On the side where V is positive, set $r-r_0=V^{\prime }(r_0){\rho }^2/4$. Its radial Euclidean metric becomes the displayed plane in polar coordinates. The would-be singularity is conical unless the angular variable $|V^{\prime }(r_0)|\tau /2$ has period exactly $2\pi$. This fixes $\beta =4\pi /|V^{\prime }(r_0)|$; a nontrivial integer multiple still has a cone excess at the tip.

The Euclidean unit ball is the closed unit ball of the Euclidean norm, centered at zero. Its radial function is identically $1$, making it a natural reference body for John ellipsoids and volume ratios.

The prime exponents $2,3,5,7,13$ give Mersenne primes and therefore perfect numbers. But $p=11$ gives $2^{11}-1=2047=23\cdot 89$. The resulting candidate $N=2^{10}\cdot 2047=2096128$ hasso $s(N)=2325392>N$. It is an abundant number, not a perfect number. Checking only that the exponent is prime would miss this obstruction.

Let $A$ be a closed operator and let $V:D(A)\to H$ be compact when the domain carries its graph norm. If $A+V$ is closed on the same domain, compact perturbation invariance of Fredholm operators applied to the maps $D(A)\to H$ proves invariance of the Fredholm essential spectrum. For self-adjoint operators this is also the essential spectrum obtained by deleting isolated finite-multiplicity eigenvalues. A bounded potential tending to zero at spatial infinity is relatively compact with respect to a second-order constant-coefficient operator: local Rellich-Kondrachov compactness theorem gives compactness on bounded intervals and the small tails give compactness globally.

For a bounded linear operator on an infinite-dimensional complex Hilbert space, define the essential spectrum by failure of the Fredholm property for $T-\lambda I$. The Atkinson theorem identifies it with the spectrum in the Calkin algebra, hence it is compact and nonempty. On a finite-dimensional Hilbert space it is empty. For a bounded real diagonal operator on a separable infinite-dimensional space it equals the subsequential cluster set of the diagonal entries, including values of infinite multiplicity.

For errors $E_i$ at distinct nonzero locations $X_i$, the evaluator associated with syndromes $S_{\ell }={\sum }_i{E}_i{X}_i^{b+\ell }$ is $\Omega (z)={\sum }_i{E}_i{X}_i^b{\prod }_{h\ne i}(1-X_{h}z)$. Its degree is smaller than the number of errors. Together with the error locator polynomial it gives the Reed-Solomon key equation and the Forney algorithm for error magnitudes.

A Chien search finds error positions by evaluating an error locator polynomial at every possible inverse location value ${\omega }^{-j}$. A zero identifies coordinate $j$. Using the recurrence between successive powers of $\omega$ avoids recomputing all monomials from scratch.

For the upper-pole integral $I={\int }_{{\mathcal{C}}_{+}}{t}^{\gamma -1}{e}^{\lambda (1-t)}(1-t{)}^{-1}dt$, let $\gamma \to \infty$ and $\lambda =\gamma +i\mu \sqrt{\gamma }+O(1)$. Scaling $t=1+s/\sqrt{\gamma }$ gives $-{\int }_{{\mathcal{C}}_{+}}{e}^{-s^2/2-i\mu s}ds/s$ at leading order. At $\mu =0$ its indentation is $i\pi$. Differentiation with respect to $\mu$ gives $i\sqrt{2\pi }{e}^{-{\mu }^2/2}$, so integration gives the displayed error function. This is an optimal truncation transition; the underlying analytic integral has no literal discontinuity. The general mechanism is discussed in the DLMF account of Stokes smoothing.

Substitution of $y=A{e}^{i{ϵ}^{-1}\int \sigma dX}$ into ${ϵ}^2{y}^{\prime \prime }+k^{2}y=0$ gives the displayed real amplitude equation and conserved imaginary-part flux. Conversely these equations produce an exact complex solution wherever $A\ne 0$. The WKB approximation is obtained by expanding this exact system where $|k|$ stays positive.

For $x\in {\mathbb{R}}^n$, Cauchy-Schwarz inequality applied to $(|x_i|)$ and $(1)$ gives the upper bound. Equal absolute coordinate values attain its factor $\sqrt{n}$. Expanding $({\sum }_i|x_i|{)}^2$ gives the lower bound, and a vector with only one nonzero coordinate attains its factor one. Thus both constants are optimal. These Lp norms define the same finite-dimensional topology despite having different unit balls.

An equivalence class representative is any set element belonging to that equivalence class. In the finite modulo-three example, either $0$, $3$ or $6$ represents $\{0,3,6\}$. Changing the equivalence class representative changes its name but preserves the equivalence class itself. Choosing $0,1,2$ gives one representative of an equivalence class from each class; the quotient set consists of the three whole sets.

For inter-renewal times with an Erlang distribution of integer shape $r$ and rate $\lambda$, the mean is $r/\lambda$ and survival function is $e^{-\lambda t}{\sum }_{j=0}^{r-1}(\lambda t{)}^j/j!$. The equilibrium first-delay density is survival divided by the mean,This is the equal mixture of Erlang distributions of shapes $1,\dots ,r$. It can be constructed by observing a Poisson clock with uniformly randomized phase modulo $r$, whose every $r$th event is a renewal.

A dissociation constant is an equilibrium constant for a reaction in which a combined species separates into components. For $AB\rightleftharpoons A+B$, a concentration convention writesIt has units of concentration. The dimensionless thermodynamic form instead uses thermodynamic activities, $K_d^{\circ }=a_A{a}_B/a_{AB}$. For ideal concentration activities, $K_d^{\circ }=K_d^{(c)}/c^{\circ }$, with specified standard concentration $c^{\circ }$. The choice of quantity and reference is therefore part of quoting the constant.^[ref]

In the simple dilute $1:1$ model with free $c_A$ held fixed, the fraction of B present as AB is $c_A/(K_d^{(c)}+c_A)$. A smaller $K_d^{(c)}$ therefore means more association at the same free concentration. The inverse association constant describes the reversed reaction only when conventions, solvent conditions and temperature match. A dissociation rate is a kinetic quantity and is not specified by this equilibrium ratio alone.

Interval discrepancy measures the difference between empirical counts and the expected count for normalized Lebesgue measure. Convolving an enlarged interval indicator function with a narrow unit-integral indicator function gives a continuous majorant. Its Fourier coefficients decay as $O(1/(\delta k^2))$, converting the count into a constant term, finite exponential sums and a controlled absolute tail.

The sequence $r(u,v)$ is equidistributed in ${\mathbb{T}}^2$ precisely when $ku+\ell v\notin 2\pi \mathbb{Z}$ for every nonzero integer pair $(k,\ell )$. Equivalently $1,u/(2\pi ),v/(2\pi )$ are rationally independent. Under that condition, nonconstant character averages tend to zero by the geometric-series formula. Uniform approximation by trigonometric polynomials gives convergence for continuous functions, and continuous upper and lower bounds give convergence for rectangles. If a character is resonant, the orbit lies in its proper closed kernel and misses a positive-area rectangle, so equidistribution fails.

For independent $n$-dimensional random vectors with densities and finite differential entropies, the entropy power inequality statesThe sum's differential entropy is required to be defined; a value $+\infty$ makes the inequality immediate. Equality for finite nondegenerate densities occurs for Gaussian distributions with proportional covariance matrices.

// Target: probability-and-statistics.bigb

A probability measure $\mu$ with a Dirichlet form ${\mathcal{E}}_{\mu }$ satisfies a logarithmic Sobolev inequality with constant $c_{\mathrm{LS}}>0$ iffor every $g$ in the domain of the Dirichlet form. The entropy functional measures how far $g^2$ is from a constant, while the Dirichlet form measures its spatial variation. For a symmetric Markov semigroup with generator $L$, the convention is ${\mathcal{E}}_{\mu }(g,g)=-\int gLgd\mu$ when this expression is defined.

For standard Gaussian measure, $L=\Delta -x\cdot \nabla$ and Gaussian integration by parts gives ${\mathcal{E}}_{\mu }(g,g)=\int \Vert \nabla g{\Vert }^{2}d\mu$. The Gaussian logarithmic Sobolev inequality has optimal constant two in this convention. Rescaling the Dirichlet form rescales the constant inversely, so a numerical constant needs its energy convention. The logarithmic Sobolev inequality implies hypercontractivity and also supports concentration inequalities through the Herbst argument.^[ref]

// Target: analysis.bigb

The displayed supremum norm is finite on every entire function, positive on every nonzero one by the identity theorem, and submultiplicative under pointwise multiplication. It is not complete: partial geometric sums ${\sum }_{j=0}^n(z/2{)}^j$ are Cauchy on the disk but their limit there extends meromorphically with a pole at two. More generally, no complete algebra norm exists on all entire functions because the coordinate function has algebraic spectrum of an element equal to the whole complex plane.

For a bipartite pure state, the squared Schmidt coefficients are the nonzero eigenvalues of either reduced density matrix. Their sum except the largest defines $E_2=1-{\lambda }_{\max }({\rho }_A)$. The largest eigenvalue is a convex function, since ${\lambda }_{\max }(\rho )={\max }_{\Vert v\Vert =1}⟨v|\rho |v⟩$; consequently $E_2$ is concave. If Alice measures locally and records outcome $r$, Bob's conditional reductions satisfy ${\sum }_r{p}_r{\rho }_{B,r}={\rho }_B$. Concavity gives ${\sum }_r{p}_r{E}_2({\chi }_r)\le E_2(\chi )$. For Bob's measurements use Alice's unchanged average reduction instead. Iteration proves average monotonicity under LOCC. For Schmidt rank at most two, $E_2$ is the smaller squared Schmidt coefficient. A conversion succeeding with probability $p$ therefore obeys $p{E}_2(\varphi )\le E_2(\psi )$, since failure branches contribute nonnegative amounts.

The relative entropy of entanglement is the smallest quantum relative entropy from a density operator to a separable quantum state, with the separable set specified for the relevant division into parties. Nonnegativity of quantum relative entropy implies $E_R\ge 0$. For a separable quantum state, choosing $\sigma =\rho$ gives $E_R=0$. For finite-dimensional bipartite states, compactness of the separable set and lower semicontinuity of the relative entropy give the converse. The support convention is $D(\rho \Vert \sigma )=+\infty$ unless $\mathrm{supp}\rho \subseteq \mathrm{supp}\sigma$.