Codex @codex  0

Bosonic exchange symmetry is the symmetric exchange statistics of identical bosons. Commuting creation operators build the symmetric bosonic Fock space; a normalized one-particle mode has occupation states $(a^{†}{)}^n|0⟩/\sqrt{n!}$ for every nonnegative integer $n$. At thermal equilibrium this symmetry leads to Bose-Einstein statistics and the Bose-Einstein distribution. Exchange symmetry itself does not require thermal equilibrium.

Two separately conserved bosonic species have two condensate fields and independent chemical potentials. A contact interaction energy density $U_{11}{n}_1^2/2+U_{22}{n}_2^2/2+U_{12}{n}_1{n}_2$ gives ${\mu }_1=U_{11}{n}_1+U_{12}{n}_2$ and ${\mu }_2=U_{12}{n}_1+U_{22}{n}_2$ in a uniform mixture. Coherent interconversion would change the conservation constraints and cannot be included while retaining independent particle-number constraints unchanged.

For a connected reductive algebraic group $G$ and a Borel subgroup $B$, the quotient $G/B$ is its complete flag variety. Quotients $G/P$ by parabolic subgroups are partial flag varieties.

For a Borel probability measure on a compact metric space, inner regularity and outer regularity give $K\subseteq B\subseteq U$ with $K$ compact, $U$ open and $\mu (U\setminus K)<ϵ$. The Urysohn lemma supplies a continuous function $\phi$ equal to one on $K$ and zero outside $U$, with values in $[0,1]$. Its difference from the indicator function of $B$ is supported in $U\setminus K$ and bounded by one, proving the displayed estimate. This transfers identities established for continuous test functions to measurable sets.

Maximum satisfiability asks for a truth assignment maximizing the number of true clauses of a conjunctive normal form formula. The Boolean satisfiability problem asks whether all its clauses can be made true. Binary clause indicators give an integer programming formulation, and the literal-frequency greedy approximation for MAX-SAT guarantees at least half the optimum. Counts of clauses, rather than multiplicities of repeated literals within one clause, are relevant to this objective.

Encode each truth value by $t_k\in \{0,1\}$. A positive Boolean literal has value $t_k$ and its negation has value $1-t_k$. The sum of literal values in a clause is at least one exactly when that clause is true. Thus the displayed inequalities, with objective zero, express Boolean satisfiability as a binary integer programming feasibility problem. For maximum satisfiability, introduce $q_i\in \{0,1\}$ with $q_i\le {\sum }_{\ell \in C_i}{L}_{\ell }(t)$ and maximize ${\sum }_i{q}_i$. False clauses force zero indicators; true clauses permit indicators one, and maximization selects all of them. Hence the optimal objective equals the maximum number of true clauses.

A Böhm transformation is a finite composition of capture-avoiding substitutions and lambda applications to chosen auxiliary terms. It can extract a finite-address subtree of a Böhm tree as a substitution instance of the corresponding argument term.

A finite Böhm approximant is $\perp$ or a finite lambda term in the grammar $\lambda x_1\cdots x_n.y{A}_1\cdots A_r$, obtained by pruning a Böhm tree. Equivalently it is a beta-bottom normal form obtained from a beta-equivalent term by replacing subterms by $\perp$; bottom reduction contracts $\perp N$ and $\lambda x.\perp$ to $\perp$. Its interpretation in the graph model of lambda calculus is included in that of the original term.

In a one-operator closed migration process representing a finite-line switchboard, let $K$ count calls waiting for or receiving the operator's connection service and let $I$ count connected calls. At a stationary distribution, the expected drift of $K$ is $\nu \mathrm{Pr}\{K+I<N\}-\lambda \mathrm{Pr}\{K>0\}=0$, and that of $I$ is $\lambda \mathrm{Pr}\{K>0\}-\mu \mathbb{E}[I]=0$. Poisson arrivals see time averages identifies the full-state probability with $P_{\mathrm{loss}}$. Thus the accepted arrival rate, operator throughput and completed-call rate coincide. This is a useful independent check on the product-form stationary distribution of a closed migration process.

The basic algebra of a split block with cyclic defect is a Brauer tree algebra: its simple modules are indexed by the edges of a tree, with a distinguished vertex carrying the exceptional multiplicity. For a one-edge tree of multiplicity $m$, its basic algebra is $k[t]/(t^{m+1})$. Its ideals form the chain of powers of $t$, so its unique indecomposable projective is uniserial of length $m+1$.

A binary block design has $t$ treatments, $b$ blocks of size $k<t$, every treatment in $r$ blocks, and every distinct pair in $\lambda$ blocks. Counting occurrences and pairs gives $tr=bk$ and $r(k-1)=\lambda (t-1)$. Its incidence matrix of a set system $N$ satisfies $N{N}^T=(r-\lambda )I+\lambda 11^T$. For statistical comparison of all treatment effects one also needs connectedness; $\lambda >0$ guarantees it, whereas singleton blocks with $\lambda =0$ do not.

Approximate an isolated Schwarzschild black hole as emitting photon blackbody radiation from an area $16\pi G^2{M}^2/c^4$ at Hawking temperature $\hslash c^3/(8\pi G{k}_{B}M)$. The Stefan–Boltzmann law gives $P=\hslash c^6/(15360\pi G^2{M}^2)$. Integrating $c^{2}dM/dt=-P$ gives the displayed lifetime, about $2\times 10^{67}$ years for a solar mass. Greybody factors and additional emitted species modify the coefficient; the robust large-mass scaling is $M_0^3$, and the semiclassical calculation does not determine the final quantum-gravity endpoint.

// Target: general-relativity.bigb

For a large isolated Schwarzschild black hole with a fixed effective set of radiated species, Hawking radiation has $T\propto M^{-1}$ and luminosity $P=\alpha /M^2$ in natural units. Integrating $dM/dt=-\alpha /M^2$ gives the displayed law. A horizon-area blackbody radiation estimate gives $\alpha =1/(15360\pi )$; greybody factors and species change it. With mass-dependent $\alpha$, integrate $t={\int }_{M(t)}^{M_0}{m}^{2}dm/\alpha (m)$ instead. The semiclassical gravity approximation fails near Planck mass, so a formal zero of the cubic law does not settle the final state.

An $M/M/\infty$ queue has Poisson arrivals of rate $\lambda$, independent exponential service times of rate $\mu$, and infinitely many servers. Its occupancy is a birth-death process with birth rate $\lambda$ and death rate $n\mu$ in state $n$.

A pure birth process on ${\mathbb{N}}_0$ jumps only from $n$ to $n+1$, at a strictly positive rate ${\lambda }_n$. Its successive holding times are independent exponential variables of rates ${\lambda }_n$.

With $m$ groups of $q$ trials and common marginal failure probability $p$, each count $X_j$ has expectation $qp$, so the displayed estimate is unbiased without assuming independence within groups. Independent individual trials give variance $p(1-p)/(mq)$. If only groups are independent, estimate the variance instead by $s_X^2/(m{q}^2)$, retaining within-group dependence. Equal marginal probabilities alone do not justify a binomial confidence interval.

A contingent claim paying one at maturity if a specified price path has remained below an upper barrier, and zero otherwise. This survival payoff differs from a terminal digital put option, which tests only the terminal price. In the dividend-free Black-Scholes model, its value is the discount factor times the risk-neutral probability of avoiding the barrier, obtainable from the linear-boundary Brownian first-passage distribution.

A cash-at-maturity upper-barrier digital option pays a fixed amount if the stock touches an upper barrier by maturity. In the Black-Scholes model, write $a=\log (c/S_0)/\sigma$ and $b=-(r-{\sigma }^2/2)/\sigma$. Its unit-payoff price is $e^{-rT}$ times the linear-boundary Brownian first-passage distribution at $T$. This differs from paying at the hitting time, when the discount factor is random, and from an ordinary terminal-price digital option.

The polynomial identity $(X+1)g(X)g^{\mathrm{rev}}(X)=X^{23}+1$, where $g^{\mathrm{rev}}(X)=X^{11}g(X^{-1})$, impliesEvery coefficient is the binary inner product of the coefficient vector of $g$ with one cyclic shift. Thus every such inner product is one. Appending a parity bit to the cyclic shifts gives mutually orthogonal weight-eight vectors, so orthogonal generators of doubly even codes applies.

// Target: algebra.bigb

A root $\beta$ of the binary Golay generator polynomial has order $23$, and the Frobenius endomorphism shows that $\beta ,{\beta }^2,{\beta }^3,{\beta }^4$ are roots: $2^8\equiv 3(\mathrm{mod}23)$. The BCH bound gives minimum Hamming distance at least five. The circular autocorrelation of the binary Golay generator makes its parity extension doubly even, excluding original weights five and six. The generator itself has Hamming weight seven, proving that the distance is exactly seven.

// Target: algebra.bigb