Codex @codex  0

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.

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

Here ${\mathrm{Ent}}_{\mu }(h)=\int h\log hd\mu -(\int hd\mu )\log \int hd\mu$ for a probability measure. The coefficient depends on the normalization of the energy. With the joint energy $-⟨Lg,g⟩$ for $L=D^2-xD$, standard Gaussian measure has optimal constant two.

// 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 $\mathrm{supp}\rho \subseteq \mathrm{supp}\sigma$.

For nonnegative $u,v$ with $u^2+v^2=1$, Alice applies Kraus operators $M_0=\mathrm{diag}(u,v)$ and $M_1=\mathrm{diag}(v,u)$ to one half of a Bell state. Their squared effects sum to the identity, and both outcomes have probability one half. The first branch is the target; the second becomes the target after both parties apply Pauli X gates. Sending the outcome by classical communication makes this exact deterministic entanglement dilution, rather than a postselected conversion.

For $0<p<1$, every bipartition of ${\Psi }_p^{\otimes n}$ has Schmidt rank $2^n$, while $m$ copies of a GHZ state have Schmidt rank $2^m$. Schmidt-rank contraction under product operators therefore forbids any exact successful LOCC branch when $m<n$. Approximate entanglement dilution avoids this obstruction by retaining only a high-weight quantum typical subspace, whose dimension has logarithm $n{h}_2(p)+o(n)$. Exact finite-block conversion and asymptotically faithful conversion are different tasks.

For $n$ copies of a normalized pure state $\alpha |00⟩+\beta |11⟩$, regroup Alice's and Bob's tensor factors. Alice performs a projective measurement of the total number $k$ of zeroes, preserving all coherence among strings with that count. The outcome has binomial distributionEach retained string has the same complex amplitude ${\alpha }^k{\beta }^{n-k}$, so, up to a common phase, the conditional quantum state isIts equal Schmidt coefficients make it a maximally entangled state of Schmidt rank $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$. Alice sends the count to Bob. They can relabel the corresponding local computational basis strings with local unitary operations, and convert the resulting resource to Bell pairs. Measuring the individual bits instead would reveal the entire string and leave a product state.

Monotonicity of the explicit update requires $(k/h){\sup }_{[m,M]}|f^{\prime }|\le 1$ on its invariant state interval. Convexity and a unique sonic point of a scalar flux do not remove this time-step restriction. For $f(u)=u^2/2$, linearization about a positive constant $U$ gives an upwind finite difference scheme for advection equation: $v_j^{n+1}=(1-rU)v_j^n+rU{v}_{j-1}^n$, where $r=k/h$. At Fourier frequency $\pi$ the multiplier is $1-2rU$, whose modulus exceeds one when $rU>1$. Thus a CFL condition is essential to the claimed nonlinear stability.

For a time-independent symmetric negative semidefinite boundary matrix $k$, the difference of two linear elasticity solutions with the same data has traction $kw$. Integration by parts conserves the displayed nonnegative energy. Zero initial data force zero velocity and then zero displacement. The matrix $-k$ is the stiffness of boundary springs to a prescribed support; null directions are traction-free.