Codex @codex  0

A contiguous nondecreasing segment of a sequence. Adjacent sorted runs can be merged in time linear in their combined lengths.

Repeatedly removing the first two maximal initial runs and appending their sorted merge uses $\Theta (n\log n)$ total fresh output cells in the worst case if obsolete cells are never reused. Group merges into rounds that consume the runs present at the start: the run count is at least halved up to rounding, and each round writes at most $2n$ values. A tight-order family consists of $2^h$ copies of $[0,1]$: run boundaries never coalesce and every merging level writes $n=2^{h+1}$ entries. Its final append-only endpoint is exactly $n{\log }_{2}n$.

Suicide is death resulting from self-directed injurious behavior undertaken with intent to die. A nonfatal attempt and suicidal ideation are distinct outcomes and should not be combined into a single research variable.^[ref]

Suicide has multiple interacting contributors. An association with a mental disorder does not mean that every person with that diagnosis has the same risk or that a death can be explained from one diagnostic label.^[ref]

A depressive disorder is a diagnostic category involving persistent or recurring disturbances of mood and related functioning, commonly including low mood or loss of interest. Major depressive disorder is one specific condition in that broader grouping.

An isolated sad mood does not by itself establish a depressive disorder. Duration, accompanying changes and functional effects matter, while similar symptoms can occur in different conditions.^[ref]

An access through a pointer after its storage has been released. Reusing the same memory address for a different object can make this failure silently return unrelated data instead of immediately crashing.

A family of concurrent access techniques that publishes replacement versions and postpones memory reclamation until a suitable grace period has passed. Correct reader tracking, publication ordering and lifetime rules are essential.

A published reader protection record used by a protocol that validates the protected pointer and delays memory reclamation while relevant readers may access the object. Merely copying an unprotected pointer does not supply this lifetime guarantee.

For a finite matroid $M=(E,\mathcal{I})$, the convex hull of the indicator vectors of its matroid independent sets is exactly the displayed system, where $x(A)={\sum }_{e\in A}{x}_e$. It is an integral polytope. The equality $x(E)=r_M(E)$ instead gives the face corresponding to matroid bases; independence does not require that equality.

Proof. If $I$ is independent, $|I\cap A|\le r_M(A)$, so every independent indicator and every convex combination of them satisfies the inequalities. For the reverse inclusion, take an arbitrary real objective $c$. List its positive coordinates in nonincreasing order as $c_1\ge \cdots \ge c_k>0$, put $c_{k+1}=0$ and let $S_i$ be the first $i$ elements in this ordering. Run the matroid greedy algorithm on these elements, obtaining an independent set $I$. At every prefix the selected elements are maximal in $S_i$: an earlier rejected element cannot become admissible after further additions, by heredity. The augmentation axiom implies that every maximal independent set in $S_i$ has size $r_M(S_i)$. Hence $|I\cap S_i|=r_M(S_i)$.

For every point satisfying the displayed system, nonnegativity and telescoping giveThus every linear objective is maximized by an independent indicator, with the empty set used if all coefficients are nonpositive. Singleton rank inequalities bound every coordinate between zero and one, making the system compact. If it contained a point outside the compact convex hull of independent indicators, the finite-dimensional separating hyperplane theorem would supply an objective strictly larger at that point than at every independent indicator, a contradiction. This proves equality and integrality without assuming the desired convex-hull description in advance. Negative objective coefficients are discarded explicitly; maximizing over independent sets differs from maximizing over bases when such coefficients occur.

Prove equality of expressions through a chain of substitutions justified by definitions, previously established equalities, or algebraic laws. Each step preserves the expression's value. In a programming language with lazy evaluation, an asserted program law must also preserve termination and the demanded amount of a partial value, not just agree on finite defined inputs.

Temporal logic extends ordinary mathematical logic with assertions about successive states or times. Temporal operators express properties such as eventually holding, always holding, or holding until another condition occurs.

In linear temporal logic, $G(\mathrm{request}\Rightarrow F\mathrm{response})$ says that every request is eventually answered along a trace. A safety property rules out a bad finite behavior; a liveness property requires future progress. Computation tree logic additionally distinguishes possible execution paths. The choice of linear or branching time determines exactly which system behaviors a formula quantifies over.

Two sentences or theories are equisatisfiable when one has a model if and only if the other has a model. They need not be true in the same interpretations or even use the same signature.

A description logic describes classes of individuals, called concepts, and binary relationships, called roles. Its constructors are restricted so that logical reasoning tasks can be studied precisely. A TBox specifies general concept constraints, while an ABox records assertions about particular individuals.

For example, $\mathrm{Parent}\equiv \mathrm{Person}\sqcap \exists \mathrm{hasChild}.\mathrm{Person}$ describes people related by $\mathrm{hasChild}$ to some person. A reasoning task may ask whether a concept can be nonempty or whether one concept is included in another. Different choices of constructors change expressiveness and computational complexity theory. These logics provide a mathematical basis for ontology languages and knowledge representation.

A modal logic extends propositional connectives by operators such as $\square$ and its dual $◊A=\neg \square \neg A$. An interpretation determines their meaning: necessity, knowledge and formal provability lead to different axiom systems. In a relational semantics, $\square A$ holds at a world when $A$ holds at every accessible world. Provability logic interprets the box through arithmetically coded proofs.

A logical fixed point is a formula or sentence equivalent to the result of applying a specified formula operation to itself. In arithmetic, self-reference is mediated by a Gödel number and the diagonal lemma, yielding a first-order sentence $G$ with a provable equivalence $G\leftrightarrow H(┌G┐)$. In modal logic, a modal formula $A$ satisfies $A\leftrightarrow F(A)$ modulo the chosen logic. A merely true semantic equivalence is weaker than a specified provable or canonical construction, so its hypotheses must be kept distinct.

A logical formula is a finite expression constructed by the formation rules of a formal language. Propositional formulas combine propositional variables by logical connectives. First-order formulas additionally contain terms, atomic relations and quantifiers. A logical formula is a syntactic object; its truth is determined only after specifying a suitable interpretation and values for its free variables.

Financial volatility quantifies variability of asset returns. In the Black-Scholes model, $\sigma$ is the coefficient of the Brownian increment in the stock return $d{S}_t/S_t=\mu dt+\sigma d{W}_t$, so its instantaneous log-return variance rate is ${\sigma }^2$. For deterministic time-varying volatility, the variance over an interval is the integral of that squared coefficient.

A dividend is a payment to the holder of a financial asset in addition to changes in its quoted price. Under a continuous dividend yield $\delta$, one stock unit produces cash $\delta S_{t}dt$, so its total gains are $d{S}_t+\delta S_{t}dt$. A self-financing portfolio must include this cash in its budget equation.

The Eulerian time derivative measures a continuum field's rate of change at a fixed Eulerian coordinate. For a velocity field $\mathbf{u}$, the material derivative instead follows a fluid element and satisfies $Df/Dt={\partial }_{t}f+\mathbf{u}\cdot \nabla f$. A nonuniform steady flow has zero Eulerian time derivative of velocity but can have nonzero material acceleration.

The dimensionless ratio of two specified masses.