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For two individual microstates, the partition function cancels:At positive temperature, a higher-energy individual microstate is less probable, and equal-energy individual microstates are equally probable. For example, a gap $E_i-E_j=k_{B}T\ln 2$, using the natural logarithm, gives $p_i/p_j=1/2$. This comparison concerns individual microstates; a whole macrostate can contain several of them.

Changing the reference for energy leaves every probability unchanged. If all energies become $E_i^{\prime }=E_i+C$ for the same constant $C$, thenThe partition function itself changes, but its common factor cancels in the Boltzmann distribution. Thus choosing a ground state energy of zero simplifies calculations without changing populations. The expected value of energy shifts by $C$, as it must when its reference changes.

For a cyclic group $G$, a rank-$d$ phase-radius-$\rho$ Bohr set contains a symmetric proper generalized arithmetic progression of rank at most $d+1$ and cardinality at least $c(d,\rho )|G|$. The constant can be taken to depend only on $d$ and $\rho$. This geometry-of-numbers consequence follows by applying successive minima to the lattice recording the simultaneous character congruences; any cyclic subgroup factor can be represented by one additional progression coordinate. It is stronger than finding just one long ordinary progression.

Choose a Hilbert-Schmidt inner product orthonormal Hermitian operator basis with final element $I/\sqrt{N}$. A density operator has real coefficients $r_k=\mathrm{Tr}({\sigma }_k\rho )$ and fixed final coefficient $1/\sqrt{N}$. The remaining coefficients form its generalized Bloch vector. Positivity restricts these real vectors to a compact convex body; for $N>2$ this body is not the entire Euclidean ball allowed by a purity bound.

The qubit density operators $\rho =(I+\mathbf{s}\cdot \mathbf{\sigma })/2$ correspond exactly to the closed unit ball: their eigenvalues are $(1\pm |\mathbf{s}|)/2$, so positivity is equivalent to $|\mathbf{s}|\le 1$. The boundary is the Bloch sphere of pure states, and the centre is $I/2$. A depolarizing channel scales this ball isotropically, possibly with a sign reversal in the larger completely positive parameter interval.

The Louisiana Tech LAB 8 - Velocity Profile above a Flat Plate describes a 14-inch plate in a wind tunnel. A Pitot-static tube of 0.10-inch outside diameter traverses normally about 12 inches downstream; its screw scale has 0.001-inch increments. A water-column manometer reads the difference between stagnation pressure and static pressure. The procedure establishes a wall position, zeros the tunnel indicator, sets fan speed, and records at least ten positions inside the boundary layer, together with room temperature and barometric pressure. The instrument list includes a flow-stream thermometer implemented with a temperature-sensitive resistor. The manual requests comparison of outer velocity with the tunnel indicator and of the recovered velocity profile with laminar or turbulent predictions; it provides no observed dataset. These are apparatus specifications and instructions, not a reported confirmation of laminar flow.

Under incompressible flow conditions, an ideal aligned Pitot tube gives $q(y)=p_0(y)-p(y)=\rho u(y{)}^2/2$. Convert the manometer head $h$ to $q=({\rho }_{\mathrm{water}}-\rho )gh$, then infer $u=\sqrt{2q/\rho }$. Here $g$ is gravitational acceleration; mass density $\rho$ can be estimated from temperature and absolute pressure using the ideal gas law. A normalized velocity profile rises from the unresolved near-wall region to a plateau, whose value defines $U$; downstream profiles should broaden and collapse when plotted against $\eta$ if the Blasius boundary layer assumptions hold. Integrating a recovered velocity profile gives displacement thickness and momentum thickness.

Additional controls for interpreting this experiment follow from the measurement model: check differential-pressure calibration with equal port pressures and known liquid heads; leak-check tubing; align the Pitot tube with the incoming velocity; convert wall contact to the inlet-centre coordinate rather than treating outside contact as $y=0$; repeat traverses and outer velocity readings to detect drift. Keep the plate smooth and aligned, and check outer static pressure at multiple downstream stations before assuming zero pressure gradient. Repeating at another station or outer velocity tests the predicted boundary-layer scaling more directly than fitting one curve. These controls are proposed checks, not actions documented as completed in the manual.

Finite probe size, wall interference, flow fluctuations, and weak near-wall dynamic pressure limit inference. Even the manual's probe-based Reynolds number criterion, ${\mathrm{Re}}_d=ud/\nu >1000$ for its ideal Bernoulli equation conversion, must be checked with local $u$, not just $U$. In the illustrative model at $x=0.3048\mathrm{m}$, ${\delta }_{99}\simeq 3.32\mathrm{mm}$; the quoted outside diameter is $2.54\mathrm{mm}$, a substantial fraction of that thickness. A real inlet's sensing footprint and correction cannot be deduced from outside diameter alone. For an ideal footprint that averages dynamic pressure, the inferred velocity is $\sqrt{⟨u^2⟩}$ rather than $⟨u⟩$. Differentiating such sparse readings to obtain wall shear stress magnifies measurement error; the momentum thickness route is less sensitive to this differentiation but still needs the unresolved near-wall contribution. The no-slip boundary condition is a model constraint here, not a direct reading from a probe centred on the wall.

The vega of an European option is the derivative of its price with respect to spot volatility, holding its other parameters fixed. A dividend-free European call option under the Black-Scholes formula has $\nu =S_0\sqrt{T}\varphi (d_1)>0$. Differentiation proves this because $S_0\varphi (d_1)=K{e}^{-rT}\varphi (d_2)$ and ${\partial }_{\sigma }(d_1-d_2)=\sqrt{T}$. Positivity proves uniqueness of Black-Scholes implied volatility for prices strictly between the limiting call-price bounds.

The claim $\max (K,S_T)$ for $K>0$ combines a guaranteed cash floor with a European call option, equivalently a stock with a put. In the Black-Scholes model, for $\tau =T-t>0$, its price is $V=S_t\Phi (d_1)+K{e}^{-r\tau }\Phi (-d_2)$, where $d_1=[\log (S_t/K)+(r+{\sigma }^2/2)\tau ]/(\sigma \sqrt{\tau })$ and $d_2=d_1-\sigma \sqrt{\tau }$. Under the risk-neutral lognormal law, the truncated stock moment is $S_t{e}^{r\tau }\Phi (d_1)$ and the probability of ending below the floor is $\Phi (-d_2)$, proving the price. Its option delta is $\Phi (d_1)$, since $S_t\varphi (d_1)=K{e}^{-r\tau }\varphi (d_2)$ cancels the derivative terms. The replicating strategy holds that many stock units and $K{e}^{-r\tau }\Phi (-d_2)/B_t$ bank-account units.

Under the Black-Scholes model with interest rate $\rho$, volatility $\sigma$ and remaining time $\Delta$, risk-neutral pricing gives $V(S)=S[\log S+(\rho +{\sigma }^2/2)\Delta ]$. This follows by differentiating the lognormal exponential moment. Its option delta is $\log S+1+(\rho +{\sigma }^2/2)\Delta$ and its second stock derivative is $1/S$.

Average constant-rate Black-Scholes model prices of a fixed terminal payoff over $\rho \sim N({\rho }_0,{\tau }^2)$. Tilting this Gaussian by the discount factor shifts its mean by $-{\tau }^{2}T$; combining the resulting log-price variance with stock variance gives the displayed effective parameters. Thus the average equals the Black-Scholes price with those parameters, whenever the expectation is well defined. This is an average of constant-parameter prices, rather than a specification of an adapted stochastic short-rate market.

For a fixed twice-differentiable terminal payoff with sufficient growth control, differentiate its lognormal distribution pricing integral. Gaussian integration by parts gives the displayed identities. Convexity implies $p_{xx}\ge 0$, so increasing spot volatility cannot lower the price. The interest rate sensitivity depends on the sign of the cash position $p-x{p}_x$. Calendar-time sensitivity is $p_t=\rho (p-x{p}_x)-\frac{1}{2}{\sigma }^2{x}^2{p}_{xx}$. Thus a convex payoff with a negative cash position has a nonincreasing price at fixed spot when $\rho \ge 0$. Negative rates invalidate that conclusion: $f(x)=x-K$ gives $p=x-K{e}^{-\rho (T-t)}$ and $p_t=-\rho K{e}^{-\rho (T-t)}>0$ when $\rho <0$.

For a regular analytic stationary asymptotically flat vacuum black hole with a connected nondegenerate horizon, the stationary Killing vector field either generates the horizon or the spacetime admits an additional commuting axial Killing vector field. In the rotating case the horizon generator is $K+{\Omega }_H\Phi$, where $\Phi$ has closed rotational orbits. Together with the appropriate staticity and black-hole uniqueness theorem, this identifies an equilibrium uncharged collapse remnant with the Kerr metric. The theorem concerns an equilibrium geometry; it does not itself prove that arbitrary dynamical collapse settles to equilibrium.

The trispectrum is the Fourier transform of a fourth-order cumulant, a fourth-order counterpart of the bispectrum. It describes connected four-point correlations; the primordial trispectrum applies it to cosmological random fields.

For distinct primes $p,q\equiv 1(\mathrm{mod}4)$, put $E=\mathbb{Q}(\sqrt{p},\sqrt{q})$ and $K=\mathbb{Q}(\sqrt{pq})$. The algebraic integers $a=(1+\sqrt{p})/2$ and $b=(1+\sqrt{q})/2$ generate an order whose basis has discriminant of elements of a number field $p^2{q}^2$: the trace pairing matrix is, up to basis order, the tensor product of the quadratic trace matrices of determinants $p$ and $q$. If its index in ${\mathcal{O}}_E$ is $j$, the discriminant-index formula for an integral lattice and the relative discriminant tower formula giveBoth $j$ and this ideal norm are positive integers, forcing $j=1$ and ${\mathfrak{d}}_{E/K}={\mathcal{O}}_K$. Thus $E/K$ is unramified at all finite primes. It is also totally real, so no real place ramifies. This quadratic abelian extension is contained in the ordinary Hilbert class field of $K$, and Artin reciprocity gives a quotient of its ideal class group of order two. The extensions $K(\sqrt{p})$ and $K(\sqrt{q})$ are equal to $E$, not two different extensions of $K$.

A crown graph is a complete bipartite graph with a perfect matching removed. For $n\ge 2$ it has chromatic number two, but greedy coloring in the order $a_1,b_1,a_2,b_2,\dots ,a_n,b_n$ uses $n$ colors: the matched pair gets the same new color, and every subsequent pair sees all earlier colors.

For upward swimming with $h=V_{s}H/D\to 0$, no-slip impermeable plates, and stationary long-wave modes with horizontal wavenumber $\kappa =h{\kappa }^{\prime }\ne 0$, the leading concentration mode is constant and the vertical velocity mode is $W_0=-{\kappa }^{\prime 2}{R}_0{z}^2(z+1{)}^2/24$. The next concentration solvability condition is ${\kappa }^{\prime 2}+{\int }_{-1}^0{W}_{0}dz=0$. Since $\int z^2(z+1{)}^{2}dz=1/30$, this gives $R_0=720$, independently of the leading gyrotactic parameter. The exactly uniform horizontal mode ${\kappa }^{\prime }=0$ instead changes conserved total cell number and does not select the instability threshold.

These two finite-sum transformations are inverses over any commutative ring. Substitute one sum into the other and use $\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{k}{j}\right)=\left(\genfrac{}{}{0ex}{}{n}{j}\right)\left(\genfrac{}{}{0ex}{}{n-j}{k-j}\right)$. The inner alternating sum is $(1-1{)}^{n-j}$ by the binomial theorem, so only $j=n$ survives. In a Mahler expansion, $b_n$ is the nth finite difference of the sampled values $a_k$.

Let $X$ count unordered vertex pairs with no common neighbour in a binomial random graph. Each other vertex provides the required pair of edges independently, with success probability $p^2$. A fixed pair has no common neighbour with probability $(1-p^2{)}^{n-2}$. Summing its indicators gives the displayed expectation.

A finite graph $H$ with at least one edge is strictly balanced when every proper nonempty subgraph $F$ has smaller edge-to-vertex ratio. It is connected and has no isolated vertices: otherwise a component of at least the average density, or deletion of isolated vertices, violates the inequality. The density condition suppresses overlapping copies at the sparse appearance scale.

Construct an independent set by choosing a candidate vertex, exposing only its incident edges, and retaining its non-neighbours as candidates. Once the chosen independent set is removed, every exposed edge has a removed endpoint. Thus the remaining internal edges still have their independent original laws, conditional on the revealed history. This special exposure rule permits fresh-binomial estimates for successive classes; an arbitrary adaptively selected vertex subset does not have that property.

For a fixed set of $r$ vertices, exclude all triangles meeting that set. Their total expectation is $r\log (n/c)+o(1)$. Distinct triangle pairs sharing an edge contribute $O_r(n^3{p}^5)=o(1)$ to the Janson dependency sum. Harris' inequality and Janson inequality bound the avoidance probability between matching exponentials, giving $(c/n{)}^r(1+o(1))$. The factorial-moment criterion for Poisson convergence then gives the displayed law.