For ${\partial }_x^2+k^2$ on $(0,\ell )$ with zero endpoint values and $\sin (k\ell )\ne 0$, let $x_{<}=\min (x,\xi )$ and $x_{>}=\max (x,\xi )$. The displayed Green function is continuous at $x=\xi$ and its first derivative has jump one, giving unit Dirac delta forcing. The formula follows from the left solution $\sin (kx)$ and right solution $\sin (k(\ell -x))$, whose Wronskian is $-k\sin (k\ell )$. At $k=0$ its limit is $-x_{<}(\ell -x_{>})/\ell$, although zero is excluded by a condition that $k$ is not an integer in the length-$\pi$ version.

The Dirichlet convolution of two multiplicative functions is a multiplicative function. For positive coprime integers $m,n$, every positive integer divisor of $mn$ has a unique form $de$ with $d\mid m$ and $e\mid n$, by prime factorization. The pairs $d,e$ and $m/d,n/e$ are each coprime integers, so the defining rules for the two multiplicative functions separate the sum:Also $(f\ast g)(1)=f(1)g(1)=1$. This short argument proves multiplicativity of both the divisor function $1\ast 1$ and the sum-of-divisors function $\mathrm{id}\ast 1$. The result need not be a completely multiplicative function, even when both inputs are: $1\ast 1=\tau$ fails at $2\cdot 2$. ^[ref]

A real Dirichlet character takes values in $\{0,1,-1\}$. Its restriction to the unit group has order at most two; if it is nonprincipal, the order is exactly two. A real nonprincipal primitive Dirichlet character is the character to which a possible Siegel zero belongs.

// Target: analytic-number-theory.bigb

A tournament is an orientation of a complete graph: between each pair of distinct vertices there is exactly one directed edge.

A cyclic ordering of a directed graph is considered unchanged by rotations or swaps of consecutive nonadjacent vertices. A representative is coherent when every backward edge $v_j\to v_i$ has a forward directed path from $v_i$ to $v_j$. Equivalently every edge is on a directed circuit that winds exactly once around the order. The number of backward edges on a circuit is invariant under the allowed changes. Every nontrivial strongly connected directed graph admits such an order.

A finite directed graph with layers $V_0,\dots ,V_n$ and edges only to the next layer is a Plünnecke graph when every edge $u\to v$ permits both commutation matchings. The successors of $v$ inject into the successors of $u$ through edges to the original successors; the predecessors of $u$ inject into those of $v$ through edges from the original predecessors. These conditions formalize interchanging adjacent summands in an additive graph. They persist on channels consisting of paths between prescribed endpoint sets.

A digraph kernel is an independent set $K$ such that each vertex outside $K$ has a directed edge to a vertex in $K$. A directed cycle of even length has a kernel, obtained by taking alternate vertices; a directed triangle has none. The kernel-perfect directed graph property requires existence of such a set in every induced subgraph.

A digraph kernel is an independent set $K$ of vertices such that each vertex outside $K$ has a directed edge from itself to a vertex in $K$. A directed graph is kernel-perfect if every induced subgraph has a digraph kernel. Preference orientations of the line graph of a bipartite graph are kernel-perfect: a stable matching in a bipartite graph is independent in the line graph, and stability supplies the required outgoing edge at every unmatched edge.

If the order of a group element is $m$ for $g$ and $n$ for $h$, with both finite, thenexactly when both $m$ and $n$ divide $k$. The smallest positive such $k$ is their least common multiple, givingThis concerns the orders of the selected elements, which need not equal the orders of their entire groups. If either component has infinite order of a group element, no positive power can return the pair to the identity element, so the pair also has infinite order.

For example, in $C_4\times C_6$, the additive element $(2,3)$ has component orders $2,2$, hence pair order $2$. The element $(1,1)$ instead has order $\mathrm{lcm}(4,6)=12$. The product has $24$ elements, so even this latter element does not generate it. This distinguishes group order from order of a group element while exposing the simultaneous-return interpretation.

For positive integers $m,n$, two finite cyclic groups combine into one cyclic group precisely when their group orders are coprime:The reason is that both cyclic counters must return to their identity elements simultaneously. If $a,b$ are generators of a group for the two factors, the order of a direct-product element gives $\mathrm{ord}(a,b)=\mathrm{lcm}(m,n)$. The pair generates all $mn$ elements exactly when this least common multiple is $mn$.

For $C_2\times C_3$, repeated addition of $(1,1)$ visitsAll six pairs appear before the return, so $(1,1)$ is a generator. In contrast, in $C_2\times C_2$ the same step gives only $(0,0),(1,1),(0,0)$. Every nonidentity element has order of a group element two, so none can generate all four pairs. This is the Klein four-group; there is no group isomorphism to $C_4$ even though both have four elements.

For a brief proof of the converse, each component order of a group element divides the corresponding factor group order. Every pair's order therefore divides $\mathrm{lcm}(m,n)$. If $\gcd (m,n)>1$, this is smaller than $mn$, so no pair can generate the whole product. The Chinese remainder theorem gives the explicit isomorphism in the coprime case: $t\mathrm{mod}mn\mapsto (t\mathrm{mod}m,t\mathrm{mod}n)$.

For a bounded domain, bounded symmetric uniformly elliptic $A$, bounded real $q\le 0$, $f\in L^2$ and a boundary lift $\psi \in H^1$, minimizing $J$ on $\psi +H_0^1$ solves $\mathrm{div}(A\nabla u)+qu=f$. The nonnegative quadratic terms are weakly lower semicontinuous, and Poincaré inequality makes the gradient norm coercive on the zero-boundary part. The first variation gives the weak equation with the forcing sign displayed above. If $q$ is positive, coercivity may fail at a Dirichlet eigenvalue; orthogonality to the homogeneous kernel is then needed for solvability.

Suppose the domain has finite measure and $F\in C^1$ is convex, with $|F(p)|\le C(1+|p{|}^2)$ and $|DF(p)|\le C(1+|p|)$. The supporting inequality $F(p)\ge F(q)+DF(q)\cdot (p-q)$, integrated with $q=Du$, gives $\int F(D{u}_k)\ge \int F(Du)+\int DF(Du)\cdot (D{u}_k-Du)$. The last term tends to zero by weak convergence of the gradients and the fixed $L^2$ test field $DF(Du)$. This proves weak lower semicontinuity without requiring pointwise convergence of gradients.

The brane embedding pulls the spacetime metric and two-form back to the worldvolume. The dilaton is evaluated at that embedding. If the quoted tension includes the constant-background factor $e^{-{\Phi }_0}$, use $\tilde{\Phi }=\Phi (X)-{\Phi }_0$; otherwise use $\tilde{\Phi }=\Phi (X)$ and a tension coefficient independent of that factor. These are normalization conventions for the same Dirac-Born-Infeld action.

For Hermitian scalar matrices, their commutator is anti-Hermitian, so $-\mathrm{Tr}[{\varphi }^I,{\varphi }^J{]}^2=\mathrm{Tr}[{\varphi }^I,{\varphi }^J{]}^{†}[{\varphi }^I,{\varphi }^J]\ge 0$. Dimensional reduction gives $F_{IJ}=i[{\varphi }^I,{\varphi }^J]$, fixing the minus sign before the literal commutator square inside the conventional overall-negative Yang-Mills action. Commuting scalar expectation values minimize this potential and describe ordinary separated branes. A plus sign instead requires interpreting the squared symbol as the positive Hermitian norm, not literal matrix multiplication.

A supersymmetric spike of a D-brane combines a worldvolume gauge field with a transverse scalar profile. Its geometry and charge represent strings ending on the brane. Electric spikes represent fundamental strings; magnetic spikes on a D3-brane represent D1-branes. The original construction is Brane Dynamics From the Born-Infeld Action.

With ${\mathbf{E}}_i=2\pi {\alpha }^{\prime }{F}_{0i}$, this normalization separates the nonlinear constitutive relation from the D3-brane tension. It is the normalized canonical electric momentum. The Gauss law constraint in gauge theory requires its divergence to vanish away from string endpoints. On the electric BIon locus it equals the scaled electric field, but this equality is not true for arbitrary configurations.

In two worldsheet dimensions, ${\delta }_{\omega }(\sqrt{g}R)=-2\sqrt{g}{\Delta }_g\omega$. Thus the standard dilaton curvature coupling does not have classical Weyl invariance for a general spacetime-varying dilaton. A constant dilaton gives the topological Euler characteristic instead. In the quantum sigma model, this classical variation combines with the loop anomaly; vanishing improved sigma-model beta functions is the full background conformal condition.

With positive diffusion coefficient $a(x)$, this diffusion equation expresses conservation with flux $-a(x)u_x$. For a differentiable coefficient, its spatial operator is $a{u}_{xx}+a^{\prime }{u}_x$, rather than either $a{u}_x$ or $a{u}_{xx}$ alone. A symmetric half-grid diffusion consistency discretization uses the difference of the two adjacent interface fluxes, preserving that divergence structure. Suitable Dirichlet boundary conditions make the spatial operator dissipative in the L2 norm.

A concentration varying over length $L$ has Laplacian of order its amplitude divided by $L^2$. Balancing the time derivative with $\kappa \Delta \chi$ in the diffusion equation gives the displayed relaxation time. A multiple-scale expansion with spatial scale $\epsilon x$ therefore uses slow time ${\epsilon }^{2}t$ when the effective diffusivity is of order one.

Rotational diffusion is the random change of a particle's rigid body orientation through Brownian motion. A suspended ellipsoid can turn while its centre hardly moves; translational diffusion instead spreads its position. Random molecular collisions supply fluctuating torques as well as forces. The rotational diffusion coefficient $D_r$ measures how quickly angular uncertainty grows, rather than a steady spin speed. Its units are ${\mathrm{s}}^{-1}$, often written $\mathrm{ra}{\mathrm{d}}^2{\mathrm{s}}^{-1}$ to distinguish angular spreading from translational spreading, whose diffusion coefficient $D_t$ has units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$. The radian is dimensionless, but writing angular variance in $\mathrm{ra}{\mathrm{d}}^2$ remains useful.

First identify what the image can reveal. A marked end of an elongated particle defines an arrow: turning it through $\pi$ reverses that arrow. An unmarked ellipsoid with identical ends reveals only its long axis, a director: angles $\varphi$ and $\varphi +\pi$ describe the same visible state. An unmarked perfect sphere has no visible rigid body orientation at all. Tracking its centre cannot measure its rotational diffusion; a surface mark or another identifiable internal direction is needed.

This distinction makes rotational diffusion useful: it measures how long a particle remembers its pointing direction. For an elongated particle, turning also changes which laboratory direction is easiest for translational diffusion. Randomization of rigid body orientation can therefore erase directional preferences in the observed displacement, even though the particle's shape has not changed.