Eliminate the juvenile variable from the stage-structured population model. Since $y_{t+1}=2z_t$,The characteristic polynomial of this linear recurrence isso $z_t=A(3/2{)}^t+B(-1{)}^t$. The initial conditions give $z_0=0$ and $z_1=(3/4)10=15/2$, hence $A+B=0$ and $(3/2)A-B=15/2$. ThereforeThe alternating term records the initial imbalance between life stages; the growing $(3/2{)}^t$ term dominates eventually.

This is a stage-structured population model with a census once per maturation interval. Only adults present at census $t$ contribute to the juvenile cohort at census $t+1$; each contributes $r$ new juveniles on average. Thus $r$ is an effective fecundity, including any survival of offspring needed to reach the next census. Juveniles present at $t$ do not contribute a separate birth term during this interval.

The term $c{y}_t$ consists of juveniles that survive and mature into adults by the next census, so $c$ is their survival-to-maturity fraction. The term $(1-d)z_t$ consists of surviving adults; $d$ is the adult mortality fraction per interval. All surviving juveniles leave the juvenile compartment in one step, adults remain adults, and there is no immigration or density dependence. The reproduction term is applied to adults counted at the start of the interval, rather than only to those surviving to its end.

Differentiate the squared-denominator population map:At the positive fixed point, $b+\bar{x}=\sqrt{r}$, so its fixed-point multiplier isSince $0<b/\sqrt{r}<1$, we have $-1<F^{\prime }(\bar{x})<1$. A small perturbation ${\epsilon }_t$ obeys ${\epsilon }_{t+1}=F^{\prime }(\bar{x}){\epsilon }_t+O({\epsilon }_t^2)$, and hence shrinks locally. Thus the positive fixed point is locally asymptotically stable by fixed point stability for an iteration.

For the squared-denominator population map, a fixed point satisfiesThere is always the extinction fixed point $\bar{x}=0$. For a positive fixed point, divide by $\bar{x}$ and use $b+\bar{x}>0$ to obtainAt equality the candidate merges with zero and is not positive.

Both candidate models fit one parameter, but their parameter symbols have different meanings. For the exponential distribution in model B,Its second derivative is $-N/{\theta }_B^2<0$, so this is its maximum-likelihood estimate. The maximized log-likelihoods, including their normalization terms, areUse the corrected Akaike information criterion because $N=4$ is small relative to the number $k=1$ of fitted parameters:The correction is $2$ for each model, givingPrefer model A, whose corrected Akaike information criterion is smaller by $7.67252$. The ordinary Akaike information criterion also gives that ranking because the two models receive equal penalties. This compares the two proposed models; it does not establish that either is a correct description of the process. The usual corrected Akaike information criterion formula is a small-sample model-selection convention, rather than an exact finite-sample guarantee for these density families.

Keep the true parameter fixed and imagine repeatedly collecting four independent and identically distributed observations under the same model. Compute the maximum-likelihood estimate, its estimated standard error, and the Wald confidence interval afresh each time. The intended coverage statement is that approximately 95% of those random confidence intervals contain the fixed true parameter, provided the normal approximation is adequate.

After these particular observations have been collected, both endpoints are fixed. The frequentist statement does not assign 95% probability to the fixed parameter lying between these particular endpoints. Nor does the confidence interval describe the range containing 95% of future observations. The small sample makes the nominal 95% coverage an approximation whose accuracy is not guaranteed.

Here $s=\log (0.4\times 0.7\times 0.8\times 0.9)=-1.601469743$. The maximum-likelihood estimate and estimated standard error from the observed Fisher information areThe 95% Wald confidence interval is thereforeThe whole reported confidence interval lies in $\theta >-1$, though its lower endpoint is close to the boundary. With only four observations, the normal approximation underlying this Wald confidence interval need not provide accurate 95% coverage.

The sample mean is $\bar{x}=0.7$. The method of moments equates this to the expected value from part (b):Hence the estimate is $\boxed{\tilde{\theta }=4/3\approx 1.33}$.

Write $s={\sum }_i\log x_i<0$. The score function isSetting it to zero gives the maximum-likelihood estimatorThis lies above $-1$. The observed Fisher information isConsequently the log-likelihood is strictly concave throughout its domain. Its stationary point is the unique global maximum, not merely a candidate local maximum. Indeed the log-likelihood tends to minus infinity as either $\theta \downarrow -1$ or $\theta \to \infty$.

For the independent and identically distributed observations, multiply the probability density functions. On the admissible parameter range $\theta >-1$, the likelihood function and log-likelihood areThese expressions presume that every observation lies strictly between zero and one; an observation outside that support would give zero likelihood under this model.

For a moment of order $q$ with $\theta +q>-1$, integration of the probability density function givesThus the expected value and variance areThe cumulative distribution function on $(0,1)$ is $F(x)=x^{\theta +1}$. Its unique median $m$ satisfies $F(m)=1/2$, giving

A probability density function must have integral one. For $\theta >-1$,The restriction $\theta >-1$ is essential: for $\theta \le -1$ the integral diverges and no positive normalizing constant exists.

The coordination number rises from six in the rock salt structure to eight in the CsCl structure. In the elementary hard-sphere ionic bonding model, accommodating eight opposite neighbours favours a larger cation/anion radius ratio: the geometrical limiting ratios are $\sqrt{2}-1$ for sixfold and $\sqrt{3}-1$ for eightfold coordination. The high-pressure change is therefore consistent with $r_K/r_{Cl}$ increasing under compression. Sincethe chloride ion is inferred to be more compressible than the potassium ion in this radius-ratio interpretation. This is a qualitative structural argument; real ionic radii depend on coordination and electronic structure, and the transition does not uniquely determine separate microscopic compressibilities.

Use only the high-pressure CsCl structure branch to extrapolate its volume per formula unit $V=a^3$. A straight-line least-squares fit to those three measurements gives approximatelywith $P$ in GPa. Thus $V(10)\simeq 39.29{\mathring{\mathrm{A}}}^3$, $a(10)\simeq 3.400\mathring{\mathrm{A}}$ and $d_{111}=a/\sqrt{3}\simeq 1.963\mathring{\mathrm{A}}$. Taking Cu K$\alpha$ wavelength $\lambda \simeq 1.541\mathring{\mathrm{A}}$ in Bragg's law givesThe estimate is model-dependent: a straight-line extrapolation of $a(P)$ instead gives $2\theta \simeq 46.0^{\circ }$. The data stop at $3.81\mathrm{GPa}$, so the absence of further phase transitions does not by itself determine an exact equation of state to $10\mathrm{GPa}$. The $111$ peak can also be weak because its basis factor is $f_K-f_{Cl}$.

Within each cubic phase, increasing pressure decreases $a$, so $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$ decreases and Bragg's law moves a given reflection to higher $2\theta$. Near $2.55\mathrm{GPa}$, peaks of the rock salt structure disappear as those of the CsCl structure appear, potentially with a coexistence interval. The former has an FCC translation selection rule, with $h,k,l$ all even or all odd; mixed-parity reflections vanish kinematically. The latter has primitive-cubic translations and basis factor $f_K+(-1{)}^{h+k+l}{f}_{Cl}$, so it has no FCC extinction rule. Odd-sum reflections may nevertheless be weak because the two ionic scattering factors are similar. Peak spacings, relative intensities and the cell volume per formula unit change discontinuously. A simple uniform shift of the entire original pattern cannot describe the structural transition.

Use mass density $\rho =ZM/(N_A{a}^3)$, with $M=74.55\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. The rock salt structure has $Z=4$ formula units per conventional cell; the CsCl structure has $Z=1$. In pressure order the densities are approximatelyThe increase across the transition is approximately $14.5\%$, much greater than the smooth compression within either branch. Comparing $a^3$ directly without correcting the different $Z$ would create a fictitious volume discontinuity.

A diamond anvil cell concentrates force onto tiny opposing culets, reaching very high pressure with small samples and useful optical access through diamond; its small volume, pressure gradients, alignment demands and difficulties with some heating or sample environments can limit experiments. A multi-anvil press supports a much larger sample and is useful for high-temperature synthesis and bulk measurements, but usually reaches lower ultimate pressures and has more restricted optical access. Hydrostaticity, attainable temperature and calibration depend on the actual apparatus and pressure medium; neither design guarantees uniform pressure.

Check stiffness and tip deflection using Young's modulus, failure from existing cracks using fracture toughness or the supplied critical energy-release rate, local crushing at supports, shear failure, buckling, fatigue and joint design. In a simple linear-elastic crack model, $K_c^2\simeq E^{\prime }{G}_c$, so toughness cannot be inferred from strength alone. Wood also has grain-direction dependence, moisture sensitivity, natural defects, property scatter and time-dependent creep; the design must control grain alignment and compression-side stability. Metals raise issues of corrosion, mass, joining, machining, residual stress and heat-treatment state. Cost, availability, reproducibility, service temperature and end-of-life treatment can decide between otherwise feasible designs. A high merit index is a screening result, not a complete safety specification.