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The bounded angular function has ordinary derivatives $D_{j}u={\delta }_{1j}/r-x_1{x}_j/r^3$ away from zero. Their magnitudes are at most $2/r$, whose $p$-th power is integrable on the unit ball exactly below the dimension threshold in this estimate. To verify the weak derivative across zero, integrate by parts outside a ball of radius $\epsilon$ against a compactly supported smooth test. The boundary term has magnitude at most $C{\epsilon }^{n-1}$, tending to zero for $n>1$, while the derivative volume term converges by integrability. This proves the stated Sobolev membership with those weak derivatives. The dimensional condition distinguishes it from the one-dimensional sign function, whose jump produces a Dirac derivative.

A send window specifies the numbered data a sender may transmit before further progress permits the range to advance.

A receive window bounds the numbered data a receiver is prepared to accept. In TCP, the receiver advertises a window that limits the sender independently of the sender's congestion window.

A fatal-infection model with vaccination proportional to infective-susceptible encounters adds $v_t=\eta qs$ and $m_t=q$. The total count including deaths is conserved. Integrating $dq/ds$ and $dv/ds$ gives $q=q_0+[s_0-s+\log (s/s_0)]/(1+\eta )$ and $v=\eta (s_0-s)/(1+\eta )$. The selected final root has $0<s_{\infty }<\min (s_0,1)$, and deaths are $q_0+(s_0-s_{\infty })/(1+\eta )$. The model is a specified deterministic reaction scheme, not a general vaccination policy rule.

In a compartmental epidemic model, a susceptible individual is one who can acquire the modeled infection. The susceptible compartment counts these individuals. An exposed but not yet infectious individual belongs to a separate latent class in an SEIR model; models that retain such individuals in a not-yet-infective compartment must state that bookkeeping convention explicitly.

An infective individual is currently capable of transmitting the modeled infection. This is distinct from a susceptible individual and, when incubation is modeled, from an exposed individual not yet infectious.

The sinc function $\sin z/z$ for $z\ne 0$, extended by value one at zero. The displayed power series has infinite radius of convergence, so the extension is entire. Its removable value follows from the first coefficient or from the sine limit. This convention differs by the argument scaling $z\mapsto \pi z$ from the normalized sinc convention.

If $K(\alpha )/K$ is a simple algebraic field extension of degree $n$, the displayed powers form a basis. The minimal polynomial of $\alpha$ reduces every higher power to a linear combination of them, while a nontrivial linear relation among the first $n$ powers would give a polynomial of degree below the minimal polynomial vanishing at $\alpha$. If $\alpha$ is an algebraic integer over the rational numbers, this is a basis of the order $\mathbb{Z}[\alpha ]$ as a free abelian group; it need not be an integral basis of the whole ring of integers.

The drift $-x{u}_x/(2t)$ exactly cancels the coordinate-motion term of the similarity solution $u=t^{-1/2}f(\eta )$, $\eta =x/\sqrt{t}$. Direct differentiation givesConsequently the nonautonomous partial differential equation reduces to the autonomous ordinary differential equation $f^{\prime \prime }=f^3-f/2$. The balance fixes the same length scale $x\sim \sqrt{t}$ as diffusion, but removes the first-derivative term that usually appears in the profile equation. A first-integral argument can then determine a whole profile family without solving a nonautonomous second-order equation.

The transformation $\tau =\log t$, $\xi =x/\sqrt{t}$, $v=\sqrt{t}u$ gives $v_{\tau }=v_{\xi \xi }+(v-v^3)/2$. Thus the stationary half-line profile $v=\tanh (\xi /2)$ yields $u=t^{-1/2}\tanh [x/(2\sqrt{t})]$, with zero value at the origin and far-field value $t^{-1/2}$. For two bounded classical solutions, their difference has reaction coefficient $[1-(v_1^2+v_1{v}_2+v_2^2)]/2\le 1/2$. Multiplying it by $e^{-\tau /2}$ therefore produces a linear parabolic equation with nonnegative absorption. A positive growing quadratic spatial barrier and the maximum principle give uniqueness on the unbounded half-line. This proof identifies the finite-time boundedness or growth hypothesis needed at spatial infinity rather than assuming a supremum is automatically attained there.

If every nontrivial closed subwalk has strictly positive weight, removing any repeated-vertex segment strictly decreases a walk's weight. Hence a minimum-weight walk must be a graph path, and it is a shortest path. Strictly positive edge weights are sufficient in an undirected graph. For a trail, positivity of every simple cycle is already sufficient: its first repeated-vertex segment is a cycle because no edge is repeated.

Under the usual unrestricted walk convention, positivity of simple cycles alone is insufficient. A graph consisting of one zero-weight edge $xy$ has no cycles, but the walk $x,y,x,y$ is a minimum-weight walk and is not a path. A zero-weight backtrack cannot be removed with a strict improvement. For nonnegative edge weights, one may erase such backtracks without increasing weight and obtain a shortest path, but cannot conclude that every minimum walk was already a path.

The infimum of total edge weights over allowed walks from a source to a target. The value is $+\infty$ when no route exists. A negative cycle reachable from the source and able to reach the target makes it $-\infty$. In a finite graph without such a cycle, a finite distance is attained by a graph path, after removing non-improving repetitions.

Compute shortest-path distances from one fixed source to every vertex, together with predecessor information when shortest paths exist. Dijkstra's algorithm handles nonnegative weights; the Bellman-Ford algorithm admits negative weights and detects reachable negative cycles. A Directed acyclic graph instead permits one pass in topological order.

Adding the same constant $C$ to every edge weight adds $C$ times the number of edges to each path's cost. Routes with different edge counts need not retain their order. For example, edges $s\to t$, $s\to u$, $u\to t$ of weights $1,-2,2$ make the two-edge path optimal with cost zero. Adding three changes the direct cost to four and the two-edge cost to six. The graph is a Directed acyclic graph and all shifted weights are positive, yet the minimizing path changes.

The Floyd-Warshall algorithm computes all-pairs shortest graph path lengths in a finite weighted directed graph with no negative-length cycles in a graph. Initialize $d_{ii}=0$, use the arc cost for each existing directed edge, and use $+\infty$ when no arc exists. For $k=1,\dots ,n$, update every ordered pair $(i,j)$ by the displayed rule. The storage is $n^2$ lengths and the running time is $O(n^3)$.

For correctness, let $d_{ij}^{(k)}$ be the shortest simple graph path length whose intermediate vertices lie among the first $k$. A shortest allowed graph path either avoids $k$ or splits at its unique occurrence of $k$, giving the recurrence. Conversely, concatenate shortest paths through $k$ and remove any repeated-vertex sections; each such closed walk has nonnegative length, so removal cannot increase the length. This proves equality. The update can be performed in place because $d_{kk}=0$ and the $k$th row and column remain unchanged during stage $k$.

If negative-length cycles in a graph are present, a shortest walk may not exist and the recurrence need not compute shortest simple graph paths. Two optimal subpaths can concatenate into a repeated-vertex walk with a negative cycle; deleting that cycle increases its length. A negative diagonal entry in the algorithm's final array detects this obstruction to finite shortest-walk distances.

A shift register accepting one sample at each clock edge and exposing every stored stage as a parallel output.