Dense saline liquid in connected pore space can drain from sea ice while fresher seawater enters. The process is controlled by buoyancy, permeability of a porous medium and network connectivity, and is a major means of growing-ice desalination.

// Target: geophysics.bigb

Surface meltwater with a positive hydraulic head can percolate through permeable sea ice, displacing brine into the ocean. Flushing can efficiently desalinate warm summer ice, but requires meltwater and connected flow paths.

// Target: geophysics.bigb

Freezing within a brine inclusion expands the solid fraction and can force concentrated liquid into neighboring pores or channels. Expulsion redistributes salt internally; net desalination requires a path crossing the ice boundary.

// Target: geophysics.bigb

A temperature gradient can move a brine inclusion by melting its warmer wall and freezing its colder wall. This slow process can redistribute salt, but is generally much less effective at net desalination than gravity drainage of sea-ice brine or meltwater flushing of sea ice.

// Target: geophysics.bigb

For $i{\psi }_t=-{\psi }_{xx}/2-|\psi {|}^2\psi$, translation of a solution at velocity $u$ must be accompanied by the displayed phase. Direct differentiation cancels the terms proportional to ${\partial }_x{\psi }_0$, leaving the original equation. A standing profile of frequency $E=-{\kappa }^2/2$ thus gives $\kappa \mathrm{sech}(\kappa (x-ut-X_0))e^{iux-i(u^2/2+E)t+i{\theta }_0}$. Its centre travels uniformly, while its L2 norm is unchanged.

Let a real localized profile satisfy $Ef=-f^{\prime \prime }/2-f^3+y^{2}f/2$. Then $\psi (x,t)=e^{i\Theta (x,t)}f(x-X(t))$ solves the cubic focusing nonlinear Schrodinger equation in the potential $x^2/2$ with the displayed phase and any harmonic-oscillator trajectory $X$. Imaginary terms require ${\Theta }_x=\dot{X}$; real terms require $\ddot{X}=-X$ and the remaining time-dependent phase. The translation is exact and causes no profile deformation, unlike a generic collective-coordinate effective Lagrangian approximation.

At an algebraic branch point, a local solution becomes meromorphic after the finite substitution $z-a=t^k$. The displayed convergent expansion has only finitely many negative terms. A square-root branch point corresponds to $k=2$; an essential singularity or logarithmic singularity need not become meromorphic under any such finite substitution.

For minimization, best-bound search expands the active branch and bound node with smallest lower bound. A feasible completed assignment supplies an incumbent upper bound. Nodes with lower bound at least that incumbent can be discarded when only one optimum is needed. Search ends with an optimality certificate when no active node can improve the incumbent.

// Target: combinatorics.bigb

An incumbent solution is the best feasible solution found so far by an optimization search. Its objective value is an upper bound on a minimization optimum and a lower bound on a maximization optimum; bounds on unexplored subproblems can be compared with it to prune the search.

// Target: combinatorics.bigb

The bisection method finds a zero of a function by repeatedly halving a root bracket. Let $f$ be a real-valued continuous function on the closed interval $[a,b]$, with $a<b$. First check for endpoint zeros of a function; otherwise require strictly opposite signs at the endpoints. Evaluate the midpoint and retain the half with strictly opposite endpoint signs, stopping immediately if the midpoint is a zero of a function. The intermediate value theorem ensures that every retained root bracket still contains a zero of a function. This is the construction in MIT's root-finding notes.^[ref]

For a worked square root calculation, use the polynomial $f(x)=x^2-2$ on $[1,2]$. Write $[a_k,b_k]$ for the root bracket after $k$ halvings, with $[a_0,b_0]=[1,2]$. The candidate evaluated during update $k$ is the old midpoint $m_{k-1}$:The initial midpoint is $m_0=1.5$. After four halvings the new midpoint is $m_4=1.40625$, and the bisection error bound is $|m_4-\sqrt{2}|\le 0.03125$. Its actual error is approximately $0.00796356237310$. The last tested candidate $m_3=1.4375$ and the reported midpoint $m_4$ are different points.

The useful feature of the bisection method is that the retained width halves on every update, independently of the shape of the function graph. The bisection error bound therefore supplies an accuracy guarantee before the zero of a function is known. No derivative is needed, and a valid initial root bracket gives convergence without a close initial guess. This makes the bisection method a dependable fallback for the faster Newton root-finding iteration.

A root bracket is a closed interval known to contain at least one zero of a function. For a real-valued continuous function on $[a,b]$, a strict change of sign between $f(a)$ and $f(b)$ certifies such an enclosure by the intermediate value theorem. An endpoint with value zero already supplies a zero of a function. For example, $[1,2]$ is a root bracket for the polynomial $f(x)=x^2-2$, since $f(1)=-1$ and $f(2)=2$. A root bracket need not contain a unique zero of a function.

For a real function $f$ bounded by one on a finite Cartesian product, the fourth power of its box norm is the mean of $f(x,y)f(x,y^{\prime })f(x^{\prime },y)f(x^{\prime },y^{\prime })$. Two applications of the Cauchy-Schwarz inequality bound every rectangular discrepancy by the box norm. Conversely, decompose each bounded real test function into positive and negative parts and use the layer cake representation. Its bilinear pairing with $f$ is at most four times the cut norm. Applying this to the row and column tests furnished by fixed $x^{\prime },y^{\prime }$ bounds the fourth moment by four times the cut norm. These dimension-independent estimates make the two notions of small discrepancy equivalent.

Give every entry of an $m$ by $n$ matrix an independent uniformly random sign. A four-entry rectangle product has expectation zero unless its two row indices coincide or its two column indices coincide; in those cases it is one. Counting the union of these diagonal cases gives $m{n}^2+m^{2}n-mn$ before normalization. Thus some sign matrix has box norm fourth power at most the displayed expectation, and its cut norm tends to zero when both dimensions tend to infinity.

Consider base-$b$ expansions using $m$ allowed digits with positive gaps between the corresponding first-level intervals. At depth $k$ there are $m^k$ cylinders of diameter comparable to $b^{-k}$, and separated cylinders cannot all be covered by fewer than a constant multiple of $m^k$ balls of that scale. Thus $N_{b^{-k}}(E)\asymp m^k$. Monotonicity between successive scales gives equal upper and lower Minkowski dimensions, both $\log m/\log b$.

Uniformly stratified, nonrotating, two-dimensional ideal Boussinesq equations in coordinates $(x,z)$ are equivalent to unstratified rotating equations independent of $Y$, with $(X,Y,Z)=(z,y,-x)$, rotation $2\mathbf{\Omega }=N{\mathbf{e}}_Z$, velocity $(U,V,W)=(w,\sigma /N,-u)$, and the same reduced pressure. The transverse rotating velocity represents buoyancy perturbation, while the rotation-axis velocity represents the stratified horizontal velocity. Matching prescribed forces may have components in the invariant plane, but no transverse component. Physical free surface conditions are not preserved: the physical vertical direction and its normal velocity are different under the mapping.

In the constant-property convection convention, the strong Boussinesq approximation supplements small relative mass density changes with constant dynamic viscosity, thermal diffusivity, specific heat capacity and thermal expansion coefficient throughout the modeled temperature interval. A uniform reference density replaces density in inertia and mass conservation, while a linear density anomaly is retained in gravitational buoyancy. The temperature obeys an advection-diffusion equation with constant coefficients. The term “strong” is not uniform across all fluid-mechanics literature: some stratified-flow treatments instead emphasize the use of a globally constant reference density. The constant-property convection usage is explained in courses.physics.ucsd.edu/2021/Spring/physics218c/Spiegel%20Convection.pdf .

The momentum thickness weights the streamwise velocity deficit by the local velocity carrying it:Thus $\rho U^2\theta$ measures a momentum flux deficit per unit span relative to transporting the same local mass flow rate at $U$. It is not $\rho \int (U^2-u^2)dy$, which compares a different reference flux. A Blasius boundary layer has $\theta \simeq 0.664115\sqrt{\nu x/U}$; the ratio of displacement thickness to momentum thickness is about $2.591$.^[ref]

For the steady flow over an impermeable flat plate at constant $U$, integrating the boundary-layer equation and using mass conservation givesThis connects growth of the momentum thickness to local wall shear stress. Consequently the drag per unit span on one face up to $x=L$ is $D^{\prime }=\rho U^2\theta (L)$ in the ideal Blasius boundary layer with $\theta (0)=0$. Measured velocity profiles can therefore estimate integrated drag without differentiating noisy near-wall readings. The identity assumes no wall suction and zero outer pressure gradient.^[ref]

For constant-mass density incompressible flow, the displacement thickness measures the lost volume flux per unit span divided by the outer velocity:An outer inviscid flow encountering the same volume flux deficit behaves, at leading order, as though the wall were displaced outward by ${\delta }^{\ast }$. In a Blasius boundary layer, numerical integration gives ${\delta }^{\ast }\simeq 1.720788\sqrt{\nu x/U}$. This equivalent displacement is smaller than the boundary layer thickness and is not the position of a physical surface.^[ref]

A cover of $S^d$ by $d+1$ closed sets has a member containing an antipodal pair. The equivalent version for open sets follows by shrinking finite open covers on a compact metric space; the reverse implication follows by enlarging antipodal-pair-free closed sets slightly. This is a covering formulation of the Borsuk-Ulam theorem.

A second indirect route makes the experimental distinction especially clear: a calorimeter can measure dissolution of a weighed salt into a known amount of water, but its final ions are hydrated, not separated gaseous ions. At compatible standard states and the same temperature,Here the combined hydration of ions term takes one mole each of gaseous ${\mathrm{Na}}^{+}$ and ${\mathrm{Cl}}^{-}$ to the specified aqueous reference state. Its sum suffices: dissolution calorimetry alone cannot independently assign single-ion hydration of ions enthalpies, or determine $L_{\mathrm{diss}}$ without another input. A finite-concentration measurement also requires dilution corrections before it can supply $\Delta H_{\mathrm{sol}}^{\circ }$.

In a simple constant-pressure solution calorimeter, an insulated vessel, stirrer and thermometer follow the temperature change after mixing. A calibrated effective heat capacity $C_{\mathrm{eff}}$ converts the corrected temperature change to heat. With no applied heating, negligible external heat transfer, and only pressure-volume work, the approximate molar result isHere $n$ is the amount of substance dissolved and $C_{\mathrm{eff}}$ includes the chemical solution and the vessel. Cooling means positive dissolution enthalpy, because the dissolving sample takes heat from its surroundings.^[ref]

An actual reference measurement used potassium chloride, not sodium chloride: a platinum-lined calorimeter with an adiabatic shield and a quartz-oscillator thermometer. Because dissolution absorbed heat, measured electrical heating prevented loss of shield control; the readouts were electrical input and the vessel/shield temperature record. Stirring was checked visually in a dummy vessel. The thermometer was compared with a calibrated resistance thermometer, and electrical calibration bracketed the reaction. The reported result was $+(17.584\pm 0.017)\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ at $298.15\mathrm{K}$, for one mole of potassium chloride in $500$ moles of water.^[ref]

Checks on residual water, particle size and final concentration, and agreement with a second calorimeter, constrained systematic errors. Imperfect shield tracking required heat transfer corrections. This experiment measures an enthalpy of solution, not a direct lattice energy; the hydration and dilution bookkeeping still has to be supplied. The full article could not be opened here; the inspected indexed text supports the stated apparatus and checks, without reconstructing its complete uncertainty budget.^[ref]