A wall of marginal stability is a locus in the parameter or vacuum space where the complex central charges in supersymmetry of possible decay products align in phase. The triangle inequality then becomes an equality, $|Z_1+Z_2|=|Z_1|+|Z_2|$, permitting a BPS state to reach the threshold for decay into other BPS states. Away from such alignment, the strict triangle inequality can prohibit the decay. Supersymmetry protects the mass-charge relation of an existing short state; it does not guarantee that the state exists or remains stable everywhere in parameter space.

For an increasing sequence converging to $L$, its absolute error is $e_n=L-a_n\ge 0$ and is nonincreasing. For a decreasing sequence, the absolute error is $e_n=a_n-L\ge 0$ and is also nonincreasing. This tells us that later terms cannot be worse, but does not supply a computable stopping index for a chosen tolerance.

Suppose an increasing lower approximation $a_n$ and a decreasing upper approximation $b_n$ enclose the same limit of a sequence $L$. Their observable gap $w_n=b_n-a_n$ givesIf $w_n\to 0$, the squeeze theorem certifies a common limit of a sequence; the midpoint has a certified absolute error at most half the gap. This is the principle behind the bisection method. In the symmetric example $a_n=1-1/n$, $b_n=1+1/n$, the gap is $2/n$, each endpoint's exact absolute error is $1/n$, and the midpoint happens to equal one exactly. The bound is valid even when such symmetry is absent.

A fixed upper bound by itself supplies less. If $a_n\uparrow L$ and $a_n\le M$ for all $n$, then $L\le M$ and $0\le L-a_n\le M-a_n$, but the last gap need not vanish. Taking $a_n=1-1/n$ and $M=2$ makes $M-a_n\to 1$. A shrinking bracket or an explicit remainder formula adds information that qualitative convergence does not provide.

Dropping either condition destroys the implication. The increasing sequence $a_n=n$ has a lower bound one but no upper bound, and no finite limit of a sequence. For any real threshold $T$, the Archimedean property gives an index $N>T$, and monotonicity gives $a_n>T$ for all $n\ge N$. Its divergence to $+\infty$ in the extended real number line is different from convergence to a real number.

Conversely, $b_n=(-1{)}^n$ is a bounded sequence, with $-1\le b_n\le 1$, but is not a monotone sequence and has no limit of a sequence. Its subsequence at even integers is constantly one, and its subsequence at odd integers is constantly minus one. These two subsequential limits cannot both equal a single limit of a sequence. ^[ref]

The ambient real numbers also matter. An original example using the floor function isEvery $q_n$ is a rational number, $q_n\le q_{n+1}$ because $\lfloor 2x\rfloor \ge 2\lfloor x\rfloor$, andThe strict lower inequality uses the irrationality of the square root of two. Hence this bounded sequence converges to $\sqrt{2}$ in the real numbers, but has no limit of a sequence in the rational numbers. Convergence to a rational candidate would also be convergence in the real numbers, contradicting uniqueness of the limit of a sequence. The formulas give $q_1=1$, $q_2=5/4$, $q_3=11/8$, $q_4=11/8$: this counterexample even has a plateau. This illustrates why completeness of the real numbers matters: the rational numbers lack the least-upper-bound property needed to contain the required supremum. ^[ref]

The supremum of an increasing bounded sequence need not be a maximum of its set of terms. For $a_n=1-1/n$, every term is strictly less than one; the supremum and limit of a sequence are one, but there is no maximum. Every candidate term is beaten by the next. The decreasing sequence $1+1/n$ similarly has infimum one without a minimum.

If an increasing sequence with limit of a sequence $L$ attains $a_k=L$, it must remain there: $L=a_k\le a_n\le L$ for $n\ge k$. The decreasing case is identical with inequalities reversed. Thus attainment of the limit of a sequence is equivalent to an eventually constant tail of a sequence for a convergent monotone sequence. The increasing sequence $0,1/2,1,1,\dots$ has both supremum and maximum one. ^[ref]

For the two sequences in bounded monotone sequence theorem,Their directions and bounds are explicit. One is a succession of lower approximations and the other a succession of upper approximations to the same limit of a sequence. To obtain absolute error strictly below $\epsilon >0$, it suffices to choose an integer $N>1/\epsilon$, and then every $n\ge N$ works. For $\epsilon =0.01$, the first such index is $101$; at $100$ the absolute error equals the tolerance.

A geometric sequence offers a faster example. The recurrence relation $c_1=1/2$, $c_{n+1}=(c_n+1)/2$ gives $c_n=1-2^{-n}$. Thus $(c_n)$ is an increasing bounded sequence with limit of a sequence one and absolute error $2^{-n}$. For $\epsilon =0.01$, $n=7$ already works, since $2^{-7}=1/128<0.01$, whereas $2^{-6}=1/64>0.01$. The bounded monotone sequence theorem guarantees both examples converge; the formulas explain their different speeds.

For a connective spectrum with ${\pi }_{0}E\cong \mathbb{Z}$, an $E$-homology isomorphism of simply connected spaces is a weak homotopy equivalence. Its simply connected homotopy mapping cone is $E$-acyclic. If it had a first nonzero integral homology group, the Hurewicz theorem and the bottom-degree generalized Hurewicz isomorphism would produce a nonzero $E$-homology group. Thus the cone is integrally acyclic and the homological Whitehead theorem applies.

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 $n$.