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An optimal value over a time interval is the supremum of immediate reward plus the conditional optimal continuation value. For a controlled state $X$ and running reward $\ell$, this reads $J(x,t)={\sup }_a\mathbb{E}[{\int }_t^{t+h}\ell (X_s,a_s,s)ds+J(X_{t+h},t+h)\mid X_t=x]$, under the usual admissible concatenation and information conditions. Conditioning and concatenating strategies proves both inequalities: every strategy's continuation is bounded by the value function, while approximately optimal continuations approach that bound. For a smooth diffusion value function, Itô formula and the limit $h\downarrow 0$ yield the Hamilton-Jacobi-Bellman equation.

Optimal substructure means that the relevant parts of an optimal solution solve their own subproblems optimally. For a 0-1 knapsack problem, removing a chosen item leaves an optimal selection among the other allowed items at the reduced capacity. For sequential mathematical optimization, this becomes the dynamic programming principle: after fixing the next decision and resulting dynamic programming state, use an optimal continuation. Optimal substructure justifies the recurrence relation; overlap between subproblems explains why storing a value function can save work.

A dynamic programming state is an element of a state space that summarizes all information needed for a remaining mathematical optimization problem. Histories can share a value function entry only when their allowed continuations and future rewards agree, or have the same conditional laws in a stochastic control problem. Time or the remaining item index may be part of the summary. If past decisions still affect feasibility or rewards, enlarge the dynamic programming state; forgetting those effects invalidates the Bellman equation.

For a specified characteristic, complete dominance means that the heterozygote has the same phenotype as one of the two homozygotes. It does not mean that the dominant allele is more common, more advantageous, or transmitted more often. Penetrance specifies the conditional probability of an expressed phenotype given a genotype and is a separate modeling assumption.

A DNA replication inhibitor interferes with DNA replication; the observed effect depends on exposure and assay conditions.

Different compounds act at different steps: hydroxyurea limits precursor production, while aphidicolin inhibits selected replicative polymerases. Similar reductions in DNA synthesis can therefore have different molecular causes. Inhibition can also generate replication stress, so the resulting state differs from an unperturbed cell simply paused in time. ^[ref]

A replicating DNA polymerase is an enzyme that extends a paired strand end with nucleotides selected against a DNA template. New DNA grows $5^{\prime }\to 3^{\prime }$ while the template is read $3^{\prime }\to 5^{\prime }$. Base pairing specifies the partner; the DNA polymerase catalyzes its incorporation. See Alberts et al.: DNA polymerase.

In semiconservative replication, each daughter duplex retains one strand of its immediate parent. Under complete copying without strand loss or exchange, one starting duplex gives $2^n$ duplexes after $n$ rounds, but still only two original strands. For $n\ge 1$, the fraction of duplexes containing an original strand is $2/2^n=2^{1-n}$: all after one round, half after two, a quarter after three. The fraction of all strands that are original is instead $2^{-n}$. A strand made in round 1 is parental when round 2 begins.

A ratio compares two quantities $a,b$, written $a:b$ and corresponding to division $a/b$ when $b\ne 0$. Equal ratios of corresponding lengths characterize the scaling of similar triangles.

For example, $6:4$ and $3:2$ specify the same ratio because $6/4=3/2$. Multiplying both entries by the same nonzero scale preserves the quotient: $(ca)/(cb)=a/b$. This is useful when comparing mixtures or scaling a recipe; doubling both ingredient amounts preserves their ratio, while adding the same amount to both generally does not. For quantities with the same physical units, their ratio is dimensionless, connecting this elementary calculation with dimensional analysis.

The quotient of $a$ by a nonzero $b$ in a field is $a/b=a{b}^{-1}$, the unique solution $q$ of $bq=a$. Thus $7/2$ is the rational quotient of $7$ by $2$.

In Euclidean division of integers, the word instead denotes the integer $q$ in $a=bq+r$ with a specified remainder range; for $b>0$, $0\le r<b$. Here $7=2\cdot 3+1$ has quotient $3$ and remainder $1$. Stating the ambient arithmetic prevents confusion between exact division and division with remainder.

An integer multiple of an integer $d$ is a product $dk$ with $k$ an integer; equivalently, $d$ is a divisor of that product. The integer multiples of $12$ are $\dots ,-24,-12,0,12,24,\dots$. Zero is an integer multiple of every integer, and the only integer multiple of zero is zero. Shared positive integer multiples lead to the least common multiple.

A differential operator $P$ is hypoelliptic when $Pu$ smooth on an open set implies that $u$ is smooth there. Every elliptic differential operator is hypoelliptic, but the heat operator is hypoelliptic without being elliptic.

A distribution on an open set $U$ is a continuous linear functional on the space of test functions $\mathcal{D}(U)=C_c^{\infty }(U)$.

A first-order irreversible transfer from fluid concentration $C$ to stored concentration $S$, measured per matching fluid storage volume, obeys the displayed balances. A sustained inlet can produce a stationary decaying fluid profile, but the stored amount then grows rather than becoming stationary. A finite pulse instead has $C\to 0$ and a finite final deposit. Decay, desorption or finite adsorption capacity is additional physics, not a consequence of the linear capture term alone.

Time-reversal symmetry reverses the time coordinate and is represented antiunitarily in quantum theory.

For a map $F:X\to X$, its iterates are defined by $F^0$ equal to the identity map and $F^{n+1}=F\circ F^n$.

The displayed map has a symmetric periodic orbit $\{\pm \sqrt{\mu +1}\}$ for $\mu >-1$, with multiplier $(-2\mu -3{)}^2>1$. For $\mu >2$, put $s_{\pm }=(\mu \pm \sqrt{{\mu }^2-4})/2$. The two additional period-two orbits are $\{\sqrt{s_{+}},\sqrt{s_{-}}\}$ and their negatives. Their multiplier is $9-2{\mu }^2$, so they are stable for $2<\mu <\sqrt{5}$ and undergo a period-doubling bifurcation at $\sqrt{5}$. At $\mu =2$ these points coincide with fixed points, not genuine two-cycles.

Away from zero this discontinuous map is odd, and its branch derivatives are $2|x|$. With the convention $\mathrm{sgn}(0)=1$, its value at zero is $-\mu$, so it is not odd at zero unless $\mu =0$. A global transition at $\mu =0$ turns two small stable fixed points into one stable symmetric two-cycle. Its unsigned dynamics can be related to the even quadratic map $g_{\mu }(x)=\mu -x^2$, provided critical hits are treated separately.

// Target: dynamical-systems.bigb

If $g$ is even and an orbit does not hit zero before its final iterate, induction gives $f^n(x)=(-1{)}^n{g}^n(x){\prod }_{k=0}^{n-1}\mathrm{sgn}(g^k(x))$. The step uses oddness of $f$ on nonzero arguments. A one-sided assigned value of $f(0)$ can invalidate that step after a critical hit; for $g(x)=\mu -x^2$, $\mu >0$, $x=\sqrt{\mu }$, the claimed second-iterate formula would give $+\mu$ while the actual value is $-\mu$.

// Target: dynamical-systems.bigb

The Hénon map is a two-parameter quadratic planar map, conventionally $(X,Y)\mapsto (1-a{X}^2+Y,b_{H}X)$. The alternative family $(x,y)\mapsto (y,\mu -bx-y^2)$ has constant Jacobian determinant $b$. When $\mu \ne 0$, the change $X=y/\mu$, $Y=-bx/\mu$ gives the conventional parameters $a=\mu$, $b_H=-b$. Fixed points in the alternative normalization satisfy $x=y$ and $y^2+(1+b)y-\mu =0$. Its invertibility requires $b\ne 0$; at $b=1$ it preserves area.

For $F(x,y)=((a+1)x-x^3+y,(b-1)y-(b-1)x^3+y^3)$, the origin multipliers are $a+1$, $b-1$. A nonzero fixed point has $S=x^2>0$, $y=x(S-a)$, and $S[1-(S-a{)}^3]=a(2-b)$. Since the map is odd, a symmetric two-cycle has $F(x,y)=-(x,y)$; it satisfies $y=x(S-a-2)$ and $S[1+(S-a-2{)}^3]=b(a+2)$. Eliminating $y$ from the two map equations gives these identities directly. They locate the side of each pitchfork or flip without relying on a potentially misleading cubic sign in the original coordinates. At $(a,b)=(-1,0)$ the second identity becomes $3S^2-3S^3+S^4=b$, giving the quartic amplitude scaling of a generalized flip bifurcation.