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If $u$ is Lebesgue integrable and $v$ is measurable on the same measure space, then $u+v$ is integrable exactly when $v$ is integrable. One implication follows from $|u+v|\le |u|+|v|$; the other follows from $|v|\le |u+v|+|u|$. Thus adding a function with finite integral of its absolute value cannot remove a failure of absolute integrability. This is the vector space property of L1 space stated as a useful test for a sum of terms with different decay rates.

Plural name of the lateral recess of the fourth ventricle.

The paired lateral aperture of the fourth ventricle openings.

For a contour integral with factor $e^{{\zeta }^4/4+z\zeta }$, the quartic term dominates the linear term on rays at large radius. Writing $\zeta =r{e}^{i\theta }$ gives real part $r^4\cos (4\theta )/4$ for the leading exponent. There are four decay sectors, centered at $\theta =\pi /4+j\pi /2$ and each of angular width $\pi /4$. On a closed subsector the integrand and all its $z$ derivatives decay faster than any exponential in $r$, locally uniformly in $z$. Contours joining two such sectors therefore allow differentiation under the integral sign and have vanishing infinite endpoint terms. Distinct geometric contours still need an independence proof; their initial moments often give a Vandermonde matrix calculation.

Every character of an algebra on ${\ell }^1(\mathbb{Z})$ with convolution multiplication is uniquely of this form for $0\le t<2\pi$. If $e_j$ is the point mass at $j$, then $e_j{e}_k=e_{j+k}$. A character has norm one and $e_1{e}_{-1}=1$, so $|\chi (e_1)|\le 1$ and $|\chi (e_1{)}^{-1}|\le 1$. Hence $\chi (e_1)$ lies on the unit circle; multiplication determines all $\chi (e_j)$ and continuity determines the displayed series. Conversely absolute convergence permits rearrangement of the double series, proving multiplicativity. The maximal ideals are exactly their kernels: each consists of sequences whose Fourier series vanishes at one fixed point of the unit circle.

A fibrocartilaginous structure within the knee that helps distribute contact loads and contributes to stability. This anatomical meaning is distinct from a liquid surface.

Glue the outer circles of two closed annuli by the identity, and their inner circles by complex conjugation. The result is a Klein bottle. To see the gluing explicitly, let $r\in [1/2,1]$ and $z\in S^1$. Parametrize the first annulus by $(z,\tau )=(z,2r-1)$, with $0\le \tau \le 1$, and the second by $(z,\tau )=(z,3-2r)$, with $1\le \tau \le 2$. The outer-circle gluing identifies their common level $\tau =1$. The remaining gluing isThis is the mapping torus of reflection of the circle. The coordinate maps and their piecewise inverses agree on every identified boundary, so the universal property of the quotient topology gives inverse continuous maps between the two quotient descriptions. Gluing collars by the reflection supplies disc charts at the seam; thus it is a closed surface. Cutting the circle coordinate at one point gives a rectangle with one matching and one reversed edge pairing, the usual Klein-bottle model. Its Euler characteristic is zero and it is a nonorientable surface, so the classification theorem identifies it with the connected sum of two real projective planes.

For a charged particle in a uniform magnetic field, the symmetric gauge $\mathbf{A}=\mathbf{B}\times \mathbf{x}/2$ makes the displayed magnetic pseudomomentum a conserved quantity when no other position-dependent potential breaks translation symmetry. It differs from the kinetic momentum $\mathbf{\pi }=\mathbf{p}-e\mathbf{A}=m\dot{\mathbf{x}}$. With $\mathbf{B}=B{\mathbf{e}}_3$, Hamilton's equations for $H=|\mathbf{p}-e\mathbf{A}{|}^2/(2m)$ give ${\dot{p}}_1=(eB/2){\dot{x}}_2$ and ${\dot{p}}_2=-(eB/2){\dot{x}}_1$, proving conservation of $K_1=p_1-eB{x}_2/2$ and $K_2=p_2+eB{x}_1/2$. The charge factor is necessary unless it has already been absorbed into the field convention. In quantum mechanics the transverse components need not commute, even though each commutes with this Hamiltonian operator; magnetic translation algebra records that distinction.

Consider an equal-density vortex sheet halfway between rigid walls, with equal layer depths $h$, base velocities $U_1,U_2$ and no restoring interfacial force. The irrotational perturbations have vertical factors $\cosh (k(y-h))$ above and $\cosh (k(y+h))$ below. Their kinematic boundary conditions and pressure continuity giveThe common depth factor cancels, yielding the displayed dispersion relation roots and Kelvin-Helmholtz instability for nonzero shear. The real phase velocity of the pattern is $(U_1+U_2)/2$ and the growing branch has temporal growth rate $|k(U_1-U_2)|/2$. This exact depth cancellation depends on equal depths and equal mass densities; a single nearby wall or unequal layers changes the inertia factors.

For two infinitely deep inviscid layers with stable density ordering ${\rho }_2>{\rho }_1$ and positive surface tension $\gamma$, the velocity jump destabilizes wavenumber $k$ precisely whenThe threshold diverges at both small and large wavenumber. Minimizing the bracket gives $|k|=\sqrt{({\rho }_2-{\rho }_1)g/\gamma }$ and the stated minimum velocity jump. A larger jump opens a bounded band of unstable wavenumbers, obtained from the two positive roots of $\gamma q^2-[{\rho }_1{\rho }_2{U}^2/({\rho }_1+{\rho }_2)]q+({\rho }_2-{\rho }_1)g=0$. Equality at threshold is marginal rather than exponential growth.

A negative signed capillary rise for a nonwetting liquid: the equilibrium contact angle exceeds a right angle, giving $\cos \theta <0$ in Jurin's law. Correcting a mercury-column measurement requires a consistent meniscus and surface condition.

Historical formulation name; composition is identified by pharmacopoeial edition.