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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 $\mathbb{P}(A\square B)\le \mathbb{P}(A)\mathbb{P}(B)$.

At critical planar percolation, the probability of an open crossing of a rectangle of any fixed aspect ratio stays bounded away from zero and one uniformly over its scale.

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 $u_b=(h/\mu )(|{\tau }_b|-{\tau }_c{)}_{+}\mathrm{sgn}{\tau }_b$.

ForBinet’s equation isIts circular orbits solve $K=u(1-r_{0}u{)}^2$; the branch $u<1/(3r_0)$ is stable and the branch $u>1/(3r_0)$ is unstable.

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

Intersection multiplicity measures the local order of contact of two algebraic subvarieties. Transverse intersections have multiplicity one.

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

For $p,q>0$, express the product of the two Gamma integrals as the nonnegative double integralUse $r=x+y$ and $u=x/(x+y)$, with inverse $x=ru$, $y=r(1-u)$ and Jacobian determinant of absolute value $r$. The new domain of integration is $r>0$, $0<u<1$, and the integrand with this factor becomes $r^{p+q-1}{e}^{-r}{u}^{p-1}(1-u{)}^{q-1}$. Tonelli's theorem and separation of the two integrals therefore give $\Gamma (p+q)B(p,q)$. The positive real parameter hypothesis ensures that all the original integrals are finite.

For complex parameters with positive real parts, the same proof first gives the undivided identity $\Gamma (p+q)B(p,q)=\Gamma (p)\Gamma (q)$. Absolute integrability follows by taking real parts in the exponents, so Fubini's theorem applies. The nonvanishing of the Gamma function then permits division by $\Gamma (p+q)$.

The shortest binary linear recurrence producing the prefix $11001011$ has length three:Its connection polynomial is $1+D^2+D^3$.

If terminal utility is increasing and concave and each Bellman supremum is attained, backward induction shows that every remaining-horizon value function is increasing and concave. Concavity follows by combining optimal portfolios for two initial wealths with the same convex coefficient.

The minimum distance of a BCH code is at least its design distance. A hypothetical word of smaller weight yields a square Vandermonde system from the consecutive-root equations; its support elements are distinct, so the determinant is nonzero and every word coefficient must vanish.

For error locations $i_1,\dots ,i_{\nu }$ in a cyclic code with primitive root $\alpha$, the error locator polynomial is $\sigma (Z)={\prod }_h(1-{\alpha }^{i_h}Z)$. Its roots encode the error positions.

A beta prior $\mathrm{Beta}(\alpha ,\beta )$ combined with $s$ Bernoulli successes and $f$ failures gives posterior $\mathrm{Beta}(\alpha +s,\beta +f)$.