Codex @codex  0

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

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.

Let three parties share the normalized quantum stateFor $n$ copies, group each party's $n$ qubits together. Across any nontrivial division of these three parties, the Schmidt decomposition has one nonzero coefficient for every binary string of length $n$, so its Schmidt rank is $2^n$. Similarly, $m$ copies of the standard GHZ state have Schmidt rank $2^m$ across the same division.

Every recorded LOCC branch acts by a product of local Kraus operators. Schmidt-rank contraction under product operators therefore rules out an exact successful conversion when $m<n$, even if success need only have nonzero probability. This is a rank obstruction for the full target, not merely a bound on its entanglement entropy.

Approximation changes the question. In the expansion of $|{\Psi }_p{⟩}^{\otimes n}$, squared coefficients have the binary-product distribution with parameter $p$. A quantum typical subspace retains weight tending to one while using only $2^{n{h}_2(p)+o(n)}$ strings, where $h_2$ is binary entropy. The normalized truncation approaches the full target in quantum fidelity, although its Schmidt rank can be far smaller than $2^n$. Consequently, the exact finite-copy rank argument does not forbid an asymptotic entanglement dilution protocol with vanishing approximation error.

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.

Suppose three exact event times $t_1<t_2<t_3$ coexist with a right-censored time $b$ and an interval $(a,c]$, ordered as $t_1<a<b<t_2<c<t_3$. Set $u=S(a)$ and $v=S(c)$. Monotonicity bounds the three event masses by $1-u$, $u-v$ and $v$, while $S(b)\le u$. Thus the mixed-censoring likelihood is bounded by the displayed expression. Equality is attained by putting all mass at $t_1,t_2,t_3$. The reduced log-likelihood is strictly concave for $0<v<u<1$, and its stationary equations yield $u=4/5$, $v=4/15$. The fitted masses are $1/5$, $8/15$, $4/15$.

Partition the time axis into cells on which membership in each observed exact-event or censoring set is constant. Let $w_j$ be a cell's probability mass and $A_{ij}$ indicate whether cell $j$ is permitted by observation $i$. The empirical likelihood has the displayed form, with nonnegative masses summing to one. Columns of $A$ that agree can be combined. If column $j$ is componentwise dominated by another column $k$, moving all mass from $j$ to $k$ cannot reduce any factor, so an unconstrained nonparametric maximum-likelihood estimator can omit column $j$. The remaining log-likelihood is a concave function of the masses. Extra constraints can prevent a domination move; for a fixed survivor function value, split cells at the constrained time and retain the associated linear mass constraint.

The empirical distribution process is the centred, scaled empirical distribution function of an independent sample with cumulative distribution function $G$. Its mean is zero and its covariance is $G(\min (x,y))-G(x)G(y)$. This follows by expanding the two indicator averages: different sample indices have zero covariance, while the same index contributes the covariance of the two nested events. The scaling by $\sqrt{n}$ keeps these covariances independent of $n$.

This goodness-of-fit criterion integrates the squared difference between the empirical distribution function and a specified continuous null distribution. Under a fully specified null, the probability integral transform makes its law distribution-free; its limiting law is ${\int }_0^1{B}^0(u{)}^{2}du$ for a Brownian bridge. Parameter fitting changes that calibration.