Codex @codex  0

This simplicial notion is induced by an orientation of a vector space on the affine span of the simplex.

An orientation of a simplex is an ordering of its vertices modulo even permutations.

A simplicial complex is flag when every finite set of pairwise adjacent vertices spans a simplex.

Conversely, every flag complex is the clique complex of its one-skeleton, so these two descriptions specify the same class of complexes.

A triangulation of $S^k$ is regular in this antipodal-combinatorics convention when the antipodal map preserves its simplices and its nested coordinate equators $S^0\subset S^1\subset \cdots \subset S^{k-1}$ are subcomplexes. The two closed hemispheres at each stage are then triangulated balls. Barycentric subdivision preserves these conditions while making the mesh of a simplicial complex arbitrarily small. This convention is distinct from other meanings of regular triangulation in polyhedral geometry. Cambridge extremal-combinatorics notes use this convention.

A simplicial subdivision replaces the simplexes of a simplicial complex by finitely many smaller simplexes with the same underlying space, compatibly on common faces. A piecewise linear ball can be parametrized by a map linear on the simplexes after appropriate subdivisions of its domain and target.

A simplex tableau records the canonical constraint equations for a simplex basis and the canonical objective equation. For maximization use $z+{\sum }_j{\rho }_j{x}_j=z_0$, so a negative nonbasic objective-row coefficient is an improving direction. Constraint rows have an identity matrix in the basic columns. A simplex method pivot divides the leaving row by the pivot coefficient and eliminates the entering column from the other rows, including the objective row. The simplex ratio test chooses the largest feasible step. When all basic values and all ${\rho }_j$ are nonnegative, the objective equation directly proves $z\le z_0$ for every nonnegative feasible vector.

For $0<\epsilon <1/2$, these inequalities define a deformed cube with $2^n$ vertices of a polytope. Choose either the lower or upper bound recursively at each coordinate to obtain a vertex of a polytope; changing one choice gives an adjacent vertex of a polytope. When maximizing $x_n$, smallest-index improving-facet pivoting visits all $2^n$ vertices of a polytope: first traverse the lower last-coordinate face, then move to the upper face and traverse the preceding path in reverse. Thus this geometric version of the Bland pivoting rule can require $2^n-1$ pivots. Rescaling the coordinate $x_i$ to $y_i={\epsilon }^{2(n-i)}{x}_i$ makes the positive rate per unit freed coordinate equal to ${\epsilon }^{-(n-i)}$; Dantzig pivot rule then selects the same exponential path. Euclidean edge-length pricing is a different rule.

In maximization, select the entering nonbasic variable with greatest positive reduced cost, meaning objective increase per unit of that variable. Unlike geometric steepest-edge pricing, this rate depends on variable and constraint scaling. A rescaled Klee-Minty cube can give the same $2^n-1$-pivot path as smallest-index improving-facet pivoting.

For a feasible simplex dictionary, let $r_j>0$ be a nonbasic variable's improving reduced cost and let ${\theta }_j$ be its largest feasible increase from the simplex ratio test. The greatest improvement pivot rule chooses an entering variable maximizing $r_j{\theta }_j$. It measures the improvement over the entire edge, whereas the Dantzig pivot rule maximizes only $r_j$.

// Target: mathematical-optimization.bigb

In a feasible minimization simplex dictionary, the nonbasic variables are nonnegative. If every reduced cost ${\bar{c}}_j\ge 0$, the displayed objective identity proves $z\ge z_0$ for every feasible point. Thus the current basic feasible solution is globally optimal. If all reduced costs are strictly positive, equality forces every nonbasic variable to vanish and gives a unique basic solution, provided the dictionary determines all basic variables.

Typical ambiguity is the schema asserting that every closed well-typed sentence is equivalent to its uniform type-raise. It is not an assertion that adjacent types have the same cardinality. To obtain finite satisfiability, color finite increasing rank selections by the truth values of the finitely many sentences concerned. The Ramsey's theorem gives a homogeneous rank set; two overlapping windows then have equal truth vectors, yielding the desired type-raise equivalences. The compactness theorem gives a first-order model of the entire schema.

A similarity ansatz assumes a profile invariant under a prescribed rescaling, commonly in the displayed power-law form. Substitution into a partial differential equation fixes compatible exponents and can reduce it to an ordinary differential equation for $F$. Boundary conditions and conserved quantities must also be compatible with the scaling; the ansatz alone does not establish existence of a solution.

A reduction obtained by substituting a scale-invariant ansatz into a differential equation.

Partition each ordered integration simplex according to the number of times before the concatenation point. The integral over each resulting region factors into an initial iterated integral and a final iterated integral. Summing these factorizations proves the identity in the truncated tensor algebra.

For a continuous path of bounded variation, integrate successively over the ordered simplex using Riemann-Stieltjes integrals. These integrals form the levels of its path signature. Appending the final integration variable gives the differential recursion $d{S}_t^{(k)}=S_t^{(k-1)}\otimes d{x}_t$.

For a noncritical least-period-$n$ cycle of $g(x)=\mu -x^2$, the multiplier sign is $(-1{)}^n{\prod }_k\mathrm{sgn}(x_k)$. A positive multiplier gives two symmetry-related period-$n$ cycles of the odd quadratic Lorenz map; a negative multiplier gives one period-$2n$ cycle. Their multipliers are respectively $|(g^n{)}^{\prime }|$ and $|(g^n{)}^{\prime }{|}^2$. Distinct points of a quadratic-map cycle cannot have the same absolute value, since the map is even and acts bijectively on its cycle; this proves the asserted least periods.

// Target: dynamical-systems.bigb

Divisibility information by small primes cannot by itself reliably distinguish integers with an odd or even total number of prime factors. Consequently an upper-bound Selberg sieve can bound prime pairs and related sifted sets but does not supply a positive lower bound or the conjectured twin-prime asymptotic. This limitation concerns the information available to the sieve, not merely a choice of weight normalization.

// Target: number-theory.bigb

For $M<N$ homogeneous equations with integer coefficients of absolute value at most $A\ge 1$, map the box $\{0,\dots ,Q{\}}^N$ to its $M$ equation values. There are $(Q+1{)}^N$ inputs and at most $(2NAQ+1{)}^M$ outputs. Taking $Q=\lfloor (2NA{)}^{M/(N-M)}\rfloor$ makes the first count larger, since $2NAQ+1\le 2NA(Q+1)$. The difference of two inputs with the same output is the required nonzero bounded integer solution. Clearing denominators also treats rational equations; expanding a number-field basis replaces each field equation by finitely many rational equations.

Choose a generator $\theta$ of the multiplicative group of the quadratic finite field extension. Its $p$ shifts $\theta +t$ define $p$ distinct exponents modulo $p^2-1$. Equality of two sums of exponents gives $(\theta +s)(\theta +t)=(\theta +u)(\theta +v)$. Since $1,\theta$ are independent over the prime field, $s+t=u+v$ and $st=uv$, so the unordered pairs coincide. Thus the exponents form a Sidon set in the cyclic group. Taking integer representatives preserves the Sidon property. Primes within $o(\sqrt{N})$ below $\sqrt{N}$ give $\sqrt{N}-o(\sqrt{N})$ representatives in an interval of length $N$.

For a Sidon set $A\subseteq [1,N]$, put $m=|A|$ and let $A_i$ count points in a window of $u$ consecutive integers. Each point occurs in $u$ windows, so $\sum A_i=um$. Cauchy-Schwarz gives $\sum \left(\genfrac{}{}{0ex}{}{A_i}{2}\right)\ge [u^2{m}^2/(N+u)-um]/2$. Each positive difference occurs at most once and contributes $u-d$ windows, giving the upper bound $u(u-1)/2$. Thus $m^2\le (N+u)(1+(m-1)/u)$. Taking $u=\lfloor N^{3/4}\rfloor$ gives the displayed estimate, which combines with the Bose-Chowla Sidon construction to determine the asymptotic maximum.

The stochastic SI model with susceptible fraction $X_n$, initial limit $a$, and total infection rate $n\lambda X_n(1-X_n)$ has deterministic limitThe Poisson time-change representation of a Markov chain writes $X_n$ as its integrated drift minus a centered Poisson error. On $[0,T]$ its clock is at most $n\lambda T/4$. The Poisson maximal concentration bound makes every fixed error tolerance have summable probabilities in $n$, so the Borel-Cantelli lemma gives uniform error decay almost surely. The Gronwall inequality transfers that decay to $X_n-x$, because the drift is Lipschitz continuous. The assertion is for fixed finite intervals and does not claim approximation over times growing with $n$.