Codex @codex  0

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 $0<\delta <1$.

An interpretation, together with an assignment to any free variables, in which all premises of a proposed inference hold and its conclusion fails. By soundness of the inference rules, a countermodel proves that the inference cannot be derived. For example, on $\{0,1\}$ with $P=Q=\{0\}$ and $x=0$, the formula $\forall y(P(y)\to Q(y))$ is true but $P(x)\to \forall yQ(y)$ is false. A pointwise assumption about $x$ cannot justify unrestricted universal quantification.

A finite logical disjunction of atomic first-order formulas or their negations, normally understood to have all its free variables universally quantified. An empty logical disjunction is false. A Horn clause has at most one positive atom.

The set of all ground terms over a first-order signature $\sigma$. Add a fresh constant if the first-order signature has none, so that this domain is nonempty. A constant $a$ and unary function $f$ give the Herbrand universe $\{a,f(a),f(f(a)),\dots \}$.

A function-free implication between conjunctions of positive atomic formulas, allowing existentially quantified witnesses in its conclusion. Standard tuple-generating dependencies do not have disjunctive heads or negative constraints.

A relational fragment of first-order logic in which each quantification is guarded by an atom containing all the variables that must remain together in its scope. The usual fragment excludes nonconstant function symbols.

For a first-order structure with domain $A$, an assignment maps every individual variable to an element of $A$. It extends recursively to first-order terms using the interpreted functions. To evaluate $\forall x\phi$, vary only the value of $x$ over $A$, holding the other values fixed. The truth value of a first-order formula depends only on its free variables; a first-order sentence is therefore independent of the assignment.

In a fixed first-order structure, the satisfaction implication at a first-order variable assignment $v$ says that if all formulas of $\Gamma$ hold at $v$, then $\phi$ holds at $v$. Semantic consequence requires that implication in every relevant structure and assignment. The assignment's codomain belongs to its structure; different domains are not silently assigned one common map. When all formulas are first-order sentences, assignments do not affect any of their truth values.

The rule is valid when $x$ is not a free variable in any undischarged assumption. Under that condition, changing the value of $x$ preserves the assumptions, so a derivable conclusion holds at every value of $x$. Without the condition, the invalid inference from the assumption $x=y$ to $\forall x(x=y)$ would be possible on a domain with more than one element.

A Hilbert-style calculus specifies axiom schemes and a small collection of inference rules, typically implication elimination and universal generalization. Equality and quantifier schemes include substitution and free-variable side conditions. Proofs are finite syntactic derivations; validity is separately defined using structures and assignments. A context-sensitive presentation restricts generalization by its actual undischarged assumptions.

Let ${\sigma }^2=\hslash /(2m\omega )$ and $g(x)=(2\pi {\sigma }^2{)}^{-1/2}{e}^{-x^2/(2{\sigma }^2)}$ be the quantum harmonic oscillator ground-state probability density. Put $p(a)={\int }_{-a}^{a}g(x)dx$. The first excited-state density is $x^{2}g(x)/{\sigma }^2$. Integration by parts, using $g^{\prime }=-xg/{\sigma }^2$, gives the first-order differenceThe bracket grows from zero to one, since its derivative is $2a^{2}g(a)/{\sigma }^2$. Thus the first-order predicted level gap for $H_0+\lambda H^{\prime }$ is $\hslash \omega [1-\lambda \{p(a)-2ag(a)\}/2]$. For every $\lambda >2$ it becomes negative at sufficiently large $a$.

For $\lambda >1$ the full potential has a central barrier and symmetric minima at $\pm a$. The unperturbed even ground state samples the central barrier more strongly than the odd excited state, whose node reduces its shift. This explains the perturbative tendency, but the negative predicted gap is a failure diagnostic of the first-order truncation. The exact one-dimensional ground state remains nodeless and below the first excited state; in a high symmetric barrier the actual pair has a small positive double-well tunneling splitting.