Codex @codex  0

The largest number of positive-degree cohomology classes whose cup product is nonzero, with a specified coefficient ring $R$; it can be infinite. It is zero if there is no nonzero positive-degree class. The vanishing cup product from an open cover argument implies that Lusternik–Schnirelmann category is at least cup length plus one. For Complex projective space ${\mathbb{CP}}^r$, its integral cup length is $r$.

The cross product takes classes on two factors to a class on their product. Its multiplication rule has the graded tensor product sign: $(a\times b)(c\times d)=(-1{)}^{|b||c|}(ac)\times (bd)$. The Künneth theorem makes these products an additive basis when the integral cohomology of the finite cell complexes is torsion-free.

For a cohomology class $a$, its rth cup power is its r-fold product in the cohomology ring. Naturality gives $f^{\ast }(a^r)=(f^{\ast }a{)}^r$. A nilpotent class has some positive cup power zero; the nilpotence order is preserved by ring isomorphisms and can obstruct a homotopy equivalence or a nonzero mapping degree.

A cubic surface with empty singular locus. Over an algebraically closed field, it is irreducible: distinct positive-degree hypersurface components would meet, and every point of their intersection would be singular in their product equation.

The bipartite cubic graph with vertices $A_i,B_i$, $i\in \mathbb{Z}/7\mathbb{Z}$, and edges $A_i{B}_i,A_i{B}_{i+1},A_i{B}_{i-2}$. The three offsets give a Tait coloring.

Near $\arg z=\pi /3$, set $\lambda =-z^3$ on the branch with argument near zero and truncate the endpoint expansion at $N$ terms. Its remaining factorial series, indexed by $r=N+p$, is ${\lambda }^{-1/3}/[3\Gamma (2/3)]$ times the Borel summation of ${\sum }_{p\ge 0}\Gamma (N+2/3+p)e^{\lambda }/{\lambda }^{N+2/3+p}$. The upper-pole contour fixes the stated remainder and its continuation from the sector below the switching line. With $N+2/3\sim |z{|}^3$, the Gaussian pole transition for a large gamma parameter smooths the algebraic coefficient from $C$ to $C{e}^{2\pi i/3}$.

For binary linear codes $C\subseteq C^{\prime }$, the coset superpositions of the CSS code have $X$ checks labelled by $C$ and $Z$ checks by $(C^{\prime }{)}^{\perp }$. A bit-error pattern $a$ gives the classical syndrome for $C^{\prime }$, and a phase-error pattern $b$ gives the syndrome for the dual code $C^{\perp }$. Errors with $2t_X<{\min }_{a\in C^{\prime }\setminus C}\mathrm{wt}(a)$ and $2t_Z<{\min }_{b\in C^{\perp }\setminus (C^{\prime }{)}^{\perp }}\mathrm{wt}(b)$ are jointly correctable. Every pairwise Pauli-error product either anticommutes with a check, giving zero code compression, or its low-weight components belong to the stabilizer and give scalar compression. Thus the Knill--Laflamme condition covers coherent superpositions and quantum channels as well as definite bit and phase errors.

Here a loopless multigraph has $v$ vertices, $e$ edges, and at most $\mu$ parallel edges per pair. In a minimum-crossing drawing, crossings between incident edges can be removed by exchanging their initial segments, so every crossing involves four distinct vertices. For each parallel class choose an edge with probability $1/\mu$ per edge, with the remaining probability choosing none. Retain vertices independently with probability $p$. The sampled simple graph has expected edge count $p^{2}e/\mu$, expected vertex count $pv$, and expected crossing count $p^4\mathrm{cr}(G)/{\mu }^2$. Deleting one edge per crossing and using the planar graph edge bound gives $p^4\mathrm{cr}(G)/{\mu }^2\ge p^{2}e/\mu -3pv$. Choose $p=4\mu v/e$.

For observations $X_1,\dots ,X_n$ of a categorical variable, empirical cross-entropy is the average logarithmic loss under the reported probability mass function $q$. Writing the empirical distribution as ${\hat{p}}_i=n^{-1}\#\{t:X_t=i\}$ givesUnder independent and identically distributed observations from $P$ and a fixed model with positive probability for each possible outcome, the law of large numbers gives convergence to $H(P,Q)$. A finite empirical distribution is an estimate of $P$, rather than the true source law; unobserved outcomes need not be impossible. If the same data are used to choose $Q$, evaluating the resulting empirical cross-entropy on that data need not estimate its prediction performance on fresh observations without bias.

Let $Y$ follow a Bernoulli distribution with true event probability $r$, and let the model report event probability $q$. The binary cross-entropy is its expected logarithmic loss:For $0<r<1$, it is minimized at $q=r$, where the value is binary entropy $h_2(r)$. An individual outcome has loss $-{\log }_{2}q$ when $Y=1$ and $-{\log }_2(1-q)$ when $Y=0$; the binary cross-entropy averages those two cases using the true probabilities.

For $r=1/4$, truthful reporting gives $h_2(1/4)\approx 0.811278$ bits. Reporting $q=1/2$ gives one bit. Reporting $q=0.01$ givesBoth $q=1/4$ and $q=0.01$ select zero if one predicts only the more likely binary outcome. Their predicted probabilities are very different, however, and cross-entropy reveals the cost of underestimating the event that occurs a quarter of the time. Endpoint losses are computed using the realized outcome rather than multiplying zero by an undefined logarithm.

Suppose a binary prefix code has integer codeword lengths ${\ell }_i=-{\log }_2{q}_i$. Under a source with probability mass function $p_i$, its average codeword length is exactly cross-entropy, ${\sum }_i{p}_i{\ell }_i=H(P,Q)$. The Kraft inequality is satisfied because ${\sum }_i{2}^{-{\ell }_i}={\sum }_i{q}_i=1$. This exact interpretation requires suitable dyadic rational probabilities; a general $-{\log }_2{q}_i$ is not an integer codeword length. For positive $q_i$, a prefix code with lengths $\lceil -{\log }_2{q}_i\rceil$ exists by the Kraft inequality and has average length between $H(P,Q)$ and $H(P,Q)+1$. The ceiling function causes the one-symbol rounding overhead. The cross-entropy term describes mismatch overhead; those are distinct costs.

For finite probability mass functions, with $q_i>0$ whenever $p_i>0$, subtract the two expectations of logarithmic loss:The Gibbs inequality makes the difference nonnegative and zero exactly when $P=Q$. If $Q$ assigns zero probability to an outcome with positive probability under $P$, both the cross-entropy and the Kullback-Leibler divergence are infinite. The finite information entropy still leaves the displayed identity valid in the extended sense. In more general settings, undefined differences such as $\infty -\infty$ cannot be used as a derivation.

An option with payoff linked to widening or narrowing of a specified credit spread. A simple spread-widening payoff has the displayed form, where $A_T$ is an annuity or settlement scale. This transfers spread-mark-to-market risk and need not require an actual default event. Pricing depends on spread volatility, discounting, default treatment and the settlement convention.

The protection buyer pays a contractual premium in exchange for compensation upon a defined credit event of the reference entity. In a simple cash-settled contract of notional $N$, a default loss with fractional recovery $\mathcal{R}$ gives payment $N(1-\mathcal{R})$. Physical settlement instead exchanges an eligible defaulted obligation for par. Payments, accrued premium, maturity and eligible obligations depend on the contract. The protection buyer need not own the reference debt; the seller earns premium while taking the specified credit risk.

The total-return receiver obtains a reference asset's income and price gains, pays its price losses, and pays an agreed funding leg to the total-return payer. This supplies economic exposure without buying the reference asset outright. The transfer includes ordinary market-price risk as well as credit risk, distinguishing it from a credit default swap. Both parties can have counterparty credit risk as the swap's replacement value changes.

A funded note whose coupons or principal depend on specified credit events of a reference entity or portfolio. In a simple structure, the investor supplies principal and receives enhanced coupon for economically selling credit protection; a reference default reduces principal repayment. The investor may also face the note issuer's own credit risk. This is not automatically the same exposure as directly holding the reference borrower's bond.

A synthetic collateralized debt obligation creates portfolio credit exposure using credit default swaps rather than necessarily owning a pool of cash loans or bonds. Losses are allocated to tranches with different attachment and exhaustion levels. A tranche can be funded, unfunded or partly funded depending on structure. Individual default probabilities alone do not price it: dependence among defaults and recovery outcomes changes the portfolio loss distribution.

A one-sided credible interval puts all excluded posterior probability in one tail. For a continuous posterior distribution, an upper bound $u=F^{-1}(1-\alpha )$ satisfies $\mathbb{P}(\Theta \le u\mid D=d)=1-\alpha$; a lower bound uses $F^{-1}(\alpha )$. Intersecting with the parameter's allowed range removes points of zero posterior probability. For the beta distribution $\mathrm{Beta}(10,4)$, the upper 95% bound is $u\approx 0.887334219$, so $[0,u]$ is the alternative credible interval shown above.

An equal-tailed credible interval uses quantiles of a continuous posterior distribution to leave posterior probability $\alpha /2$ in each tail. It is also called a central credible interval; central refers to equal tail probabilities, not equal distances from the posterior mean. For the beta distribution $\mathrm{Beta}(10,4)$ in credible interval, the 95% endpoints are approximately $0.461868461$ and $0.909079605$, whose midpoint $0.685474033$ differs from the posterior mean $5/7$.

If a common Kaplan–Meier estimator $\hat{S}$ is fitted and the same data are transformed to $u_i=-\log \hat{S}(x_i)$ with censoring indicators retained, event ordering and risk sets are preserved under compatible tie conventions. At a transformed event time $u_j$, the residual product-limit estimate equals $\hat{S}(x_j)=e^{-u_j}$. Consequently agreement of the pooled residual curve with the exponential target is largely built into the construction. Separate group curves or a model fitted with covariates can reveal differences concealed by the pooled fit. Residuals become infinite if the fitted survivor reaches zero.

// Target: survival-analysis.bigb