Abstract
Compact, copy-paste quick reference for SSM function lifts. Each entry states the magnitude rule (ASCII), domain/range, monotonicity, alignment policy, key guards, and a stable numeric formula where applicable. Under collapse phi(m,a) = m, all rows reduce to their classical counterparts.
Legend (columns)
Name | Lift on (m,a) | Domain → Range | Monotonicity | Alignment | Guards | Stable numerics
Elementary & rational
| Name | Lift on (m,a) | Domain → Range | Monotonicity | Alignment | Guards | Stable numerics |
|---|---|---|---|---|---|---|
| Power k (integer) | ( m^k , a ) | R → R | inc on m≥0 if k≥1; even k nonmonotone | carry | declare branchwise for even k | — |
| Root r (principal) | ( m^(1/r) , a ) | m≥0 → m≥0 | increasing | carry | error if m<0 | scale before pow for large m |
| Reciprocal | ( 1/m , a ) | m≠0 → R | decreasing on (0,∞) | carry | error at m=0 | fuse as 1/(scaled m) |
| Rational p/q | ( p(m)/q(m) , a ) | q(m)≠0 | piecewise | carry | poles where q=0 | use Horner; guard near poles |
Inverse trigonometric (principal)
| Name | Lift on (m,a) | Domain → Range | Monotonicity | Alignment | Guards | Stable numerics |
|---|---|---|---|---|---|---|
| asin | ( asin(m) , a ) | [-1,1] → [-pi/2, +pi/2] | increasing | carry | error if | m |
| acos | ( acos(m) , a ) | [-1,1] → [0, pi] | decreasing | carry | error if | m |
| atan | ( atan(m) , a ) | R → (-pi/2, +pi/2) | increasing | carry | — | atan via atan2(m,1) |
| atan2 (y,x) | ( atan2(y,x) , a’ ) | (x,y)≠(0,0) → (-pi,+pi] | quadrant-aware | a’ from weights | error at origin | use builtin atan2 |
Hyperbolic & inverse hyperbolic
| Name | Lift on (m,a) | Domain → Range | Monotonicity | Alignment | Guards | Stable numerics |
|---|---|---|---|---|---|---|
| sinh | ( 0.5*(e^m – e^-m) , a ) | R → R | increasing | carry | — | exp-splitting; one-sided scaling for |
| cosh | ( 0.5*(e^m + e^-m) , a ) | R → [1,∞) | dec on (-∞,0], inc on [0,∞) | carry | — | exp-splitting; branchwise order |
| coth | ( cosh(m)/sinh(m) , a ) | R{0} → (-∞,-1)∪(1,∞) | decreasing on each side | carry | undefined at 0 | use 1/tanh(m) |
| asinh | ( log(m + sqrt(m^2+1)) , a ) | R → R | increasing | carry | — | for |
| acosh | ( log(m + sqrt(m-1)*sqrt(m+1)) , a ) | m≥1 → [0,∞) | increasing | carry | error if m<1 | for m≫1 use log(2*m) |
| acoth | ( 0.5*log((m+1)/(m-1)) , a ) | m | >1 → (-∞,0)∪(0,∞) | odd; sign-monotone | carry |
Piecewise, indicator, rounding
| Name | Lift on (m,a) | Domain → Range | Monotonicity | Alignment | Guards | Stable numerics |
|---|---|---|---|---|---|---|
| sgn | ( -1, +1 ) / (0,+1) / ( +1,+1 ) | R → {-1,0,1} | step | constants +1 | tie policy at 0 | — |
| Heaviside H | ( 0,+1 ) / (1/2,+1) / (1,+1) | R → {0,1/2,1} | step | constants +1 | choose H(0) | — |
| ramp / ReLU | ( m,a ) if m>0 else (0,+1) | R → [0,∞) | nondecreasing | carry or +1 at 0 | tie policy at 0 | — |
| clip [L,U] | (L,+1)/(m,a)/(U,+1) | L≤U | nondecreasing | carry-from-source | swap or error if L>U | — |
| floor | ( floor(m) , a ) | R → Z | step | carry | — | — |
| ceil | ( ceil(m) , a ) | R → Z | step | carry | — | — |
| trunc | ( trunc(m) , a ) | R → Z | step | carry | — | — |
| round | ( round(m) , a ) | R → Z | step | carry | rounding_mode | ulp-aware half detection |
Selectors & order
| Name | Lift on (m,a) | Domain → Range | Monotonicity | Alignment | Guards | Stable numerics |
|---|---|---|---|---|---|---|
| min_beta | select by S_beta | R^2 → R | lattice | carry from selected | tie rule | — |
| max_beta | select by S_beta | R^2 → R | lattice | carry from selected | tie rule | — |
| clamp_beta | min_beta(max_beta(x,L),U) | L≤U | bounded | from selected | swap or error | — |
S_beta(m,a) = m*(1 – beta*(1 – a)), beta in [0,1].
Smooth nonlinearities
| Name | Lift on (m,a) | Domain → Range | Monotonicity | Alignment | Guards | Stable numerics |
|---|---|---|---|---|---|---|
| sigmoid | ( 1/(1+e^-m) , a ) | R → (0,1) | increasing | carry | — | 0.5*(1+tanh(m/2)) |
| softplus | ( log(1+e^m) , a ) | R → (0,∞) | increasing | carry | — | piecewise: m+log1p(e^-m) / log1p(e^m) |
| softsign | ( m/(1+ | m | ) , a ) | R → (-1,1) | increasing | carry |
Series, analytic, transforms
| Name | Lift on (m,a) | Domain → Range | Monotonicity | Alignment | Guards | Stable numerics |
|---|---|---|---|---|---|---|
| Power series | ( Σ c_n (m-m0)^n , a ) | m-m0 | <R | as f | carry | |
| exp | ( Σ m^n/n! , a ) | entire | increasing | carry | — | truncation with remainder bounds |
| log(1+m) | ( Σ (-1)^{n+1} m^n/n , a ) | m | <1 | increasing | carry | |
| Z-transform | ( Σ m[n] z^{-n} , a_Z ) | classical ROC | — | rapidity mean | ROC guard | — |
| Laplace | ( ∫ m(t) e^{-s t} dt , a_L ) | Re(s)>sigma0 | — | rapidity mean | ROC guard | — |
Alignment summary (transforms): a_T = tanh( Σ w u / Σ w ), with u=atanh(a), weights typically |m|^gamma.
Special functions
| Name | Lift on (m,a) | Domain → Range | Monotonicity | Alignment | Guards | Stable numerics |
|---|---|---|---|---|---|---|
| Gamma | ( Gamma(m) , a ) | m>0 → (0,∞) | unimodal (min near 1.4616) | carry | error if m≤0 (default) | use lgamma; Stirling for large m |
| Beta | ( B(x,y) , a_B ) | x>0,y>0 | decreasing in each arg | rapidity mean over inputs | domain guard | use betaln then exp if safe |
| erf / erfc | ( erf(m)/erfc(m) , a ) | R | inc / dec | carry | — | series small |
| Bessel J,Y | ( J_nu(m), Y_nu(m) , a ) | m≥0 / m>0 | osc. | carry | Y singular at 0 | scaled forms, recurrences |
| Bessel I,K | ( I_nu(m), K_nu(m) , a ) | m>0 | I inc, K dec | carry | singular at 0 | scaled IK to avoid overflow |
| Polylog Li_s | ( Li_s(m) , a ) | m | <1 (principal) | — | carry | |
| Lambert W_0 | ( W_0(m) , a ) | m≥-1/e | increasing | carry | error outside | series small; log-log for large m |
| Lambert W_-1 | ( W_-1(m) , a ) | -1/e≤m<0 | decreasing | carry | branch select | asymptotic near 0- |
Multivariate, inverse, inequalities
| Name | Lift on inputs | Domain → Range | Monotonicity | Alignment | Guards | Stable numerics |
|---|---|---|---|---|---|---|
| Aggregator f* | ( f(m_i) , a’ ) | per f | per f | rapidity mean with w_i= | ∂f/∂m_i | * |
| Inverse of monotone f | ( f^{-1}(y) , a_y ) | y in range | inc/dec maps | carry from target | branch select if nonmonotone | Newton with guards |
| Implicit F=0 | ( g(p) , a’ ) | IFT region | as g | rapidity mean with s_i= | ∂g/∂p_i | |
| Inequalities | f*(x) | piecewise | preserves/reverses on magnitudes | alignment metadata only | declare domains | — |
Vectors, matrices, convolution
| Name | Lift | Domain → Range | Monotonicity | Alignment | Guards | Stable numerics |
|---|---|---|---|---|---|---|
| A+B | entrywise | matrices | linear | rapidity-weighted sum per entry | — | weights |
| c*A | entrywise | matrices | linear | constants +1 | — | — |
| A*B | product | shapes match | bilinear | per-term M2 then weighted mean per entry | — | weights |
| tr(A), | A | _F | scalar | any | ||
| det(A) | scalar | nonsingular | poly | cofactor-weighted mean | warn near singular | adjoint weights |
| exp(A), log(A) | spectral | log on SPD | per eigencurve | eigen-channel aggregation | domain guards | scaling & squaring |
| Convolution | classical on m | mode/causal as set | linear | rapidity mean with | k | * |
| Correlation | classical on m | — | bilinear | per-term M2 then mean | — | — |
Takeaway
Use the rows above as a practical index: apply the classical magnitude formula, carry or aggregate alignment exactly as stated, enforce the guard, and prefer the stable numeric identity. Collapse (a=+1) reproduces the standard function.
Navigation
Previous → Convolution & correlation (scalar kernels) (2.48)
Next → Manifest templates & examples (2.50)
Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: mult_mode = M2 for products, rapidity-weighted sums for additions/aggregations, clamp_eps = 1e-6, |a| < 1. All formulas are presented in plain text. Collapse uses phi(m,a) = m.
SHUNYAYA 