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In a series RLC circuit, the same electric current $i=q^{\prime }$ flows through the resistor with electrical resistance $R$, the inductor, and the capacitor. Choose $q$ as the signed electric charge on the capacitor plate into which positive $i$ flows. With terminal drops measured in that direction, Kirchhoff's voltage law givesHere $V(t)$ is the applied electric potential difference; assume constant $L>0$, $C>0$, and $R>0$. With the applied source set to zero and the loop kept closed, the free response satisfiesThe angular frequency ${\omega }_0$ is the value for lossless free oscillation; $\alpha$ is the decay parameter. The three possibilities are underdamped RLC response for $\alpha <{\omega }_0$, critically damped RLC response for $\alpha ={\omega }_0$, and overdamped RLC response for $\alpha >{\omega }_0$. ^[ref].

For a fair comparison, start each circuit with the same electric charge $q(0)=q_0$ and zero electric current $q^{\prime }(0)=0$. Set $\tau ={\omega }_{0}t$ and $a=\alpha /{\omega }_0$; then $x=q/q_0$ obeys $d^{2}x/d{\tau }^2+2adx/d\tau +x=0$. The plot compares $a=0.5,1,2$. These dimensionless curves apply to any $L,C$ after rescaling time and electric charge.

The stored energy is the sum of magnetic energy and capacitor energy:Differentiating and using the circuit equation gives the complete power balance,In the free response, $E^{\prime }=-R{i}^2\le 0$: Joule heating accounts exactly for the lost energy. For the plotted initial conditions, $E(0)=q_0^2/(2C)$ and $E/E(0)=x^2+(dx/d\tau {)}^2$. Since every free solution tends to zero for $R>0$,In the lossless limit $R=0$, energy stays constant and shuttles between the capacitor and inductor. Thus an oscillation of electric charge and a decrease of total energy are compatible statements.

The analogy with a damped harmonic oscillator is exact within these ideal models: electric charge corresponds to displacement, electric current to velocity, inductance to mass, $1/C$ to spring constant, and $R$ to the coefficient of a resisting force proportional to velocity. The two stored energy terms play the roles of kinetic energy and elastic potential energy. ^[ref].

Differentiating the driven equation also gives $L{i}^{\prime \prime }+R{i}^{\prime }+i/C=V^{\prime }$. When using this equation to solve for electric current, retain the original constraint $L{i}^{\prime }(0)+Ri(0)+q(0)/C=V(0)$: the initial electric charge fixes the initial rate of change of electric current.

The direct product of a family of rings consists of all tuples of their elements, with coordinatewise addition and multiplication. Unlike a direct sum, tuples may have infinitely many nonzero coordinates.

A ring is von Neumann regular if every element $a$ has an element $b$ with $a=aba$. In a commutative ring, this is $a=a^{2}b$. An arbitrary direct product of rings whose factors are fields has this property by taking inverses in each nonzero coordinate. Every localization at a prime ideal of a commutative von Neumann regular ring is a field.

A topological ring is a ring equipped with a topology making addition, additive inversion and multiplication continuous functions. An adic topology is obtained from powers of an ideal.

If an algebra $B$ is generated by a subalgebra $A$ that is a right Noetherian ring and one element $x$, and $A+xA=A+Ax$, then $B$ is a right Noetherian ring. The proof uses $F_n={\sum }_{i\le n}A{x}^i={\sum }_{i\le n}{x}^{i}A$ and, for each right ideal $I$, the ascending right ideals $L_n=\{a:a{x}^n\in I+F_{n-1}\}$. After these stabilize, finitely many lifts of generators for $L_0,\dots ,L_N$ generate $I$ by induction on degree. Neither unique normal forms nor an automorphism moving coefficients is required.

A noncommutative domain whose right ideals satisfy the ascending chain condition is a right Noetherian domain. Its right regular module is a uniform module, so any two nonzero principal right ideals intersect. Thus its nonzero elements satisfy the right Ore condition and it embeds in a division ring by Ore localization.

A right Artinian ring is a right Noetherian ring. Its right regular module therefore has finite composition length. Its Jacobson radical is a nilpotent ideal, and its quotient by that Jacobson radical is a semisimple ring. In particular every finitely generated right module over such a ring has finite composition length.

For a smooth map $f:M\to N$, the pullback of a Riemannian metric is $(f^{\ast }g{)}_p(u,v)=g_{f(p)}(d{f}_{p}u,d{f}_{p}v)$. It is positive definite precisely when $d{f}_p$ is injective at every point, that is, when $f$ is an immersion. In coordinates, $(f^{\ast }g{)}_{ij}=g_{ab}(f(y)){\partial }_i{f}^a{\partial }_j{f}^b$.

A Riemannian metric gives inverse vector bundle isomorphismsIn coordinates they have matrices $g_{ij}$ and $g^{ij}$. Every smooth manifold admits a Riemannian metric by a partition of unity, so its tangent and cotangent bundles are isomorphic as real smooth vector bundles. This isomorphism depends on the metric, not merely on the manifold.

The product of Riemannian manifolds has metricThe tangent summands are orthogonal, and positivity follows because a nonzero product tangent vector has a nonzero component. Its Levi-Civita connection is the product affine connection of the two factor Levi-Civita connections: mixed torsion is zero and metric compatibility follows from independence of the two coordinate sets.

For every integer $n\ge 1$, the Riemann zeta function satisfies $\zeta (1-n)=(-1{)}^{n-1}{B}_n/n$. One proof applies meromorphic continuation of a Mellin transform from an asymptotic expansion to $(e^y-1{)}^{-1}$, whose transform is the Bose integral $\Gamma (s)\zeta (s)$, and divides residues by the residues of the Gamma function. The formula includes $\zeta (0)=-1/2$.

The study of deformations and markings of Riemann surfaces, using quasiconformal maps, holomorphic quadratic differentials, extremal length and the mapping class group.

A surface with an atlas outside finitely many cone points whose changes of coordinate are translations. On a compact surface it is equivalent to a nonzero holomorphic one-form. A zero of order $m$ has cone angle $2\pi (m+1)$.

In a holomorphic coordinate, a length element $\rho (z)|dz|$. Smooth positive densities define ordinary Riemannian metrics; extremal length also permits nonnegative measurable densities with finite positive area $\int {\rho }^{2}dxdy$.

An orientation-preserving homeomorphism with locally $L^2$ weak derivatives and essentially bounded infinitesimal distortion. In holomorphic coordinates, its Beltrami coefficient satisfies $\Vert \mu {\Vert }_{\infty }<1$, and the maximal dilatation is $(1+\Vert \mu {\Vert }_{\infty })/(1-\Vert \mu {\Vert }_{\infty })$.

In a two-treatment urn design, begin with $a$ balls of each type, draw and replace a ball to allocate treatment, then add $b$ balls of the same type after success or of the opposite type after failure. This response-adaptive randomization favours treatments with more observed successes. For RPW$(1,1)$, type-0 probability after counts $(s_0,f_0,s_1,f_1)$ is $(1+s_0+f_1)/(2+s_0+f_0+s_1+f_1)$.

For positive-semidefinite kernels $k_j$ with Reproducing kernel Hilbert spaces ${\mathcal{H}}_j$, the Reproducing kernel Hilbert space of $k={\sum }_j{k}_j$ consists of sums $f={\sum }_j{f}_j$, withThe minimum is attained uniquely. Indeed, in the Hilbert space direct sum $E={⨁}_j{\mathcal{H}}_j$, the tuples summing to zero form a closed subspace $N$: they are the intersection, over inputs $x$, of the kernels of the continuous maps $(f_j)\mapsto {\sum }_j{f}_j(x)$. Every affine fibre has a unique representative in $N^{\perp }$ by orthogonal decomposition by a closed subspace. The vector $(k_j(\cdot ,x){)}_j$ lies in $N^{\perp }$ and represents evaluation in the quotient; its evaluation at $y$ is ${\sum }_j{k}_j(y,x)$, identifying the positive-semidefinite kernel with $k$.

For the chain $S_0\subset \cdots \subset S_n$, the Gelfand–Tsetlin algebra in $\mathbb{C}{S}_n$ is generated by the centers of $\mathbb{C}{S}_r$ for $0\le r\le n$. Multiplicity-free restriction makes its joint eigenspaces one-dimensional in each irreducible representation. Products of the central primitive idempotents along restriction paths give its minimal projections. Thus it is a maximal commutative subalgebra and a semisimple algebra, isomorphic to a finite product of copies of $\mathbb{C}$.

Delete $n$ from the cycle notation of a permutation in $S_n$ to obtain a permutation in $S_{n-1}$. Extend this set map linearly to ${\Pi }_n:\mathbb{C}{S}_n\to \mathbb{C}{S}_{n-1}$. It is an $S_{n-1}$-bimodule map and the identity on $\mathbb{C}{S}_{n-1}$, but is not in general an algebra homomorphism. In particular,For $n\ge 5$ it is the unique bimodule-equivariant set map $S_n\to S_{n-1}$; at $n=4$ another such map exists because $C_{S_3}(S_2)=S_2$.

For $|\lambda |=n$, $|\mu |=m$, and $k=n-m$, defineThe copy of $S_k$ acting on the last $k$ letters commutes with $S_m$ and acts on this multiplicity space. Iterated restriction branching rule for a symmetric group supplies a basis indexed by standard skew Young tableaux. Every such basis vector is a cyclic vector for a group representation: admissible swaps connect all tableaux, and their off-diagonal coefficients in the Young orthogonal form are nonzero.