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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.

The normalized sinc function is $\sin (\pi x)/(\pi x)$ away from zero and one at zero. The value at zero removes the apparent singularity, giving an entire function. Its nonzero integer zeros make it useful for Sampling Theory.

With the cycles-per-unit Fourier transform convention, the inverse transform of the unit-width rectangle isThe factor of $\pi$ is therefore tied to a frequency convention. The unnormalized sinc function uses $\sin x/x$ instead, so the two names must be distinguished when comparing formulas.

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.

The minimum interval before an active clock edge during which the data input of a flip-flop must remain stable for specified operation. It constrains the latest arrival of new data at the receiving flip-flop.

The minimum interval after an active clock edge during which the data input of a flip-flop must remain stable for specified operation. It constrains how quickly a newly launched value may reach the receiving flip-flop.

An equality between sets expressed in terms of arbitrary input sets that holds for every allowed choice of those inputs. A double inclusion proof proves a set identity by proving both set inclusions. Membership equivalences give another proof: the expressions are equal if and only if each element belongs to either both or neither.

A solution to a partial differential equation that factors into functions of separate variables, for example $u(x,t)=X(x)T(t)$. For the heat equation $u_t=D{u}_{xx}$, substitution yields $T^{\prime }/(DT)=X^{\prime \prime }/X$, so both sides are a common constant wherever division is valid. The resulting ordinary differential equations must be solved together with the original boundary conditions. Individual separated solutions generate more general solutions by superposition when the equation and boundary data are linear.

A nonzero polynomial is factorwise separable if every irreducible factor is a separable polynomial. Unlike the convention that all roots must be distinct, this condition allows repeated irreducible factors. Its splitting field is a separable field extension precisely when the polynomial is factorwise separable: in one direction every root has a separable minimal polynomial; in the other, adjoining finitely many roots with separable minimal polynomials gives a separable extension. Thus the implication from a separable splitting extension to a polynomial with distinct roots needs the extra squarefree hypothesis. The example $(X-1{)}^2$ already distinguishes the conventions.

Consider an implicit centered heat stencil of width $\Delta x$ weighted by $\theta$, combined with an explicit centered stencil of width $2\Delta x$ weighted by $1-\theta$. With $\mu =\kappa \Delta t/(\Delta x{)}^2$, its amplification factor is $G(\xi )=[1-\mu (1-\theta ){\sin }^2\xi ]/[1+4\mu \theta {\sin }^2(\xi /2)]$. For $0\le \theta <1/2$, contraction in the discrete ${\ell }^2$ norm holds exactly under the displayed condition. It is unconditional for $1/2\le \theta \le 1$. Put $v={\sin }^2(\xi /2)$: the condition $G\ge -1$ reduces to $4\mu v[(1-2\theta )-(1-\theta )v]\le 2$. Maximizing this concave quadratic proves the threshold; the Parseval identity converts the multiplier bound into contraction.

Choosing uniformly among currently unused neighbours gives a self-avoiding walk with a path-dependent probability: $a_j$ is the number of choices available at step $j$. In the square lattice, a four-step straight path has probability $1/(4\cdot 3^3)$, whereas the path $(0,0),(1,0),(1,1),(0,1),(-1,1)$ has probability $1/(4\cdot 3^2\cdot 2)$. There is no trapping at that length, and these unequal probabilities disprove uniformity.

When a trapped attempt is restarted, the final law is the one-attempt law conditional on successful completion. This divides successful-path probabilities by a common normalizing constant and does not remove their unequal products. The local uniformity of each move therefore does not imply uniformity over complete paths.

A self-avoiding polygon on a lattice is a closed nearest-neighbour walk whose vertices are distinct except for the final repetition of the starting point. On the square lattice a nondegenerate polygon has even perimeter at least four, since each step changes the parity of the sum of the coordinates. One can generate such a polygon by generating a self-avoiding walk with $M-1$ edges, retaining it only when its endpoint is a neighbour of its origin, and appending the closing edge. The resulting distribution inherits the bias of the walk sampler unless an additional correction is applied.

A nonnegative second variation requires its highest-derivative coefficient $P=f_{vv}$ to be nonnegative. If it were negative on an interval, a smooth compactly supported factor times $\sin (nx)$ would have negative derivative energy of order $n^2$, while its zero-order energy stays bounded. Large $n$ would contradict nonnegativity. This is a local necessary condition; it does not rule out negative directions caused by the lower-order terms.