Hearing loss arising from inner-ear sensory structures or auditory neural pathways. It differs from failure to conduct sound through the outer or middle ear, although both mechanisms can coexist.

The cochlear nerve fibres carrying auditory information from hair cells toward the brain.

Audiometry measures aspects of hearing under specified stimulus and response conditions. A tone-detection threshold depends on frequency, presentation route and the task. It is a psychophysical measurement rather than a complete description of every auditory ability.

Traumatic brain injury is disruption of brain structure or function caused by an external mechanical force. The damage can be focal, distributed or a combination, and its effects need not be captured by the appearance of the external head injury.

Concussion is one clinical form. Distinguishing immediate symptoms from later structural or functional findings helps avoid treating all traumatic injuries as the same mechanism or assuming that loss of consciousness is required.^[ref]

The Haskell type class providing the overloaded operations == and /=. A signature Eq a => [a] -> [a] -> Bool requires the corresponding operations for the element type. Correctness arguments treating them as mathematical equality additionally assume that the instance supplies the intended equality relation.

Choose the order of pending input-output requests. Physical head movement, rotational latency, request arrival times and fairness can favour different orders. A seek distance calculation is a simplified model; real seek time is not necessarily linear in track displacement.

The time required to move and settle the disk head over the desired track. It is distinct from seek distance and from waiting for a sector's rotation.

The wait for the requested sector to reach the disk head after positioning on its track. For a uniformly distributed arrival phase and rotation period $T_{\mathrm{rot}}$, its expected value is half a rotation. At $r$ revolutions per minute, this is $30/r$ seconds.

Bleeding within the cranial cavity. The involved compartment—such as brain tissue or surrounding spaces—affects pressure, tissue injury and the interpretation of imaging.

Bleeding into lung tissue or air spaces. It differs from a blood collection in the pleural cavity and can impair gas exchange as well as cause visible haemoptysis.

Small, pinpoint areas of hemorrhage in skin or mucosal tissue. They are a pattern of bleeding with several possible causes, not a specific proof of one disease or mechanism of death.

For a subgroup $H\le G$ and a normal subgroup $N⊴G$, the map $h\mapsto hN$ from $H$ onto $HN/N$ is a group homomorphism with kernel $H\cap N$. The first isomorphism theorem gives the displayed group isomorphism. The product $HN$ is a subgroup because $N$ is normal. In particular, if $HN=G$ and $H\cap N=1$, then $H\cong G/N$.

For the quadrant $x,y>0$, the displayed Green function satisfies $\Delta G={\delta }_{(\xi ,\eta )}$, zero value on $y=0$ and zero outward normal derivative on $x=0$. The method of images uses the logarithmic fundamental solution of the Laplace equation with coefficient $1/(2\pi )$: keep the source sign when reflecting across the Neumann side and reverse it when reflecting across the Dirichlet side. The four image signs are therefore $+,+,-,-$ for sources $(\xi ,\eta ),(-\xi ,\eta ),(\xi ,-\eta ),(-\xi ,-\eta )$.

On $y=0$, equal numerator and denominator give zero. Evenness in $x$ gives the Neumann condition at $x=0$. The outward normal derivative on the Dirichlet side is the positive kernelGreen's second identity consequently gives $u(P)={\int }_{D}Gf+{\int }_{y=0}g{\partial }_{n}G$ for $\Delta u=f$ with these mixed boundary data, assuming any remote-boundary contribution vanishes. The sign convention is $\Delta G=+\delta$; changing to $-\Delta G=\delta$ changes both the Green function and the representation signs.

Suppose a linear ordinary differential operator satisfies $L(qv)=rMv$ on an interval, where the functions $q,r$ never vanish and homogeneous endpoint boundary conditions for $y=qv$ correspond to those used for $v$. If $G_M$ is a Green's function for $M$, then $Ly=f$ is equivalent to $Mv=f/r$, soThus the displayed kernel is the Green's function for the original, possibly non-normalized operator. The source factor belongs at $s$, while the dependent-variable factor belongs at $x$.

For $Ly=x^2{y}^{\prime \prime }-2x{y}^{\prime }+(x^2+2)y$ on $[\pi /2,\pi ]$ with zero endpoint values, $q=x$ and $r=x^3$ reduce the equation to $Mv=v^{\prime \prime }+v$. The kernel of $M$ is $\cos (x_{<})\sin (x_{>})$, where $x_{<}=\min (x,s)$ and $x_{>}=\max (x,s)$. Therefore $G_L(x,s)=x\cos (x_{<})\sin (x_{>})/s^3$. Its first derivative has jump $1/s^2$, as required by the leading coefficient $s^2$ of $L$.

For $L=a_2(x)D^2+a_1(x)D+a_0(x)$ with $a_2(\xi )\ne 0$, a Green's function solving $L_{x}G=\delta (x-\xi )$ is homogeneous on either side of $\xi$, continuous there, and has first derivative jump $[G_x{]}_{-}^{+}=1/a_2(\xi )$. A nonzero jump in $G$ itself would produce an unwanted distributional derivative of the Dirac delta. Once continuity removes that term, integrating the equation across $\xi$ fixes the derivative jump. Endpoint boundary conditions complete the classical characterization.

An adjoint Green's function solves the corresponding adjoint point-source problem, with boundary or terminal conditions chosen so that integration by parts gives a representation for a solution of the original operator. For $L={\partial }_t-{\partial }_x^2+c$ on the real line, the formal adjoint is $L^{\ast }=-{\partial }_t-{\partial }_x^2+c$. The backward kernel with terminal Dirac delta at $(\xi ,\tau )$ is the translated heat kernel with constant absorption, defined for $t<\tau$. Its pairing with a homogeneous forward solution is constant in $t$; taking the terminal limit evaluates that solution at $(\xi ,\tau )$.

variation of parameters with the fundamental solutions $x,x^2$ givesThe kernel is continuous across $x=t$ and its first derivative jumps by $t^{-2}$, which gives a unit-strength Dirac delta function source after multiplication by the leading coefficient. It satisfies both endpoint conditions. The homogeneous solution $Ax+B{x}^2$ satisfies them only when $A=B=0$, proving uniqueness.

A continuous forcing gives the solution with zero initial data $z(x)=-{\int }_0^x\sin (x-s)f(s)ds$. Every solution vanishing at zero is $A\sin x+z(x)$. If $\sin L\ne 0$, the remaining endpoint determines $A$ uniquely, with Green's functionIts first derivative has jump $-1$, the correct sign for $-y^{\prime \prime }-y=f$. If $L=m\pi$ with a positive integer $m$, solvability instead requires ${\int }_0^{L}f(s)\sin sds=0$, and the coefficient $A$ is free. This is the Fredholm alternative with a one-dimensional homogeneous kernel.

Multiplying the equation for $v=y^{\prime }$ by the integrating factor $e^x$ gives $v(x)=-e^{-x}{\int }_x^1{e}^{t}f(t)dt$. Integrating once more and reversing the order of integration gives the Green's functionIt is continuous across $x=t$, its first derivative has jump $+1$, and it satisfies the mixed Dirichlet boundary condition and Neumann boundary condition. Both homogeneous solutions, $1$ and $e^{-x}$, are eliminated by these two conditions, giving uniqueness. For $f=e^{-x}$ the solution is $-x{e}^{-x}$.