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A quotient operator is a surjective bounded linear operator that is an open map. It is equivalent to requiring a ball about zero in the image of the open unit ball. Translation and scaling then prove being an open map at every point. For a Banach space domain, a bounded linear operator is a quotient operator exactly when its Banach-space adjoint is an isomorphic embedding: a lower bound on the adjoint gives approximate ball-surjectivity by Hahn-Banach separation, and successive approximation for quotient operators gives exact preimages.

Take $X=[0,1]$ with its usual topology and $Y$ the Sierpiński space with open singleton $\{1\}$. The map $f(0)=0$, $f(x)=1$ for $x>0$ is continuous, since $f^{-1}(\{1\})=(0,1]$ is relatively open in $X$. The sets $K_n=[0,1/n]$ are nonempty, nested and compact, butThe fiber of $1$ is not closed, which is exactly where the compact-intersection proof fails. This distinguishes the valid Hausdorff-codomain theorem from a false extension to arbitrary codomains.

For a stock with constant continuous dividend yield $\delta$, holding one fixed share and depositing each dividend in a bank account at rate $r$ gives the displayed wealth process. Since $d{A}_t=r{A}_{t}dt+\delta S_{t}dt$, this is a self-financing portfolio: its only cash inflow to the bank is the dividend earned by its stock holding. A claim paying $n{X}_T$ is replicated by $n$ copies and costs $n{X}_t$, including the known accrued-dividend balance at time $t$.

A different claim paying only $n{S}_T$ costs $n{S}_t{e}^{-\delta (T-t)}$ in the Black-Scholes model with dividends. It is replicated by $n{e}^{-\delta (T-t)}$ shares with dividends reinvested in new shares: the share count grows at rate $\delta$ and reaches $n$ at expiry. Thus banking dividends and reinvesting them produce distinct terminal payoffs and distinct current values, even though both strategies are self-financing.

For smooth inequalities $c_j\ge 0$, the displayed intersection of half-spaces contains the contingent tangent cone. This follows by differentiating each active feasibility inequality along a feasible tangent sequence. The linear independence constraint qualification, and more generally the Mangasarian–Fromovitz constraint qualification, ensure equality in the inequality-only setting. Without a qualification the inclusion can be strict: for $c(x)=-x^2\ge 0$, the feasible set is $\{0\}$ and its tangent cone is $\{0\}$, while $c^{\prime }(0)=0$ makes the linearized cone the whole real line.

An operation associated with a context-free grammar production, performed when that production is recognized. Semantic actions can construct an abstract syntax tree by combining the trees attached to its child nonterminal symbols.

A context-free grammar is ambiguous when some accepted token sequence has two distinct parse trees. Giving a language an unambiguous context-free grammar requires specifying grouping in the productions themselves, rather than relying on an unstated parser conflict rule.

Store the start nonterminal symbol on a stack. An epsilon transition nondeterministically replaces its top nonterminal symbol by a production rule right-hand side. A top terminal symbol can be removed only while consuming the same input symbol. With a protected stack bottom marker, enter a final control state when that marker is exposed. Requiring all input to have been read implements acceptance by final state. The stack records the unprocessed suffix of a leftmost derivation, proving equality of the recognized and generated formal languages.

A contact line is pinned when its position remains fixed while the surrounding drop shape or contact angle changes. Surface heterogeneity or deposited material can sustain such a state over a range of angles. Evaporation from a pinned drop then requires compensating liquid motion and can generate outward solute transport associated with the coffee-ring effect. A model with a freely receding line and fixed contact angle has a different boundary constraint and need not predict the same flow direction.

A rooted tree whose leaves describe primitive regions and whose internal nodes describe Boolean operations. A half-space leaf can hold an implicit equation and an inside-sign convention. Evaluate membership from the leaves toward the root; interval arithmetic can propagate conservative inside/outside bounds over a box instead of a single point.

A constructible line is a line through two distinct constructible points. If the available coordinates lie in a real field $F$, a line through $(a,b)$ and $(c,d)$ has equation $(d-b)(x-a)-(c-a)(y-b)=0$, with coefficients in $F$ and a nonzero normal vector. Intersecting two distinct such lines introduces no new field elements, because solving a nonsingular pair of linear equations uses only field operations.

A constructible circle has a constructible point as its centre and radius equal to the distance between two constructible points. When those coordinates lie in a real field $F$, its equation $(x-a{)}^2+(y-b{)}^2=r^2$ has coefficients in $F$ because the squared radius is a sum of squared coordinate differences. Intersecting it with a constructible line reduces to a quadratic equation. Intersecting two distinct circles reduces to the same situation by subtracting their normalized equations to obtain the radical axis. Thus each isolated real intersection requires at most a real quadratic field extension.

The hypergraph whose vertices are the variables of a constraint satisfaction problem and whose hypergraph edges are its constraint scopes. Several constraints can have the same scope. An incidence drawing uses extra scope nodes for display; these do not become extra variables of the original problem. Ordinary treewidth of a constraint hypergraph refers to treewidth of its primal graph, equivalently to a tree decomposition covering each whole scope, not to generalized hypergraph width measures.

A collection of propositional formulas is consistent if it does not derive a contradiction. Equivalently, in a standard classical calculus, it does not derive both a logical formula and its negation.

For a nontrivial connected graph, return the internal vertices of a rooted tree in a depth-first search tree. Undirected depth-first search has only ancestor-descendant edges, so every edge meets an internal vertex; the internal vertices are connected along parent paths. For a maximum-size proper matching in a rooted tree $M$ of that tree, existence of which is assumed, $|I|=|V|-|L|\le |V|-|U(M)|=2|M|$. Every vertex cover has size at least $|M|$, proving the factor two. The implementation needs only depth-first search, hence linear time complexity; it does not compute the analysis matching in a graph.

The torus and Klein bottle are not homeomorphic because only the former is orientable. Adding a projective plane to either by connected sum of surfaces makes a nonorientable surface with Euler characteristic minus one. The classification of compact surfaces identifies both with three projective planes. Thus connected-sum cancellation can fail even for closed connected surfaces.

For an infinite set $X$ with its cofinite topology, every infinite subset is connected, but $X\times X$ has an infinite disconnected topological subspace. Choose $a\ne b$ and use $D=\{a,b\}\times X$. The open sets $(X\setminus \{b\})\times X$ and $(X\setminus \{a\})\times X$ have disjoint nonempty traces $\{a\}\times X$ and $\{b\}\times X$ on $D$, covering it. Thus $D$ is a product of a discrete two-point subspace with an infinite cofinite factor. The ambient open rectangles may intersect; only their traces must be disjoint for a separation of $D$.

For a normal subgroup $K$ of $G$, each $c_h:k\mapsto h^{-1}kh$ is a group automorphism of $K$. Its inverse is $c_{h^{-1}}$, but the dependence on $h$ is a group antihomomorphism under ordinary function composition, because $(gh{)}^{-1}k(gh)=h^{-1}(g^{-1}kg)h$. The assignment $h\mapsto (k\mapsto hk{h}^{-1})$ instead is a group homomorphism to the automorphism group. Equivalently the first convention is a right group action, while the second is a left group action.

Assume the Strengthened Legendre condition $P>0$ and let $q(\phi )={\int }_a^b(P{\phi }^{\prime 2}+Q{\phi }^2)$. Then $q$ is strictly positive on all nonzero fixed-endpoint Lipschitz variations exactly when the normalized Jacobi equation solution with $w(a)=0$, $P(a)w^{\prime }(a)=1$ has no zero in $(a,b]$. A zero at $c$ gives a nonzero variation equal to $w$ up to $c$ and zero thereafter, with $q=0$. Conversely, if no zero occurs, the Jacobi identity for the second variation writes $q$ as a nonnegative square. Equality forces $\phi =Cw$, and the endpoint at $b$ forces $C=0$.

Nonzero vectors are conjugate directions with respect to a symmetric positive-definite matrix $A$ when they are pairwise orthogonal in the inner product $⟨u,v{⟩}_A=u^{T}Av$. They need not be orthogonal in the ordinary Euclidean inner product. They are linearly independent: multiplying a vanishing linear combination by $p_j^{T}A$ isolates its $j$th coefficient times $p_j^{T}A{p}_j>0$.

For a symmetric positive-definite matrix, choose an orthonormal eigenbasis and write the exact solution as $x={\sum }_i{\alpha }_i{v}_i$. Starting at zero, the conjugate gradient method chooses an admissible polynomial $p(0)=1$ of degree at most the iteration count that minimizes the displayed error. The weights depend on both the eigenvalues and the solution coefficients. For two eigenvalues $1,L$, the first polynomial is $1-\eta t$, with $\eta =({\alpha }^2+{\beta }^2{L}^2)/({\alpha }^2+{\beta }^2{L}^3)$. Comparable nonzero coefficients and large $L$ give $\eta \approx 1/L$, so the high-eigenvalue error is removed first. A sufficiently tiny high-mode coefficient changes this behavior; the spectrum alone does not determine the first step.