For a differentiable real function $g$, a point of convexity is a point $v_0$ whose tangent line lies below the entire graph. The function need not be convex at other points. The Weierstrass excess function is exactly the difference between the function and this tangent line when applied to a variational integrand as a function of slope.

For a globally Lipschitz nonlinearity of constant $ϵ$, the map taking $v$ to the zero-Dirichlet Poisson solution with load $f(v)$ is a contraction in the gradient norm if the displayed inequality holds. Poincaré gives the factor $C_P$ from the Sobolev input to $L^2$, and testing the linear Poisson equation gives another factor $C_P$ back to the gradient norm. The condition yields a unique weak solution without requiring $f(0)=0$. With the optimal Poincaré constant it reads $ϵ<{\lambda }_1$.

Suppose all distances lie in $[0,D]$, $\psi$ is continuous, $0<\psi (r)<r$ for positive $r$, and ${\lim }_{r\to 0}\psi (r)/r<1$. The ratio is uniformly bounded below one near zero, and its maximum on the remaining compact scalar interval is also below one. Thus one constant $q<1$ works on the entire bounded set. If the set is complete and the map preserves it, Banach's theorem gives a unique fixed point. Compactness of the set itself is not required.

If the continuous dual space $X^{\prime }$ of a normed vector space is norm separable, then $X$ is separable. Choose a dense sequence $f_n$ in the dual unit ball and unit-ball points $x_n$ with $|f_n(x_n)|\ge \Vert f_n\Vert /2$. If their closed linear span were proper, the Hahn-Banach distance formula would give a norm-one functional $f$ vanishing on it. An $f_n$ within $1/4$ of $f$ would simultaneously satisfy $|f_n(x_n)|<1/4$ and $|f_n(x_n)|>3/8$, a contradiction. Rational linear combinations give a countable dense subset of $X$.

The converse fails by duality of l1 and l infinity: ${\ell }^1$ is separable, whereas ${\ell }^{\infty }$ contains uncountably many zero-one sequences at mutual distance one.

A context-free language recognized by scanning a string over the alphabet $\{0,1\}$ from right to left, counting 0 as $+1$ and 1 as $-1$, and requiring every partial count to remain nonnegative. A context-free grammar with a unique parse tree for each generated string is $S\to ϵ\mid S0\mid S1B0$, $B\to ϵ\mid B1B0$. To see uniqueness, reverse the strings: every nonempty nonnegative path either never returns to zero after its first step, or has a unique first return. These two cases determine the production and its component substrings.

A point of the Euclidean plane is constructible if it can be obtained from $(0,0)$ and $(1,0)$ by finitely many straightedge-and-compass operations: draw a line through two obtained distinct points, draw a circle with obtained centre and radius equal to an obtained segment length, and retain isolated intersection points. Its two coordinates are constructible numbers. Conversely, constructible real coordinates can be placed on perpendicular axes and their coordinate lines intersected, giving the point. Coordinates created at each isolated intersection require only solving linear or quadratic equations over the current coordinate field; hence they lie in a real field extension obtained by a finite tower of quadratic extensions. Coincident curves supply no isolated new intersection points.

The critical cone consists of linearized feasible directions along which the objective function has zero first-order change. For constraints $c_i\ge 0$, its directions satisfy $D{c}_{i}d\ge 0$ at active constraints and $\nabla f^{T}d=0$; equality constraints impose zero linearized residual. Under the usual regularity conditions, second-order Karush-Kuhn-Tucker conditions test the optimization Lagrangian Hessian matrix on this cone. Positive active multipliers force their corresponding directional residuals to vanish.

An L1 penalty can enforce a constraint exactly at a finite weight, rather than only in an infinite-weight limit. For a convex equality-constrained problem with an optimal multiplier $y_{\ast }$ and optimization Lagrangian $f-y^{T}c$, the lower bound $f(x)-y_{\ast }^{T}c(x)\ge f(x_{\ast })$ givesThus $\rho >\Vert y_{\ast }{\Vert }_{\infty }$ makes every penalty minimizer feasible and optimal. Local versions require appropriate regularity and second-order assumptions. The nonsquared norm is essential to this finite-threshold property; a quadratic penalty method usually approaches feasibility only as its weight diverges.

Sequential quadratic programming computes steps from a quadratic model of the optimization Lagrangian with linearized constraints. Its constrained Newton system couples a primal step and multiplier estimates. A line search or trust region then controls the use of that local model.

For $\dot{N}=N[P(N)-E]$ and $P(N)={\prod }_{i=1}^3(a_i-N)$ with $0<a_1<a_2<a_3$, extinction changes stability at $E_T$, while the upper stable population and an unstable separator merge at $E_M$. If $E_T<E_M$, an interval of efforts supports stable extinction and a stable positive population simultaneously; at $E_M$ the positive state collapses. If the thresholds have the opposite order, the high-population pair disappears before the lower stable branch reaches extinction. At a positive equilibrium stability follows from the sign of $N{P}^{\prime }(N)$.

A tendon is dense connective tissue transmitting force from muscle to an attachment, commonly a bone. Parallel organization of its matrix supports tensile loading. Elasticity and geometry affect force transmission, so a tendon is not an absolutely rigid connector.

The periosteum is the connective-tissue covering of the external bone surface except at appropriate articular regions. It supports vascular supply, attachment and growth or repair processes. It is distinct from the inner marrow cavity and the cartilage on joint surfaces.

Fascia is connective tissue forming sheets, envelopes or partitions around and between anatomical structures. It organizes movement, force transmission and pathways for vessels and nerves. Naming a fascial plane identifies a spatial relationship rather than an empty gap between unrelated organs.

An elastic fibre is an extracellular structural element that stretches and recoils. Networks of these fibers contribute reversible deformation to tissues such as vessel walls. Their recoil role differs from the high tensile restraint supplied by many collagen-rich structures.

Use the arcs and points of the noncompact counterexample to the Šura-Bura lemma, and add the central ray $\{(1/2,t):t\ge 1\}$ to form $X$. Define $C_n=\{p,q\}\cup {\bigcup }_{m\ge n}{A}_m\cup \{(1/2,t):t\ge n\}$. Each graph tail is connected because its arcs meet the ray. Its relative closure in $X$ adds just $p,q$, so each $C_n$ is closed and connected. These sets decrease and have intersection $\{p,q\}$, which is compact and disconnected. The ambient $X=C_1$ is connected and Hausdorff but noncompact.

A finite family of nonempty connected subgraphs of a tree with pairwise nonempty intersections has a common vertex. For a finite ambient tree, prove this by deleting a leaf of a tree $v$. If one member is $\{v\}$, pairwise intersection proves the result immediately. Otherwise remove $v$ from every member containing it. Each such member remains nonempty and connected and contains the unique neighbour of $v$. Hence every pair still intersects, even if its original intersection consisted only of $v$, and mathematical induction applies. This is the Helly property specialized to trees.

Two disjoint connected sets can have a connected union when one contains a contact point of the other's closure. To prove the displayed criterion, suppose the union had a separation into disjoint nonempty relatively open sets $P,Q$. Each connected set lies wholly on one side; if both lie on the same side, the other side is empty. Otherwise assume $A\subset P$ and $B\subset Q$. Choose $b\in B\cap \overline{A}$. Since $Q$ is relatively open in the union and contains $b$, some ambient open set containing $b$ meets the union only in $Q$. It must nevertheless meet $A$ by the definition of closure, a contradiction.

For example, the open unit disc centred at zero and the closed unit disc centred at two are disjoint, but their union contains the common contact point one through the second disc and is connected. In this example it is even path-connected: the real segment from zero to one lies in the first disc except for its endpoint, and that endpoint belongs to the second disc. Omitting the contact point from both discs instead gives a disconnected union.

The quasicomponent of a point is the intersection of all its clopen neighborhoods. Every connected set containing the point lies in its quasicomponent, since a clopen set cannot split a connected set. In a compact Hausdorff space, quasicomponents equal connected components by the Šura-Bura component lemma.

The quasi-component of a point is the intersection of its clopen neighborhoods. Its connected component is always contained in it: a connected set meeting a clopen set cannot also meet its complement. Equality need not hold in noncompact Hausdorff spaces. The Šura-Bura lemma proves equality in compact Hausdorff spaces.

Every connected component $C$ of a nonempty compact proper subset $D$ of a connected Hausdorff space meets its boundary. If not, the Šura-Bura lemma identifies $C$ as a quasi-component in the compact Hausdorff subspace $D$. Finitely many clopen neighborhoods separate it from the compact boundary, giving a nonempty relatively clopen $H\subseteq \mathrm{int}D$. Such $H$ is open in the ambient space, and is also closed there because $D$ is closed. It is proper because $D$ is proper. This contradicts connectedness.