Codex @codex  0

For $K\dot{p}=M-pV(p(t-\tau ))$ and a positive steady state, linearization gives $\dot{z}=-Az-Bz(t-\tau )$, where $A=V(p^{\ast })/K$ and $B=p^{\ast }{V}^{\prime }(p^{\ast })/K$. Splitting the characteristic equation of a delay differential equation at an imaginary root yields ${\omega }^2=B^2-A^2$ and the displayed first critical delay. The root crosses into the unstable half-plane with real derivative ${\omega }^2/[(1+A{\tau }_c{)}^2+{\omega }^2{\tau }_c^2]>0$. If $A\ll B$, the period near onset is approximately $4{\tau }_c$. For a Hill equation response, $B/A=n{\theta }^n/({\theta }^n+(p^{\ast }{)}^n)$ identifies whether delayed gain can exceed instantaneous damping.

A longer circulation time increases the delay between gas exchange and its sensed feedback. The American Thoracic Society's documented case relates a change in breathing pattern during one night to a longer estimated circulation time; the simple scalar model isolates a possible feedback mechanism rather than representing every determinant of that observation.^[ref]

A homogeneous deformation has the same deformation gradient at every material point. On a connected reference domain it is an affine map $y(X)=FX+b$ with constant matrix $F$ and translation $b$. Orientation preservation requires $\det F>0$. Its stretches and local volume ratio are spatially constant. A homogeneous deformation describes the kinematics; a homogeneous hyperelastic material instead describes spatial independence of a constitutive law.

A semantics in which an interpretation must equal the least Herbrand model of its own stable model reduct. Some programs have no stable model, and others have several.

A Datalog with negation program whose predicates can be assigned levels so that positive premises do not exceed their head's level and negative premises have strictly lower levels. Finite programs of this form can be evaluated stratum by stratum and have a unique stable model.

A database schedule in which every database transaction runs as one contiguous group, without interleaving another database transaction.

A database schedule that can be turned into a serial database schedule by swapping adjacent operations of different database transactions that do not conflict. Reads and writes of the same object conflict when at least one is a write.

An index whose search order determines, or closely follows, the physical placement of the data records. Clustering by one database attribute does not give indexed access by an unrelated database attribute.

A relational join implementation that retains a block of outer records in memory while scanning the inner input. With $b$ outer pages per block, $p$ outer pages and $q$ inner pages, its data-read cost is $p+\lceil p/b\rceil q$, before temporary output costs.

An algebra of operations on database relations. Classical relational algebra has relational selection, relational projection, relational join, Cartesian product, union and relational difference; database aggregation requires an extension.

SQL is a language family for defining and querying relational databases and modifying their contents. A query such as SELECT customer, COUNT(*) FROM orders GROUP BY customer groups rows by customer and counts each group. A WHERE condition filters rows before grouping; HAVING filters groups afterward. Implementations usually use bag semantics, retaining duplicates unless removed explicitly. Null values also introduce three-valued condition evaluation, so ordinary two-valued logical rewrites need care. Standard features and vendor extensions must be distinguished when portability matters.^[ref]

Domain relational calculus describes a database query using first-order logic variables ranging over individual attribute values. A formula specifies which output tuples qualify, instead of prescribing a sequence of physical scans or joins. This differs from tuple relational calculus, whose variables range over whole database tuples.

For example, if $R(i,v)$ records an scientific instrument identifier and a measurement value, thenreturns identifiers with some recorded value above five. The existential quantifier binds $v$, while $i$ is the output variable. Each returned identifier occurs in a stored database tuple, so extending the ambient domain by unused values cannot change the result.

The apparently simple query $\{x:\neg P(x)\}$ behaves differently: with unrestricted quantification, it can return every unused domain value as well as absent recorded values. Database query safety and domain independence of a database query formalize why a finite input must not silently turn such a query into an unbounded enumeration. A finite candidate relation $C$ gives the safer intended query $\{x:C(x)\wedge \neg P(x)\}$.

A data type with exactly one value. In Haskell, both the data type and its sole value are written (); it is often the result of a computation whose purpose is an effect.

A relation between abstract and concrete states used to prove data refinement. Establish it initially; show that each concrete operation simulates the specified abstract operation and restores the relation; show that returned observations agree. For a rank collection, $u[\mathrm{0..}n)=\mathrm{sort}(xs)$ and $n=\mathrm{length}(xs)$ justify constant-time indexed queries on a sorted array.

A data-link frame is a delimited unit of link-level transmission containing data and any associated control or checking fields.

Framing identifies the boundaries of data-link frames in a transmitted stream, allowing a receiver to locate separately interpretable link-level units.

A younger instruction needs a value produced by an older instruction before the ordinary writeback/read path would deliver it. Register forwarding or a pipeline stall can enforce the true dependence.

A d-ary heap in which each parent key is no greater than each child key. Transitivity makes the root a minimum. Removing the root requires choosing a smallest child at each downward step, taking $O(d{\log }_{d}n)$ time in the usual $2\le d\le n$ range. Decreasing a key or inserting a new leaf only compares along one chain of parents, taking $O({\log }_{d}n)$ time in that range. These priority queue bounds assume constant-time key comparisons and available array capacity. An indexed priority queue must also update its item-to-index table after every swap.

A cup applied to the skin with reduced internal pressure. Ancient bronze and horn versions are described by Celsus. ^[ref]

A critical point whose Hessian matrix is a singular matrix. The second-derivative test does not completely classify such a point. Higher-order terms can produce a local minimum, a local maximum, or a saddle point of a scalar function.