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.

The patellar ligament connects the patella to the tibial tuberosity and continues the force-transmission route of the quadriceps apparatus. The quadriceps tendon is proximal to the patella. The patella therefore lies within a mechanical chain transmitting knee-extension force.

The oblique popliteal ligament reinforces the posterior capsule of the knee. It is associated with an expansion from the semimembranosus tendon. This shows how a muscle tendon can contribute both to active movement and to a passive supporting structure.

The tibial collateral ligament is the medial supporting ligament of the knee, connecting femur and tibia. It helps resist medial opening under valgus loading and has close relations to the medial meniscus. Those attachments differ from the more separate lateral collateral arrangement.

The fibular collateral ligament runs on the lateral side of the knee from femur to fibula. It helps resist lateral opening of the joint under varus loading. Its fibular attachment distinguishes it from the medial tibial collateral ligament.

The anterior cruciate ligament links tibia and femur within the knee and helps restrain anterior tibial translation and rotation. Its oblique fiber arrangement means tension changes with knee position. It acts together with other ligaments, menisci and muscles.

The posterior cruciate ligament links tibia and femur within the knee and restrains posterior translation of the tibia relative to the femur. Its effect depends on joint position and loading. It complements rather than duplicates the anterior cruciate ligament.

The infrapatellar fat pad is a fat-containing structure behind the patellar ligament and below the patella. It occupies changing space near the front of the knee and is closely related to the synovial lining. Intracapsular position does not make it part of the cartilage-bearing joint surface.

The suprapatellar recess is an extension of the knee synovial cavity above the patella, deep to the quadriceps tendon. Its communication with the joint explains why joint fluid can accumulate superior to the patella. It differs from the superficial prepatellar bursa.

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.

Spiritus Juniperi is an alcoholic aromatic preparation associated with juniper oil. Its solvent-based form differs from intact juniper material.^[ref]

Pulvis Jalapae Compositus is a compound jalap powder. Its additional ingredients distinguish it from powdered jalap alone.^[ref]

If $w$ is a nonvanishing solution of $-(P{w}^{\prime }{)}^{\prime }+Qw=0$, expanding the right-hand side gives the displayed identity. Integrating expresses a scalar second variation as a square plus a boundary term. A simple zero of $w$ at a fixed endpoint causes no difficulty for Lipschitz endpoint-vanishing variations, since $\phi =O(x-a)$ makes the boundary term tend to zero. This identity is the scalar form of a ground-state transform for a weighted Dirichlet energy.

Let $T$ be a self-map of a nonempty closed subset of a Banach space, with the displayed bounds. Then $T$ has a unique fixed point $p$ and every orbit $x_n=T^n{x}_0$ converges to it. Indeed, $\Vert x_{n+1}-x_n\Vert \le {\alpha }_n\Vert x_1-x_0\Vert$ makes the orbit Cauchy by summing successive increments. The first bound makes $T$ continuous, so the limit is fixed. Since ${\alpha }_n\to 0$, the bounds also imply uniqueness. The tail sum gives $\Vert p-x_n\Vert \le {\sum }_{k\ge n}{\alpha }_k\Vert x_1-x_0\Vert$ for $n\ge 1$. A Volterra integral equation supplies factorially decreasing bounds even when the first iterate is not a contraction mapping.

For a simple module $M$ and a central element $z$ of an algebra $A$, the image $zM$ is a submodule. If it is nonzero, it equals $M$, and iterating gives $z^{r}M=M$ for every $r$. This is impossible when $z$ is nilpotent. Hence $zM=0$. It follows, for example, that the only simple module for $k[X]/(X^r)$ is the one-dimensional quotient on which $X$ acts by zero.