.6, 6.8
- Cauchy–Schwarz inequality, 5.5, 5.5, 6.2
- Cauchy-Riemann operator, 5.3, 5.3, 4.3.1, A.2
- Cayley transform, 10, 10.4.2, 5.5
- Clifford
- algebra, 1.6, 4.3, 4.4, 6.4, 4.2, 4, A.2, A.4, B.5
- cycle matrix, B.5
- Möbius map, B.5
- matrix similarity, B.5
- Clifford algebras, 1.6
- calculus
- contravariant, 4.3, A.3
- covariant, 4.3, 1.5, 1.6, 1.6, A.3
- functional, 4.3, 1, A.3, A.3
- covariant, 6.7
- Riesz–Dunford, 4.3
- support, see spectrum
- symbolic, see covariant calculus
- umbral, 4.1
- cancellation
- formula for cross ratio, 4.5
- formula for cross-ratios, 9.5
- cancellative semigroup, 4.1
- canonical
- case
- categorical viewpoint, 1.3, 6.1
- category theory, 1.3
- causal, 4.1.2
- centre
- cycle, of, 1.2, 3.4, 4.1, 4.1, 5.1, 5.2, 7.1, 7.1, 9.4, 10.3, B.2, B.3
- ellipse, of, B.2
- hyperbola, of, B.2
- length from, 1.6, 7.2, 7.2, 7.3, 10.1, 10.1, 10.2, 10.2, 10.3, 10.3
- character, 3.1
- character of a group, 1.1
- character of representation, 1.1
- characteristic
- circle, 4.1, B.2
- imaginary, 4.1, 5.1, 5.2, 5.3
- parabolic (Yaglom term), 6.2, B.3
- unit, see elliptic unit cycle
- classes of equivalent representations, 1.1
- classic
- Fock–Segal–Bargmann representation, 4.5.1
- probability, 4.5.3
- classical
- classical mechanics, 4.2.1
- co-adjoint
- representation, 3.3
- representation of ax+b group, 6.3
- coaxial
- coefficients
- matrix, 2.1
- representational, 2.1
- coherent state, see wavelet
- coherent states, 2.1
- commutation
- relation
- Heisenberg, 2, 2.2
- Weyl, 2
- commutation relation
- commutative
- commutator, 2.3.3, 4, 4.1.1
- commuting operator, 1.3
- compactification, 8.1
- complex
- complex numbers, 6.4
- computer algebra system, see CAS
- concentric, 1.2, 4.1, 5.2, 7.5, 10.3
- concurrent cycles, 6.2
- concyclic, 4.5
- condition
- cone, 1.1, B.2
- configuration
- confocal, 7.5
- conformal
- infinitesimally, 7.6
- space-time, 8.1
- conformality, 1.6, 7.3
- congruence
- conic
- conjugated subgroups, 2.2.1
- conjugation, 2.2.1, 4.4.2
- constant
- construction
- continuous
- continuous group, 2.1
- contour line, 3.5, 7.3, 7.4, 7.4, 11.2
- contravariant
- contravariant transform, 4.4, 5.7
- convolution, 2.4, 4.1.1
- convolution operator, 2.4
- correspondence
- coset space, 2.2.2
- covariant
- calculus, 4.3, 1.5, 1.6, 1.6, A.3
- functional calculus, 6.7
- pencil, 1.6
- spectral distance, 1.5
- symbol, 4.3
- symbolic calculus, 4.3
- transform, 4.1, 6.4, 6.8
- induced, 5.1
- inverse, see contravariant transform
- cover
- creation operator, see ladder operator
- cross ratio, 4.5, 12.3
- cross-ratio, 9.5
- cancellation formula, 9.5
- projective, 9.5
- curvature, 3.5
- curve
- cycle, 1.1, 4.1, 4.2, 13.2.1, 1.6, B.2
- σc-product, 5.3, 6.1, 7.1
- Cayley transform, 10.4
- centre, 1.2, 3.4, 4.1, 4.1, 5.1, 5.2, 7.1, 7.1, 9.4, 10.3, B.2, B.3
- concentric, 7.5
- confocal, 7.5
- conjugation, 1.5, 7.5
- cross ratio, 6.6, 12.3, 14.7
- diameter, 7.1, B.3
- equation, 1.2, 4.1, 4.2
- f-ghost, 6.6
- flat, 4.1
- focal length, 5.1, 7.1
- focus, 1.2, 3.4, 5.1, 7.1, 9.4, 10.3, B.2
- ghost, 6.3, 6.5
- infinitesimal radius, of, 7.5, 7.5, 7.6, 10.4.1
- inversion, 6.5, 6.5, 6.5, 6.6, 8.1
- isotropic, 1.4, 6.2
- matrix, 1.2, 4.2
- normalised, 1.3, 5.2
- det-, 5.2, 5.4, 5.4, 6.6, 7.1, 8.1, C.4
- k-, 1.6, 5.2, 5.4, 5.4, 7.1, 7.1, 8.1, C.4
- norm, 5.4
- orthogonality, 4.3, 7.5
- parabolic (Yaglom term), B.3
- positive, 5.3, 5.3, 9.1, 9.1
- radius, 1.6, 5.2, 7.1, C.4
- reflection, 1.5, 6.5, 6.5, 6.5, 10.4.1
- selfadjoint, 6.2, 6.6, 9.5, B.3, B.3
- similarity, 6.4, 10.4.1, C.4
- space, 1.2, 4.2, 8.1, 13.2.1, B.3
- unit
- zero-radius, 1.3, 1.4, 4.4.1, 5.3, 5.4, 5.4, 5.5, 6.2, 6.2, 6.2, 7.1, 7.5, 8.1, 8.1, 9.4, B.3
- elliptic, 5.4
- hyperbolic, 5.4
- infinity, at, 8.1
- parabolic, 5.4
- cycles
- coaxial, 3.6, 5.2
- concurrent, 6.2
- disjoint, 5.5, 6.2, 6.2
- f-orthogonal, 1.5, 1.6, 6.6, 7.4, 10.4.1, C.4
- intersecting, 5.5, 6.2, 6.2
- orthogonal, 1.2, 1.4, 4.1, 6.1, 6.1, 6.3, 6.5, 6.5, 9.1, 10.4.2, C.4
- pencil, ??, 5.2, 5.4, 5.5, 5.5, 5.5, 6.2, 6.2, 6.2, C.3.4
- radical axis, 3.6, 5.2
- tangent, 5.2, 5.5, 6.2
- Descartes–Kirillov condition, 5.5
- cyclic vector, 1.2
- De Donder–Weyl formalism, A.4
- Descartes–Kirillov
- condition for tangent circles, 5.5
- decomposable
- decomposition
- defect operator, 4.3
- degeneracy
- dequantisation
- derivation
- derived action, 1.1, 2.3.3, 3.1, 3.5
- det-normalised cycle, 5.2, 5.4, 5.4, 6.6, 7.1, 8.1, C.4
- determinant, 1.3
- diameter
- dicrete series, 2.2
- directing functional, 4.2
- directrix, 1.2, 3.5
- discrete
- analytic function, A.2
- geometry, A.1
- spectrum, A.3
- discriminant, 5.3
- disjoint
- disk
- distance, 1.6, 7.1, 7.1, 7.3, 10.1, 10.1, 10.2, 10.2, 11.1
- conformal, 7.3
- covariant spectral, 1.5
- inversive between cycles, 5.4
- distinct, essentially, 4.5, 9.5
- divisor
- zero, 1.1, 3.3.2, 3.4, 3.7, 3.7, 4.5, 8.1, 11.2, 4.4.5, B.1
- domain
- non-simply connected, 5.5
- simply connected, 5.5
- double, 1.1
- double numbers, 3.3.3, 6.4
- doubling condition, 6.3
- dual, 1.1
- dual numbers, 3.3.2
- dual object, 1.2
- dual space, 1.2
- dyadic
- dynamics
- EPAL, see Erlangen programme at large, 1.7
- EPH classification, 1.1, 3.4, 9.4
- Erlangen
programme, 1
- Erlangen programme, ??, ??, ??, 1
- Euclidean
- Euler
- e-centre, 4.1
- e-focus, 5.1
- eigenvalue, 3.3, 3.3, 1.3, 1.5
- generalised, 1.6
- quadratic, 1.6
- element
- ellipse, B.2
- elliptic
- equation
- equivalent representations, 1.1
- essentially distinct, 4.5, 9.5
- exact representation, 1.1
- exponential map, 2.3.1, 3.2
- extension
- extragalactic astronomy, 8.2, 8.4
- Fillmore–Springer–Cnops construction, 1.2, 4.2, 10.3, A.1
- Fock–Segal–Bargmann
- representation, 3.3, 3.5, 3.5, 4.5.4, 4.5.4, 5.4
- representations, 4.2.1
- space, 4.2, 3.5, 4.3.1, 4.5.5, A.4
- transform, 5.4
- FSB
- representation, see Fock–Segal–Bargmann representation
- space, see Fock–Segal–Bargmann space
- FSCc, see Fillmore–Springer–Cnops construction
- f-ghost cycle, 6.6
- f-orthogonality, 1.5, 1.6, 6.6, 7.4, 7.5, 9.4, 10.4.1, C.4
- factorisation
- faithful representation, 1.1
- fiducial operator, 4.1, 6.4
- fiducial vector, 2.1
- field
- finite dimensional
- fixed point
- flat cycles, 4.1
- focal
- length, 3.6, 7.1, 7.5, 8.4
- cycle, of, 5.1
- parabola, of, 5.1
- orthogonality, see f-orthogonality
- focus, 5.3, 7.1
- cycle, of, 1.2, 3.4, 5.1, 9.4, 10.3, B.2
- ellipse, of, B.2
- elliptic, 5.1
- hyperbola, of, 3.5, B.2
- hyperbolic, 5.1
- length from, 1.6, 7.2, 7.2, 7.3, 7.6, 10.3
- parabola, of, 1.2, 3.5, 5.1, B.2
- parabolic, 5.1
- form
- formalism
- formula
- Baker–Campbell–Hausdorff, 2.2
- Campbell–Hausdorff, see Baker–Campbell–Hausdorff formula
- cancellation for cross ratio, 4.5
- cancellation for cross-ratios, 9.5
- Euler, 11.2, B.2
- reconstruction, 4.4, 6.5
- Sokhotsky–Plemelj, 6.2
- function
- characteristic, 4.3
- Gauss, see Gaussian, 2.3
- Heaviside, 1.4, 3.1, 6.3
- Hermite, see Hermite polynomial
- norm of, 4.2
- orthogonality of, 4.2
- parabolic cylinder, see Weber–Hermite function
- polyanalytic, 5.3
- special, 1.7
- Weber–Hermite, 4.4.4, 4.4.5
- functional
- analysis, 4.2
- calculus, 4.3, 1, A.3, A.3
- covariant, 6.7
- Riesz–Dunford, 4.3
- support, see spectrum
- directing, 4.2
- invariant, 4.4, 6.5
- model, 4.3, 1.1, 1.5, A.3
- functions
- Galilean
- Gårding space, 2.1
- Gauss
- function, see Gaussian
- measure, 2.3
- Gaussian, 2.1, 2.3, 2.3, 4.2, 5.1, 6.4, 4.2.3, 4.4, 4.4.3, 5.2, 5.4
- Gelfand–Naimark–Segal construction, 1.4, 5.3, 5.3
- GiNaC, see CAS, C.2.3
- GNS construction, 1.4, see Gelfand–Naimark–Segal construction
- GNU, C
- General Public License (GPL), C
- Linux, C
- GPL, see GNU General Public License
- generalised
- generator
- quadratic, 3.5
- subgroup, of, 3.3
- geodesics, 9, 9.3, 9.5, 10.4.2, 11.1, 11.6.2
- geometrical quantisation, 4
- geometry, A.1
- birational, 11.5
- commutative, ??
- discrete, A.1
- Euclidean, ??, 7.4
- hyperbolic, 3.4
- Lobachevsky, ??, 3.4, 4.4, 5.2, A.1
- non-commutative, ??, 1.7, 1.6
- Riemann, 3.4, 9.1
- representational, 4.2
- ghost cycle, 6.3, 6.5
- grand maximal function, 4.2, 6.4
- ground state, 2.1
- group, 1, 2.1
- SU(1,1), 4.3, see also SL2(ℝ)
- SL2(ℝ), 1, 2.1, 2.2.1, 6.1, 6.3, 6.4, 6.8
- Hardy representation, 1.1
- Lie algebra, 2.3.3
- one-dimensional subgroup, 2.3.1
- [2], 4, see also SL2(ℝ)
- ax+b, 1, 2.1, 3.1, 3.5, 4.2, 4.4, 4.4, 5.1, 5.1, 5.2, 6.3, 6.4, 6.5, 6.6
- abelian, see commutative group
- abstract, 2.1, 2.2
- affine, see ax+b group, see ax+b group
- automorphism of, 2.3
- characters, of, 1.1
- commutative, 2.1
- continuous, 2.1
- Heisenberg, 1, 1.7, 2.1, 2, 2.1, 2.4, 4, 4, 4.5.5, A.4
- centre, 2.1
- Fock–Segal–Bargmann
representation, 5.4
- Fock–Segal–Bargmann representation, 3.3
- Haar measure on, 2.4
- induced representation, 3.1
- invariant measure, 3.1
- Lie algebra, 2.3.2, see Weyl algebra
- one-dimensional subgroup, 2.3.1
- polarised, 2.1
- Schrödinger representation, 1.1
- Heisenberg–Weyl, see Heisenberg group, 2
- Lie, 2.1, 2.3
- law, see group multiplication
- locally compact, 2.1
- multiplication, 2.1
- non-commutative, 2.1
- representation, 1.1
- Schrödinger, 2.3, 4.3
- symplectic, 2.1, 2.3, 3.5, 4, 4.3.1
- transformation, 2.1, 2.2
- unimodular, 3.1
- Weyl, see Heisenberg group, 2
- Haar measure, 2.4, see invariant measure, see invariant measure
- ax+b group, on, 2.4
- Heisenberg group, on, 2.4
- Hamilton
- Hamiltonian, 8.3.2
- classical mechanics, 2.1
- quadratic, 8.3.2
- Hardy
- pairing, 4.4, 4.4, 6.5
- space, 1.7, 4, 4.2, 4.5, 5.3, 5.5, 6.2, 6.3, 6.4, 6.7, 1.2, 1.5, A.2
- Hardy–Littlewood
- Heaviside
- Heaviside
function, 6.3
- Heaviside function, 1.4
- Heisenberg
- commutation relation, 2.3.3, 2, 2.2
- equation, 4, 4.2.2
- group, 1, 1.7, 2.1, 2, 2.1, 2.4, 4, 4, 4.5.5, A.4
- centre, 2.1
- Fock–Segal–Bargmann representation, 3.3, 5.4
- Haar measure, 2.4
- induced representation, 3.1
- invariant measure, 3.1
- Lie algebra, 2.3.2
- one-dimensional subgroup, 2.3.1
- polarised, 2.1
- Schrödinger representation, 1.1
- Heisenberg group, 1
- Heisenberg–Weyl
- group, see Heisenberg group, 2
- Hermite
- Hilbert
- Huygens
- h-centre, 4.1
- h-focus, 5.1
- half-plane, 1.6
- boundary effect, 1.3, 5.1, 5.4, 6.1, 6.3, 6.6
- invariance, 3.3.1, 3.3.2, 8.2, 8.2
- lower, 2.2.1, 3.6, 3.7
- non-invariance, 3.3.3, 8.2, 8.2, 8.2
- upper, 2.2.1, 3.6, 3.7, 10, 5.6
- harmonic
- analysis, 1.7
- oscillator, 8.3.2, 2.3, 3.5, 4, 4.1.1, 4.2.2, 4.3, 4.4.2, 4.5.2
- heat
- homogeneous space, 2.2.1, 2.2.1, 2.2.2
- homomorphism
- horizon, 9.4
- horocycle, 1.3, 5.4, 5.5
- hyperbola, 4.1, B.2
- hyperbolic
- Cayley transform, 10.1
- case, 1.1, 3.4
- Fock–Segal–Bargmann representation, 4.4.1
- geometry, 3.4
- harmonic oscillator, 4.4.4
- Moyal bracket, 4, 4.4.2
- plane
- probability, 4.4.3
- rotation, 10.2
- unharmonic oscillator, 4.4.2
- unit
- unit (є), 1.1
- upper half-plane, 3.3.3, 8.2
- zero-radius cycle, 5.4
- hyperboloid, 8.1
- hypercomplex
- hypercomplex numbers, 6.3
- hyperreal
- IPython, C.3
- Iwasawa decomposition, 1.1, 3.2, 3.5, A.1
- ideal
- idempotent
- identity, 2.1
- approximation of the, 6.3
- Jacobi, 2.3.3
- Pythagoras’
- imaginary
- indefinite
- index
- induced
- induced representation, 1.4, 2.3
- induced wavelet transform, 2.3, 2.3
- induction, 3
- inequality
- infinite dimensional representation, 1.1
- infinitesimal
- infinitesimally conformal, 7.6
- infinity
- inner
- automorphism of group, 2.3
- derivation, 4, 4.1.1
- product, 4.2
- inner product
- integrability of discrete equation, B.4
- integral
- Bergman, 1.7
- Cauchy, 1.7, 4.2, 5.3, 5.4, 5.4, 5.5, 6.2, 6.4, 6.8, 1.2, A.2
- Poisson, see Poisson kernel, see Poisson kernel
- interference, 4.1.2, 4.2.3, 4.5.3
- intersecting
- intertwining map, 4.3
- intertwining operator, 1.3, 4.1, 4.1, 4.5, 5.1, 5.1, 6.6, 6.6, 6.7, 1.1, 1.2, 1.6, 5.4
- interval
- invariant, 4.4, 6.5
- functional, 4.4, 6.5
- measure, 4, 4.2, 4.4, 4.4, 6.3, 6.4, 6.5, 6.5
- metric, 9, 9.5, 11.3.1
- pairing, 4.4, 6.5
- subset, 2.2.1
- subspace, 6.7
- vector fields, 2.3.2
- invariant measure, 2.4
- invariant subspace, 1.2
| - inverse, 2.1
- covariant transform, see contravariant transform
- inverse wavelet transform, 2.1
- inversion
- circles, in (Yaglom term), B.3
- first kind, of the (Yaglom term), 6.5
- in a cycle, 6.5, 6.5, 6.5, 6.6, 8.1, 8.1
- second kind, of the (Yaglom term), 6.5, B.3
- inversive distance between cycles, 5.4
- irreducible
- irreducible representation, 1.2
- isotropic cycles, 1.4, 6.2
- isotropy subgroup, 2.2.1, 3.7, 3.7.1, 10.2, 10.3, 10.3, 10.3
- Jacobi identity, 2.3.3
- Jordan
- Jordan’s normal form of a matrix, 1.2
- jet, 1.3, 1.4, 1.5, 4
- joint
- Kirillov
- correspondence, 4.4.1
- Descartes–Kirillov condition for tangent circles, 5.5
- orbit method, 4.4.2
- Krein
- Kroneker delta, 1.1
- kernel
- Laplace
- Laplace operator, see Laplacian
- Laplacian, 5.3, 5.3, A.2
- Lebesgue measure, 2.4
- Lidskii theorem, 1.5
- Lie
- Littlewood–Paley
- Littlewood–Paley theory, 6.2
- Lobachevsky
- Lorentz–Poincare
- ladder operator, 3.3, 3.3, 5.3, 5.3, 5.4, 4.3, 4.3.2, 4.4, 4.4.4, 4.4.5, 4.5.4, 4.5.5
- left regular representation, 1.2, 3.1
- left shift, 2.1, 2.2.1, 2.2.2, 2.3.2
- lemma
- length, 1.6, 7.2
- conformal, 7.3
- curve, of, 9.1
- focal, of a cycle, 5.1, 7.1
- focal, of a hyperbola, 3.6, 8.4
- focal, of a parabola, 5.1
- from centre, 1.6, 7.2, 7.2, 7.3, 10.1, 10.1, 10.2, 10.2, 10.3, 10.3
- from focus, 1.6, 7.2, 7.2, 7.3, 7.5, 7.6, 10.3
- lifting, 3.2
- light
- cone, 3.7, 5.4, 6.2, 6.4, 7.5, 8.1, 8.2, 8.4
- speed, of, 8.4
- limit
- line
- contour, 3.5, 7.3, 7.4, 7.4, 11.2
- Menger, 9.3
- parallel, 5.5
- special (Yaglom term), B.3
- spectral, 8.3.3, 8.4
- linear
- linear-fractional transformations, see Möbius map
- linearization procedure, 1.1
- locally compact group, 2.1
- loop, 2.3
- lower half-plane, 2.2.1, 3.6, 3.7
- lowering operator, see ladder operator
- Möbius map, 1
- Maslov
- Menger line, 9.3
- Mexican hat
- Minkowski
- Möbius map, 1, 2.1, 2.2.1, 4.2, 4.3, 5.2, 5.5, 1.5, 1.6, 1.6
- Clifford algebra, B.5
- on cycles, 1.2, 4.3
- on the real line, 3.1
- Moyal bracket, 4.2.2, 4.2.3
- main problem
- of representation theory, 1.2
- map
- p, 2.2.2, 8.3.3, 1.4
- r, 8.3.3
- s, 2.2.2, 8.3.3, 1.4
- M, 4.2, 8.1
- Q, 4.2, 6.2, 8.1
- r, 2.2.2, 1.4
- exponential, 2.3.1, 3.2
- intertwining, 4.3
- Möbius, 1, 2.1, 2.2.1, 4.2, 4.3, 5.2, 5.5, 1.5, 1.6, 1.6
- Clifford algebra, B.5
- on cycles, 1.2, 4.3
- on the real line, 3.1
- preserving orthogonality, 6.2
- mathematics
- matrix
- matrix coefficients, 2.1
- matrix elements, 1.1
- maximal function
- maximal functions, 6.3
- Hardy–Littlewood, 4.2, 4.5, 6.2, 6.4, 6.7, 6.8
- non-tangential, 4.4, 6.5, 6.8
- vertical, 4.4, 6.5, 6.8
- measure, 2.4
- Carleson, 6.8
- Gauss, 2.3
- Haar, 2.4, see invariant measure, see invariant measure
- invariant, 4, 4.2, 4.4, 4.4, 6.3, 6.4, 6.5, 6.5
- Lebesgue, 2.4
- left invariant, 2.4
- mechanics
- metaplectic representation, see oscillator representation
- method
- metric, 9.1
- minimal
- model
- modulus
- monad, 7.6
- monotonous
- mother wavelet, 1.7, 2.1, 4.2, 6.4, 1.2
- multiresolution analysis, 4.2
- netbook, C.1
- nilpotent
- nilpotent unit, see parabolic unit
- non-Archimedean analysis, see non-standard analysis
- non-commutative
- non-commutative geometry, 1.6
- non-linear dynamics, 11.5, B.4
- non-locality, 1.4, 1.5, 6.1, 6.3
- non-simply connected domain, 5.5
- non-standard
- non-tangential
- non-trivial invariant subspaces, 1.2
- norm
- norm-normalised cycle, 5.4
- normal
- normalised cycle, 1.3, 5.2
- det-, 5.2, 5.4, 5.4, 6.6, 7.1, 8.1, C.4
- k-, 1.6, 5.2, 5.4, 5.4, 7.1, 7.1, 8.1, C.4
- norm-, 5.4
- nucleus, 4.4, 6.5
- number
- double, 4, 4.4, 4.4.5
- dual, 4, 4.5, 4.5, 4.5.4
- hypercomplex, 4.5.5
- infinitesimal, 7.5, 7.6
- prime, 1.2
- system, 3.3.4
- numbers
- complex, 3.3.1, 6.4, B.1
- double, 1.1, 3.3.3, 6.4, B.1
- dual, 1.1, 3.3.2, B.1
- hypercomplex, 6.3
- split-complex, see double numbers
- numerical
- observable, 4.1.1
- one-dimensional subgroup, 2.3.1
- open
- operator
- annihilation, see ladder operator
- Casimir, 5.3
- Cauchy–Riemann, 1.7, 5.1, 6.6, 6.8
- Cauchy-Riemann, 5.3, 5.3, 4.3.1, A.2
- commuting, 1.3
- convolution, 2.4
- creation, see ladder operator
- defect, 4.3
- Euler, 3.5
- Fillmore–Springer–Cnops construction, 1.6
- fiducial, 4.1, 6.4
- integral
- intertwining, 1.3, 4.1, 4.1, 4.5, 5.1, 5.1, 6.6, 6.6, 6.7, 1.1, 1.2, 1.6, 5.4
- Laplace, 1.7, 5.1, see Laplacian, 6.6
- Littlewood–Paley, 4.2
- ladder, 3.3, 3.3, 5.3, 5.3, 5.4, 4.3, 4.3.2, 4.4, 4.4.4, 4.4.5, 4.5.4, 4.5.5
- lowering, see ladder operator
- power bounded, A.3
- pseudo-differential, 2.3.2
- pseudodifferential, 4.3
- quasinilpotent, A.3
- raising, see ladder operator
- Toeplitz, A.2
- unitary, 2.4, 1.1
- optics, 2.3
- orbit, 2.2.1
- co-adjoint representation, 3.3
- isotropy subgroup, 3.7, 5.4
- method, 6.3, 4, A.1
- method of Kirillov, 4.4.2
- subgroup , of, 3.7
- subgroup A, of, 1.1, 3.5, 10.4.2
- subgroup K, of, 1.1, 3.5, 3.5, 3.5, 3.6, 3.7, 5.1, 8.2, 8.2, 10.4.2
- subgroup N, of, 1.1, 3.5, 10.4.2
- subgroup N′, of, 3.7
- orders
- orientation
- orthocenter, 6.2
- orthogonal
- cycles, see orthogonality of cycles
- pencil of cycles, 6.2, 6.4, 6.5
- orthogonality
- cycles, of, 1.2, 1.4, 4.1, 4.3, 6.1, 6.1, 6.3, 6.5, 6.5, 7.5, 9.1, 10.4.2, C.4
- focal, see f-orthogonality
- function, of, 4.2
- Galilean, 7.4
- preserving map, 6.2
- relation, 6.8
- second kind, of the, see f-orthogonality
- orthonormal basis, 1.1
- oscillator
- harmonic, 8.3.2, 2.3, 3.5, 4, 4.1.1, 4.2.2, 4.3, 4.4.2, 4.5.2
- representation, 3.5
- unharmonic, 4.1.1, 4.2.2, 4.5.2
- outer
- automorphism of group, 2.3
- PDO, see pseudo-differential operator, 4.3
- Planck
- Plato’s cave, 1.2, 1.2, 5.1
- Poincaré extension, 4.4.2, 14.2
- Poisson
- Poisson kernel, 4.2, 5.3, 6.2, 6.4, 6.8
- Pontrjagin
- PT-symmetry, 4
- Pythagoras’
- Python, C.2.3, C.2.3, C.3.1
- Pyzo, C.3
- p-centre, 4.1
- p-focus, 5.1
- p-mechanics, 4.1
- p-mechanisation, 4.1.1
- packet
- pairing
- parabola, 4.1, B.2
- directrix, 1.2, 3.5, 5.1, B.2
- focal length, 5.1
- focus, 1.2, 3.5, 5.1, B.2
- vertex, 1.2, 5.1, B.2
- parabolas
- parabolic
- Cayley transform, 10.3
- case, 1.1, 3.4
- circle (Yaglom term), 6.2, B.3
- cycle (Yaglom term), B.3
- cylinder function, see Weber–Hermite function
- degeneracy, 6.3, 10.3
- Fock–Segal–Bargmann representation, 4.5.1
- modulus, 11.3.1
- norm, 11.3.1
- Pythagoras’ identity, 11.2, 11.3.1
- probability, see classic probability
- trigonometric functions, 11.1
- unit
- unit (ε), 1.1
- upper half-plane, 3.3.2, 3.7
- zero-radius cycle, 5.4
- parallel
- parallelism, Benz, 6.2
- paravector formalism, B.5
- paraxial
- pencil
- covariant, 1.6
- cycles, of, ??, 5.2, 5.4, 5.5, 5.5, 5.5, 6.2, 6.2, 6.2, C.3.4
- periodic table, 3.3.4
- chemical elements, of, ??
- perpendicular, 1.6, 7.4
- phase
- plane
- point
- cycle, of, 5.4, 7.1
- infinity, at, 8.1
- power, of, ??, 3.6, 4.1, 5.4, 7.1, 7.1, C.4
- space, 1.2, 4.2, 13.2.1, B.3
- polar
- projection, see stereographic projection
- polyanalytic function, 5.3
- polynomial
- positive
- power
- cycle, of, 5.4, 7.1
- point, of, ??, 3.6, 4.1, 5.4, 7.1, 7.1, C.4
- Steiner, see power of a point
- power bounded operator, A.3
- primary
- primary representation, 1.2
- prime number, 1.2
- principal
- principle
- probability
- product
- projection
- polar, see stereographic
- stereographic, 8.1
- projective
- projective space, 1.2, 4.2, 4.4.1, 4.5, 11.4.2, B.1
- prolongation, 1.3
- pseudo-differential operator, 2.3.2
- pseudodifferential operator, 4.3
- pulling, 3.2
- pyGiNaC, C.2.3
- Quaternions, 1.6
- quadratic
- quadric, B.2
- quantisation
- quantum
- quantum mechanics, 4, 4
- quasi-regular
- representation of ax+b group, 6.3
- quasinilpotent
- quotient
- Radon transform, 4.2
- Riemann
- radical axis
- radius
- raising operator, see ladder operator
- range
- ray, 8.3.1, 9
- real line, 1
- reconstruction formula, 4.4, 6.5
- red shift, 8.2, 8.4
- reduced wavelet transform, see induced wavelet transform, 5.1
- reducible representation, 1.2
- reference
- reflection
- refractive
- regular representation, 1.2
- relation
- relation of equivalence, 1.1
- relative
- relative convolution, 5.7
- relativity, 8.4
- representation, 1.1
- SL2(ℝ) group, 3.2, 3.2
- ax+b group, 4.2
- co-adjoint, 6.3
- quasi-regular, 6.3
- adjoint, 4.4.2, 1.1
- character of, 1.1
- co-adjoint, 3.3
- coefficients, see wavelet transform, see wavelet transform
- complementary series, 4.2, A.3, A.3
- continuous, 1.1
- decomposable, 1.2
- discrete series, 3.2, 4, 4.2, 4.2, A.3, A.3
- exact, 1.1
- Fock–Segal–Bargmann, 3.3, 3.5, 3.5, 4.5.4, 4.5.4, 5.4
- FSB, see Fock–Segal–Bargmann representation
- faithful, 1.1
- finite dimensional, 1.1
- group
- Heisenberg group
- induced, 4.4.2, 8.3.3, 11.5, 1.4, 2.3, 3, 1.2, 4
- infinite dimensional, 1.1
- irreducible, 1.2, 6.3, 6.7
- left regular, 1.4, 3.1
- linear, 1.1
- linearization, 1.1
- metaplectic, see oscillator representation
- primary, 1.2, 6.3, 1.2, 1.3, 1.4, 1.6
- principal series, 4.2, A.3, A.3
- reducible, 1.2
- regular
- Schrödinger, 1.1, 3.5, 4.2.1, 4.2.1, 4.3.1, 4.4.4, 5.2
- Shale–Weil, see oscillator representation
- series
- square integrable, 2, 2.2, 4, 4.2, 4.4, 4.4, 5.1, 6.4, 6.5, 6.8
- trivial, 1.1
- unitary, 1.1
- representation of a group, 1.1
- representation space, 1.1
- representation theory
- representational coefficients, 2.1
- representational geometry, 4.2
- representations, 1.1
- equivalent, 1.1
- Fock–Segal–Bargmann, 4.2.1
- linear, 3
- reproducing kernel, 2.2
- repulsive
- harmonic oscillator, 4.4.4
- resolvent, 4.3, 1.1, 1.2, 1.6
- restriction of representation, 1.2
- right shift, 2.1, 2.2.1, 2.3.2
- rotation, 3.7, 9.1
- elliptic, 10.1
- hyperbolic, 10.2
- parabolic
- Schrödinger
- Schrödinger representation, 1.1
- Schur’s lemma, 1.3
- Shale–Weil representation, see oscillator representation
- SIO, see singular integral operator
- seesingular integral
operator, 6.7
- Sokhotsky–Plemelj formula, 6.2
- Steiner power, see power of a point
- scalar product, 1.1
- section, 2.2.2
- selfadjoint cycle, 6.2, 6.6, 9.5, B.3, B.3
- semiclassical
- semigroup
- set
- shift
- signature
- similarity
- simply connected domain, 5.5
- singular
- source
- space
- Bergman, 1.7, 4, 4.2, 5.5, A.2
- commutative, ??
- configuration, 4.2.1, 4.2.3, 4.4.1, 4.5.1, 4.5.4
- cycle, 13.2.1
- cycles, of, 1.2, 4.2, 8.1, B.3
- Euclidean, 7.4
- Fock–Segal–Bargmann, 4.2, 3.5, 4.3.1, 4.5.5, A.4
- FSB, see Fock–Segal–Bargmann space
- Gårding , 2.1
- Hardy, 1.7, 4, 4.2, 4.5, 5.3, 5.5, 6.2, 6.3, 6.4, 6.7, 1.2, 1.5, A.2
- Hilbert, 4.2, 5.3
- Krein, 5.3, 3.2, 4
- linear, 4.2
- Minkowski, 5.3
- non-commutative, ??
- Pontrjagin, 5.3
- phase, 8.3.2, 3.5, 4, 4, 4.2.1, 4.2.3, 4.3, 4.4.1
- point, 1.2, 4.2, 13.2.1, B.3
- projective, 1.2, 4.2, 4.4.1, 4.5, 11.4.2, B.1
- tangent, 2.3.1, 4.4.2, 6.6, 9.1
- space-like interval, 8.4, 9.1
- space-time, 8.1, 8.4
- special
- function, 1.7
- line (Yaglom term), B.3
- spectral
- spectrum, 1.1, 1.3
- speed
- sphere
- split-complex numbers, see double numbers
- square integrable
- square integrable representation, 2, 2.2
- squares
- stability
- contravariant spectrum, 1.5
- state, 4.1.2
- coherent, see wavelet
- ground, 2.1
- vacuum, see mother wavelet, see mother wavelet
- stereographic
- stopping time argument, 6.6, 6.8
- subalgebra
- subgroup, 2.2
- , 9.1
- A, 1.1, 3.2
- A′, 3.2, 3.7, 8.4, 10.2, 11.2
- A″, 3.7
- F, 3.1
- F′, 3.1
- F′, 3.1
- K, 1.1, 1.7, 3.2, 9.1, 11.2
- orbit, 1.1, 3.5, 3.5, 3.5, 3.6, 3.7, 5.1, 8.2, 8.2, 10.4.2
- N, 1.1, 3.2
- N′, 3.2, 3.7, 8.4, 9.1, 11.2
- conjugated, 2.2.1
- generator, 3.3
- isotropy, 2.2.1, 3.7, 3.7.1, 10.2, 10.3, 10.3, 10.3
- normal, 2.2.2
- one-dimensional, 2.3.1
- subrepresentation, 1.2
- subset
- subspace
- summation
- support
- functional calculus, see spectrum
- swiGiNaC, C.2.3
- symbol
- contravariant, 4.3
- covariant, 4.3
- symbolic calculus, see covariant calculus
- symplectic
- form, 4.5, 4.5, 8.3.1, 8.3.2, 2.1, 2.2, B.5
- group, 2.1, 2.3, 3.5, 4, 4.3.1
- transformation, 3.1, 3.1, 4, 4.4.4
- system
- Taylor
- Toeplitz
- tangent, 1.4, 5.1, 6.1, 6.1, 6.3, 6.3, 6.3, 7.4, 9.4
- tent, 6.2, 6.4, 6.8
- theorem
- automatic proving, C.3.3
- Lidskii, 1.5
- main of arithmetic, 1.2
- Riemann mapping, 5.5
- spectral mapping, 1.1
- theory
- time
- time-like interval, 8.4, 9.1
- token, 4.1
- total set, 2.1
- trace, 1.3, 1.1
- transfer
- transform
- Cayley, 10, 10.4.2, 5.5
- contravariant, 4.4, 6.5, 6.8, 5.7
- covariant, 4.1, 6.4, 6.8
- induced, 5.1
- inverse, see contravariant transform
- Fock–Segal–Bargmann, 5.4
- Hilbert, 6.7, 6.7
- Radon, 4.2
- wavelet, 1.7, 2.1, see wavelet transform, 6.4, 6.8
- continuous, 6.4
- induced, 2.3
- reduced, see induced wavelet transform
- transformation
- canonical, 8.3.2
- Galilean, 8.4
- group, 2.1, 2.2
- Lorentz–Poincare, 8.4
- linear-fractional, see Möbius map
- Möbius, see Möbius map
- symplectic, 3.1, 3.1, 4, 4.4.4
- transitive, 1, 2.2.1, 2.2.2, 3.5
- triangle
- trigonometric
- trivial representation, 1.1
- tropical
- umbral calculus, 4.1
- unharmonic
- unimodular
- unit
- circle, see elliptic unit cycle, 5.5
- cycle
- disk, 10, 5.5
- hyperbolic (є), 1.1
- hypercomplex (ι), 1.1, B.1
- imaginary (i), 1.1
- nilpotent, see parabolic unit
- parabolic (ε), 1.1
- unitary equivalent representations, 1.1
- unitary operator, 2.4, 1.1
- unitary representation, 1.1
- upper half-plane, 2.2.1, 3.6, 3.7, 10, 5.6
- boundary effect, 1.3, 5.1, 5.4, 6.1, 6.3, 6.6
- elliptic, 3.3.1, 3.7
- hyperbolic, 3.3.3, 8.2
- parabolic, 3.3.2, 3.7
- Vahlen condition, 6.4, 6.5
- vacuum state, see mother wavelet, see mother wavelet
- vacuum vector, 2.1
- vacuum vector of quantum mechanics, 2.3
- value
- vector
- vector fields
- vector formalism, B.5
- vertex
- vertical
- Weber–Hermite function, 4.4.4, 4.4.5
- Weyl
- algebra, 2, 2.2
- commutation relation, 2
- group, see Heisenberg group, 2
- Wiener–Hopf factorisation, 4.2
- wave, 9
- wavelet, 1, 4, 4.1, 4.2, 5.1, 5.2, 6.4, 6.4, 1.2
- admissible, 4, 4.2, 4.4, 6.4, 6.5
- Mexican hat, 6.4
- mother, 1.7, 2.1, 4.2, 6.4, 1.2
- transform, 1.7, 4.2, 6.4, 6.8, 1.2, 4.3.1
- continuous, 6.4
- induced, see induced covariant transform
- reduced, 5.1
- wavelet transform, 2.1
- induced, 2.3, 2.3
- inverse, 2.1
- reduced, see induced wavelet transform
- wavelets, 2.1
- Yaglom term
- inversion
- circles, in, B.3
- second kind, of , B.3
- parabolic
- special line, B.3
- zero
- divisor, 1.1, 3.3.2, 3.4, 3.7, 3.7, 4.5, 8.1, 11.2, 4.4.5, B.1
- order, of, 1.4
- zero-radius cycle, 1.3, 1.4, 4.4.1, 5.3, 5.4, 5.4, 5.5, 6.2, 6.2, 6.2, 7.1, 7.5, 8.1, 8.1, 9.4, B.3
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