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Let $V$ be a finite-dimensional real inner-product space and let $F:V\to V$ be continuous. If $⟨F(x),x⟩<0$ on the sphere $\Vert x\Vert =R$, then $F$ has a zero in the open ball. Otherwise the radial projection of $F$ gives a map of the ball that contradicts the Brouwer fixed-point theorem.

When an inverse Laplace integrand contains $p^{-1/2}$ on the principal branch, closing the Bromwich contour to the left encloses isolated poles and wraps a branch cut along the negative real axis. Pole residues give persistent oscillatory terms, while the jump across the two sides of the cut gives a real decaying integral.

Letwhere $P$ has a simple zero at $a$, and let $A$ be the limiting value of $W$ at $a$ on one sheet. Analytic continuation once around $a$ changes the sign of the square root and hence of $W^{\prime }$. The continued primitive agrees at $a$ with the original one, so it is

A phonon is a bosonic quantum of a crystal's normal modes. Because phonon number is not conserved in thermal equilibrium, its chemical potential is zero.

Bose-Einstein condensation is the macroscopic occupation of the lowest one-particle state by bosons below a critical temperature.

The unit ball in ${\mathbb{R}}^n$ can be covered by at most $9^n$ balls of radius $1/4$. A maximal $1/4$-separated set supplies the centres, and disjoint radius-$1/8$ balls around them fit inside the ball of radius $9/8$.

Kahn and Kalai represented uniform subsets by quadratic sign vectors so that diameter pairs correspond to one forbidden intersection size. The Frankl-Wilson theorem bounds each smaller-diameter part, forcing exponentially many parts while the ambient dimension grows only quadratically.

For independent rate-one exponential variables $X_n$ and $M_n={\max }_{m\le n}{X}_m$,almost surely. The first follows by applying both Borel--Cantelli lemmas to thresholds $(1\pm \epsilon )\log n$; the second uses
$\mathbb{P}(M_n\le (1-\epsilon )\log n)\le e^{-n^{\epsilon }}$.

For a nonnegative random variable $X$, Tonelli theorem givesConsequently, for each $a>0$, $\mathbb{E}[X]=\infty$ exactly when ${\sum }_{n\ge 1}\mathbb{P}(X>an)=\infty$; compare the integral on successive intervals of length $a$.

A nonzero Boolean integral domain has only the elements zero and one, so it is ${\mathbb{F}}_2$. The quotient criterion then makes every prime quotient of a Boolean ring a field, and every prime ideal maximal.

A Boolean function is a function on a Boolean hypercube, often with codomain $\{-1,1\}$ or $\{0,1\}$.

The edge boundary of a vertex set $A$ consists of graph edges with exactly one endpoint in $A$. In a $d$-regular graph,where $e(A)$ counts edges with both endpoints in $A$.

If a nonzero function on the Boolean hypercube has degree at most $k$, then its support has measure at least $2^{-k}$. This sharp bound follows by induction on the dimension.

One form of the hypercontractive inequality is $\Vert T_{1/\sqrt{3}}f{\Vert }_4\le \Vert f{\Vert }_2$, where $T_{\rho }$ is the noise operator on the Boolean hypercube.

When $N$ is prime, simultaneous approximation of the frequencies shows that $B(\Gamma ,\rho )\subseteq \mathbb{Z}/N\mathbb{Z}$ contains a centered arithmetic progression of length at least

A Bohr set of rank $d$ and width $\rho$ satisfiesThe proof partitions each circle coordinate into arcs and applies translation averaging and the pigeonhole principle.

Three Hermitian expectation values determine an arbitrary qubit state exactly when the traceless parts of the three observables span the three-dimensional space generated by the Pauli matrices. Mere linear independence as Hermitian matrices is insufficient: $I,{\sigma }_x,{\sigma }_z$ are independent but cannot detect the sign or magnitude of the Bloch $y$ component.

The Black-Scholes implied volatility is the volatility parameter that makes the Black-Scholes price equal an observed claim price.

A delta hedge holds ${\Delta }_t={\partial }_{s}V(t,S_t)$ units of the risky asset. In the Black-Scholes model, this choice cancels the claim's Brownian exposure.

The option delta is one of the Greeks (finance); hedging is the trading procedure that uses this sensitivity.

The price $V(t,s)$ of a sufficiently regular European claim satisfies