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A mixed boundary condition imposes different kinds of boundary conditions on different portions of the boundary. A common elliptic example combines a Dirichlet boundary condition on one portion with a Neumann boundary condition on the remainder.

A Neumann boundary condition prescribes the outward normal derivative of the unknown function on the boundary.

A Dirichlet boundary condition prescribes the value of the unknown function on the boundary.

An ideal energy measurement returns an eigenvalue of the Hamiltonian with probability equal to the squared norm of the state's projection onto its eigenspace. After obtaining a nondegenerate energy, the state is the corresponding energy eigenstate.

A Boolean function $f:\{0,1{\}}^n\to \{0,1\}$ is $(\epsilon ,p,r)$-quasirandom when conditioning any set of at most $r$ coordinates to any values changes its ${\mu }_p$-expectation by at most $\epsilon$.

A $J$-junta is a function whose value depends only on coordinates indexed by $J$. A $k$-junta depends on at most $k$ coordinates.

For $f:\{-1,1{\}}^n\to \mathbb{R}$, the characters ${\chi }_S(x)={\prod }_{i\in S}{x}_i$ form an orthonormal basis, and the Fourier-Walsh expansion is

On the finite Boolean hypercube, these coefficients are the Hadamard transform of the vector of function values, with expectation normalization.

A Boolean function $f:\{0,1{\}}^n\to \{0,1\}$ is a monotone function when $x_i\le y_i$ for every $i$ implies $f(x)\le f(y)$.

For increasing events $A,B$ in a product percolation measure, their disjoint occurrence satisfies $\mathbb{P}(A\square B)\le \mathbb{P}(A)\mathbb{P}(B)$.

For independent bond percolation on the square lattice at $p=1/2$, the Russo-Seymour-Welsh theorem gives nontrivial crossings at every scale. For each fixed aspect ratio $\rho \ge 1$, there is $c_{\rho }>0$ such thatwhere $h_p(m,n)$ is the left-to-right rectangle crossing in bond percolation probability. The constant can depend on $\rho$, but not on the rectangle's size.

Reflection extensions and transverse crossings in overlapping rectangles permit longer crossings to be assembled from square crossings. The Harris-FKG inequality controls intersections of these increasing events; dual bond percolation supplies the complementary crossing bound. These are the geometric ingredients of the original square-lattice crossing analysis.^[ref]

For example, a rectangle twice as wide as it is high retains a positive crossing probability as both dimensions grow. The bounds imply positive probabilities for open or closed circuits in fixed-shape annuli and support the one-arm extension estimate. Their critical-parameter assumption is essential: a corresponding uniform lower bound would contradict exponential decay of subcritical percolation.

The percolation susceptibility is the expected size of the open cluster containing a specified root:

If a bounded real sequence has the property that every convergent subsequence converges to the same number $L$, then the full sequence converges to $L$. Otherwise, a subsequence stays at least some fixed distance from $L$; Bolzano--Weierstrass gives it a convergent subsubsequence, contradicting the assumed uniqueness.

If $I_1\supseteq I_2\supseteq \cdots$ are nonempty closed intervals with finite endpoints, then at least one real number belongs to every member. If their lengths tend to zero, exactly one real number belongs to every member. The result turns an unlimited sequence of consistent enclosures into an existence guarantee.

Write $I_n=[a_n,b_n]$. Nesting makes the left endpoints nondecreasing and the right endpoints nonincreasing. Their bounds give finite supremum and infimum, withIndeed, every $b_n$ is an upper bound for all the $a_j$, so the least-upper-bound property gives ${\sup }_j{a}_j\le b_n$ for every $n$. This proves the displayed interval is nonempty. If $b_n-a_n\to 0$, its two endpoints must coincide. This is the connection to the completeness of the real numbers discussed in Günther Hörmann and David Langer, Einführung in die Analysis, §§3.15–3.16: nested intervals and order completeness.

The distinction between existence and uniqueness is useful in computation. If an unknown point $x_{\ast }$ lies in every $[a_n,b_n]$, the midpoint $m_n=(a_n+b_n)/2$ satisfiesThus the enclosure itself certifies an absolute error without knowing the answer. The bisection error bound is the special case where each update halves the width.

Closedness and a finite initial enclosure matter. The open intervals $(0,1/n)$ have no common point, while the closed but unbounded intervals $[n,\infty )$ also have empty intersection. A picture of a few intervals cannot certify the intersection of an infinite family; the hypotheses of the nested interval theorem do that work.

A thin Bingham layer of thickness $h$ between a stationary wall and a moving body gives $u_b=(h/\mu )(|{\tau }_b|-{\tau }_c{)}_{+}\mathrm{sgn}{\tau }_b$.

Consider a point mass $m>0$ in the central potentialHere $r_0$ is a positive length offset, and $\ell =m{r}^2\dot{\theta }\ne 0$ is the conserved angular momentum. With reciprocal radius $u=1/r$, the Binet equation becomeswhere primes denote derivatives with respect to the orbital angle $\theta$. A circular orbit requires $K=u_c(1-r_0{u}_c{)}^2$, with $0<u_c<1/r_0$. The right side has its maximum $4/(27r_0)$ at $u_c=1/(3r_0)$. Thus there are two circular orbits for $0<K<4/(27r_0)$ and none above that maximum.

Writing $u=u_c+\delta u$ and retaining linear terms givesThe outer branch $r_c>3r_0$ is radially stable at fixed angular momentum, while $r_0<r_c<3r_0$ is unstable. At the meeting point $r_c=3r_0$ the linear restoring coefficient vanishes, so the linear test is degenerate. Stating the sign and domain assumptions is essential to this branch classification.

For a stationary disk of radius $a$ in the far-field shear $u_{\infty }=\gamma y{e}_x$, using $u_r={\psi }_{\theta }/r$ and $u_{\theta }=-{\psi }_r$, the exterior solution isIt satisfies no slip at $r=a$ and approaches the imposed shear in velocity.

For $\psi =r^{2}f(\theta )$, biharmonicity gives
$f^{(4)}+4f^{\prime \prime }=0$. In the wedge $-\alpha <\theta <0$, no slip at the lower wall and no penetration plus tangential stress $S$ at the upper surface giveThe resulting surface speed is

For two plane curves $C=V(f)$ and $D=V(g)$ meeting only at an isolated point $p$ locally on a smooth surface over an algebraically closed field $k$, their intersection multiplicity isIn local plane coordinates centered at $p$, this is also ${\dim }_{k}k[[x,y]]/(f,g)$. The quotient is finite-dimensional because the local intersection is isolated. This length counts contact that the underlying point set alone cannot detect.^[ref]

The coordinate axes have quotient $k[[x,y]]/(x,y)\cong k$ and multiplicity one. The curves $y=0$ and $y=x^m$, $m\ge 1$, instead givewith basis $1,x,\dots ,x^{m-1}$, hence multiplicity $m$. For $m\ge 2$ the higher value records tangency. If the curves share a local component, this quotient has infinite length and the isolated-intersection formula does not apply. Summing the local multiplicities for projective plane curves without a common component gives the product of degrees in Bézout's theorem.

A projective plane curve is a projective curve embedded in the projective plane, usually given by one homogeneous equation.

For $0<a<2$,Differentiating at $a=1$ and expanding the secant gives