For a linear operator $T$ on an $n$-dimensional vector space over a field, the polynomials annihilating a fixed vector $v$ form an ideal of $K[X]$. It is nonzero because $v,Tv,\dots ,T^{n}v$ are linearly dependent. Since $K[X]$ is a principal ideal domain, its unique monic generator $m_v$ exists and has degree at most $n$. It divides $f$ exactly when $f(T)v=0$. For $v=0$ it equals one. Its degree is the dimension of the cyclic subspace generated by $v$: the vectors up to $T^{d-1}v$ are independent by minimality and span an invariant space by the annihilating recurrence.

For a linear operator on a finite-dimensional vector space, an eigenvector gives $m_T(T)v=m_T(\lambda )v=0$, proving the forward implication from eigenvalue to root. Conversely write $m_T(X)=(X-\lambda )q(X)$. If $T-\lambda I$ were invertible, the equation $(T-\lambda I)q(T)=0$ would imply $q(T)=0$, contradicting minimal degree. Thus $T-\lambda I$ has a nonzero kernel of a linear map. The equivalence concerns roots in the scalar field; roots in an extension describe eigenvalues after extending scalars.

For a linear operator on a finite-dimensional vector space over a field $K$, all polynomial kernels stabilize at their first power exactly when its minimal polynomial $m$ has no repeated irreducible polynomial factor in $K[X]$. This formulation does not require those factors to be linear.

To prove it, the Bézout's identity for polynomials gives $\ker q(T)=\ker \gcd (q,m)(T)$: a greatest common divisor is a combination of $q$ and $m$, while $q$ is its multiple. If $m$ is square-free, $\gcd (p^n,m)=\gcd (p,m)$, proving the displayed equality. Conversely, if an irreducible factor $f$ occurs to exponent $e\ge 2$, its primary decomposition theorem summand has minimal polynomial $f^e$. On this summand $f(T)$ is nonzero, while $f(T{)}^e=0$. Thus the kernels of those two powers differ. An eigenbasis follows from this criterion precisely when the distinct factors of $m$ are also linear over $K$; a real rotation through a right angle has square-free $m=X^2+1$ but no real eigenvalues.

A finite conjunction from the signed atomic diagram of $A$ becomes, after existentially closing its new names, a sentence in its existential theory. Joint satisfiability with $T$ therefore makes the full diagram finitely satisfiable with $T$. Compactness gives the embedding, with inequalities ensuring injectivity and signed atomic formulas ensuring both preservation and reflection of relations as required for a structure embedding. The reverse implication is preservation of existential truth under embeddings.

A communication object with an output port used to send values and an input port used to receive them. Its contract specifies ordering, capacity, sharing and closure. In synchronous message passing, a send completes through a matching receive; a buffered message-passing channel can retain accepted values until a receiver is ready.

A periodic indication that a peer is still communicating even when it has no application data to send. Absence beyond a deadline can suggest a failure but does not distinguish a crash from excessive delay.

Sharing one message-passing channel among multiple logical streams by attaching a stream identity to each transferred value. Independent cancellation requires a protocol that conveys which stream is ending; bounded per-stream flow control can prevent head-of-line blocking from a single unready consumer.

A message-passing channel carrying protocol instructions, such as cancellation or permission to send, rather than ordinary application payloads. A reverse control channel can communicate consumer closure back to a remote producer.

The cork cambium is a lateral meristem producing protective outer tissues during secondary growth. Continued production of new surface layers helps accommodate enlargement of stems and roots. It differs in position and products from the vascular cambium.

The +2 oxidation state of mercury. The actual aqueous species can depend on pH, hydrolysis and ligands; a bare ionic formula is an analytical shorthand.

The mercury(I) chloride historically called calomel. Reduction can give metallic mercury.

The paired cation $\mathrm{H}{\mathrm{g}}_2^{2+}$, containing a mercury–mercury bond and average oxidation state +1.

A mercury compound contains the chemical element mercury bonded or combined with other elements. The name refers to elemental chemistry, independently of the planet Mercury. Formula and oxidation state matter: mercury(I) ion contains a bonded pair $\mathrm{H}{\mathrm{g}}_2^{2+}$, whereas mercury(II) denotes the +2 oxidation state.

Consequently a common anion need not give the same stoichiometry in the two cases. Combining $\mathrm{H}{\mathrm{g}}^{2+}$ with sulfate, $\mathrm{S}{\mathrm{O}}_4^{2-}$, gives mercury(II) sulfate, $\mathrm{HgS}{\mathrm{O}}_4$, while the dimeric mercury(I) cation requires a different formula. These charge balances identify a composition; they do not by themselves determine the structure or aqueous speciation.

An operation constraining specified memory-access orderings. The strength and scope of a memory barrier matter; it does not by itself provide mutual exclusion or safe memory reclamation.

An architectural memory model preserving store-to-store and load-to-load ordering while allowing a later load to a different address to bypass a locally buffered store. This permits the two-zero store-buffering outcome while requiring no extra hardware fence for simple immutable pointer publication.

A value loaded by one instruction directly contributes, through register computation, to a later memory access's memory address. Loading a pointer and then reading fields through that saved pointer is a natural example. This differs from a control dependency.

A memory operation's execution is conditional on a preceding value. A control dependency alone does not generally order two reads on POWER architecture or the relevant ARM memory models; it must not be substituted for an address dependency in pointer-publication reasoning.

For a measure space $(\Omega ,\mathcal{F},\mu )$ and $E\in \mathcal{F}$, the restriction of a measure to a measurable subset is the measure on $(E,\mathcal{F}{|}_E)$ given by the displayed formula. The trace sigma-algebra consists of the measurable sets contained in $E$. The empty-set axiom and countable additivity are inherited directly from $\mu$. An associated measure on the original sigma-algebra is $\nu (A)=\mu (A\cap E)$ for $A\in \mathcal{F}$; the two constructions have different underlying measurable spaces.

For measurable sets $A\subseteq B$, the disjoint union $B=A\bigsqcup (B\setminus A)$ and finite additivity give $\mu (B)=\mu (A)+\mu (B\setminus A)\ge \mu (A)$. This uses nonnegativity and is valid with infinite values; it does not subtract infinities. Monotonicity of a measure implies that increasing measurable sets have nondecreasing measures, but the equality of their limit with the measure of their union uses the stronger axiom of countable additivity.

A Lebesgue measurable function $f:\mathbb{R}\to \mathbb{R}$ has a Lebesgue measurable set as the preimage of every open subset of $\mathbb{R}$. Equivalently, use every Borel set in the target: the sets whose preimages are measurable form a sigma-algebra containing all open sets. The source carries the Lebesgue sigma-algebra and the target its Borel sigma-algebra. A Borel measurable function on $\mathbb{R}$ instead requires the preimages to be Borel in the source as well. Restricting the source to a Lebesgue measurable set uses its induced sigma-algebra.