Mathematical notation¤
These conventions connect the probability model, numerical methods, and sensitivity calculations. Chapters define their local quantities where used; API parameter names and Python examples retain their code spelling. Use this page as a reference: the model introduces the biological process, then computational tasks explains why the numerical representations are needed.
Population model¤
These symbols describe allele counts, biological parameters, and one-generation transitions. The matrix orientation fixes how propagation and expectation relate.
| Concept | Symbol |
|---|---|
| Diploid effective population size | \(N_e\) |
| Population gene-copy count | \(M=2N_e\); \(M_s,M_d\) for source and destination |
| Sample gene-copy count | \(n\) |
| Source and destination allele counts | \(i,j\) |
| Allele frequency | \(x\), with \(x_i=i/M_s\) |
| Offspring frequency after the full frequency map | \(q_\theta(x)\); omit the subscript when parameters are implicit |
| Model parameter collection and its dimension | \(\theta,n_\theta\) |
| Selection coefficients and dominance | \(s_{\mathrm{het}},s_{\mathrm{hom}},h\) |
| Multiplicative fitness ratio | \(\rho\) (API parameter r) |
| Directional mutation rates | \(\mu_{0\to1},\mu_{1\to0}\) |
| One-generation transition and its numerical approximation | \(P,\widetilde P\) |
| Population mass column vector | \(v_t\) |
| Observable or destination-value vector | \(f\) |
| Generation index and interval duration | \(t,g\) |
| Exact and represented interior transitions | \(Q,\widetilde Q\) |
| Expected-visits vector | \(u\) |
| Immigration/source vector | \(b\) |
Rows of \(P\) index source states; columns index destination states. Vectors are columns unless transposed. The model chapter develops the transition, propagation, and expected-visits equations.
The fastDTWF paper uses the same column-distribution and transpose convention, but names its transition matrix \(\mathbf M\) and its haploid population size \(N\). Here those correspond to \(P\) and the gene-copy count \(M\), respectively. We retain \(N_e\) for diploid size.
Numerical representations¤
These symbols distinguish computational dimensions from biological sizes and durations. The computational overview introduces the representations; each algorithm chapter defines its construction.
| Concept | Symbol |
|---|---|
| Prepared forward operator; input and output working states | \(A;\ z,y\) |
| Represented forward population operator | \(\widetilde P^T\) |
| Prepared exponent; reconstruction map and offset | \(k;\ R,c\) |
| Working dimension and Krylov basis dimension | \(d,m\) |
| Krylov basis and projected operator | \(V_m\), \(H_m\) |
| Jacobi coefficient transform and evaluation | \(E,D\) |
| Reduced Jacobi expectation operator | \(H_{\mathrm J}\) |
| Retained interior Jacobi dimension and assembly padding | \(r,r_{\mathrm{pad}}\) |
| Polynomial degree and coefficient count | \(L,n_{\mathrm{modes}}=L+1\) |
| Quadrature, sampling, and evaluation counts | \(n_{\mathrm{quad}},n_{\mathrm{nodes}},n_{\mathrm{eval}}\) |
| Retained binomial stencil width | \(W\) |
| Angular coordinate | \(\vartheta\) |
| Centered Jacobi coordinate | \(\xi=2q-1\) |
| FFT log-odds spacing | \(\Delta_{\mathrm{logit}}\) |
Use \(k\) for the prepared exponent and \(g\) for physical generations. A tilde marks a numerical approximation. \(P\) and \(\widetilde P\) act on population observables; \(\widetilde P^T\) propagates population mass. \(A\) is the operator in the coordinates used for the power calculation: \(\widetilde P^T\) in population coordinates, or \(H_{\mathrm J}^T\) in reduced Jacobi coordinates.
Polynomial families retain the standard notation \(T_j,U_j\) for Chebyshev, \(P_j^{(\alpha,\beta)}\) for Jacobi, and \(C_j^{(\lambda)}\) for Gegenbauer; \(\phi_j\) denotes a normalized basis. These conventions follow NIST DLMF. A polynomial approximation may be called \(p_L\); it is not a population mass.
Derivatives and local notation¤
Derivative notation distinguishes perturbations of a calculation from sensitivities propagated backward from its output.
A dot denotes a directional derivative; a bar denotes the cotangent of the same named quantity. Subscripts identify coordinates or projection spaces; they do not change the underlying symbol's role.
| Concept | Symbol |
|---|---|
| Scalar objective | \(\ell\) |
| Population, working-state, and operator cotangents | \(\bar v_t,\bar z,\bar y,\bar A\) |
| Coupled tangent/state operator and vectors | \(A_{\mathrm{blk}},z_{\mathrm{blk}},y_{\mathrm{blk}}\) |
| Coupled Krylov basis and projected operator (at most \(2m\) columns) | \(V_{2m,\mathrm{blk}},H_{2m,\mathrm{blk}}\) |
| Backward/forward adjoint bases (at most \(m\) columns each) | \(V_{m,\mathrm L},V_{m,\mathrm R}\) |
| Their projected operators and seeds | \(H_{m,\mathrm L},H_{m,\mathrm R};\ \bar y_{m,\mathrm L},z_{m,\mathrm R}\) |
| Projected operator-cotangent core and lifted left factor | \(C,F\) |
| Rational Krylov shift and shifted operator | \(\tau,S=I+\tau(I-A)\) |
| Expected-visits incoming cotangent and adjoint solve variable | \(\bar u,\eta_u\) |
The autodiff chapter defines the labelled constructions. It uses \(\bar\theta_A\) for the contribution to the parameter cotangent through \(A\) alone; other parameter-dependent inputs contribute separately.
Write \(\mathcal L_\varphi(A,\Delta A)\) for the Fréchet derivative of a matrix function \(\varphi\), and \(\mathcal L_k\) for the power function \(\varphi(A)=A^k\). The matrix-function literature commonly uses \(E\) for the perturbation; we use \(\Delta A\) to avoid confusion with the Jacobi transform. See Kandolf et al..
Local summation indices and auxiliary functions may vary, but must be defined. In particular, sample allele counts may use \(k\) locally without referring to a prepared power. Research notes retain their derivation notation and include explicit crosswalks to these conventions.