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Suppose every continuous periodic function $f$ has a unique continuous periodic $Tf$ with Fourier coefficients $\widehat{Tf}(n)={\alpha }_n\hat{f}(n)$. Uniqueness first makes $T$ linear. If $f_j\to f$ and $T{f}_j\to g$ uniformly, continuity of each coefficient functional gives $\hat{g}(n)={\alpha }_n\hat{f}(n)$ for every fixed $n$, hence $g=Tf$. Thus $T$ has a closed graph and the closed graph theorem makes it bounded. Applying it to normalized exponentials gives $|{\alpha }_n|\le \Vert T\Vert$.

For $e_n(t)=(2\pi {)}^{-1/2}{e}^{int}$, the bounded functional $\varphi (f)=\sqrt{2\pi }(Tf)(0)$ satisfies $\varphi (e_n)={\alpha }_n$. The normalization factor follows from $e_n(0)=(2\pi {)}^{-1/2}$.

For the discrete dynamical system $N_{n+1}=F(N_n)$ withevery positive starting value satisfying $F(N_0)>0$ has positive later iterates. The useful invariant interval is $(0,3/2)$.

First, for every $N>0$,because the quadratic has positive leading coefficient and discriminant $49-56<0$. Thus a positive first image lies below $3/2$.

Second, on $[0,3/2]$ the cubic product can be negative only for $1/2<N<1$. On that interval its magnitude is at most $(1/2)(1/2)(3/2)=3/8$, so $F(N)\ge 1/8$ throughout $[0,3/2]$. Together the bounds give $F((0,3/2))\subset (0,3/2)$. Mathematical induction then keeps all later values in that positive bounded interval.

The hypothesis on the first image is necessary: a positive starting population outside the invariant region need not have a positive image. The interval argument supplies positivity and boundedness; it does not by itself classify the eventual orbit.

An invariant interval for a scalar map is an interval whose points remain in that interval under every forward iterate. Here invariance means forward invariance and does not assert surjectivity onto the interval.

The deprotonated counterpart of formic acid, a product of alkaline chloral cleavage.

Describe operations through laws relating their observable results, without selecting an internal representation. For an abstract data type, the laws must state the operation domains and error results as well as ordinary cases. A disjoint union can distinguish a successful value from an error with the same textual spelling.

Specify behavior by constructing a mathematical model and defining the operations on that model. Finite sequences model a FIFO queue: insertion appends, the head is the first element, and successful deletion removes it. An axiomatic specification can then be checked against the model without requiring the implementation to use that representation.

A conclusion is formally derivable from premises when a permitted formal proof exists with those premises as its undischarged assumptions. This is a syntactic relation, distinct from semantic entailment.

A formation rule specifies how well-formed expressions can be constructed from simpler expressions in a formal language. The formal grammar of propositional formulas has rules for propositional variables, negation and binary logical implication.

Blue fluorescence of suitably acidified quinine solutions, used in identifying quinine and illustrating fluorescence.

A small flower, especially one forming part of a dense inflorescence such as a flower head.

A prescribed minimum amount on each directed edge can be removed by substituting $f_e={\ell }_e+x_e$. The new capacity constraints are $0\le x_e\le c_e-{\ell }_e$. For a circulation with vertex demands using net inflow $d_v$, the adjusted demand is $d_v^{\prime }=d_v-({\sum }_u{\ell }_{uv}-{\sum }_w{\ell }_{vw})$. Solving the adjusted problem and adding $\ell$ back gives a one-to-one correspondence of feasible solutions. In particular, removing a lower bound on $u\to v$ increases $u$'s residual demand and decreases $v$'s by that amount.

A dependency by which an unready receiver delays upstream acceptance of further work. In synchronous message passing it can propagate through a chain of blocked sends; a finite buffer postpones this propagation until its capacity is used.

A sixteen-bit floating-point representation with one sign bit, five exponent bits and ten stored fraction bits. For sign $s$, stored exponent $E$ and fraction integer $F$, normal finite values areThe implicit leading one gives eleven significant binary digits. With $E=0$, the value is $(-1{)}^{s}F{2}^{-24}$, including signed zero at $F=0$ and subnormal numbers otherwise. The stored exponent $E=31$ is reserved for infinities at $F=0$ and NaN otherwise. The positive normal range is $2^{-14}$ through $65504$; the least positive subnormal number is $2^{-24}$.

To add two finite floating-point numbers, compare their exponents, shift the smaller significand to a common exponent, add or subtract the signed significands, normalize the result, and apply floating-point rounding. Extra low-order bits are needed to determine correct rounding when shifting or cancellation discards information. A carry beyond the leading digit increases the exponent; cancellation can decrease it. Unlike exact real addition, this operation need not be associative. Changing the child evaluation order in a fixed binary expression tree need not change its parenthesization, whereas reassociation changes which intermediate results undergo floating-point rounding.

A floating-point operation is one arithmetic operation on values represented in floating-point arithmetic. Algorithmic counts commonly count addition, subtraction, multiplication and division separately. A fused multiply-add can therefore count as two arithmetic operations even when executed by one hardware instruction. Such counts describe work, while elapsed time also depends on memory traffic, parallelism and the implementation.

The tapeworm group, lacking an alimentary canal and including species with tissue larvae and intestinal adults.

The fluke group, including internal parasites with typically complex life cycles.

The specialized external covering of parasitic flatworms.

For $u_t=u_{xx}+u(1-u^q)$, $q>0$, the displayed decreasing travelling wave has speed $c=(q+4)/\sqrt{2(q+2)}$, with $a>0$ setting its translation. Writing $Y=U^{q/2}$ gives $U^{\prime }=-kU(1-Y)$, $k=\sqrt{2/(q+2)}$, which verifies the wave equation by equating constant and $Y$ coefficients. The explicit speed increases with $q$, the leading-edge decay rate decreases, and the filled-state decay rate $q/\sqrt{2(q+2)}$ increases. Aligning fronts at half height distinguishes these changes from a translation.

In $u_t=u(1-u-\beta v)+u_{xx}$, $v_t=-\delta uv+v_{xx}$, the line $v=1-u$ is invariant when $\beta =1-\delta$. On it $u$ satisfies Fisher's equation with growth coefficient $\delta$. The reduction also requires compatible initial and boundary data: parameter tuning alone does not reduce arbitrary two-field solutions. Positive concentrations and positive $\beta$ in this normalization require $0<\delta <1$.