A critical point of a smooth map is an input; a critical value is an output reached from at least one such input. For the fold $F(x,y)=(x,y^2)$, all inputs $(a,0)$ are critical points of a smooth map, and all outputs $(a,0)$ are critical values. The identical coordinate pairs live in different copies of ${\mathbb{R}}^2$. To classify an output $(a,b)$, inspect its entire fiber of a function:For $b>0$ both inputs have matrix rank two, so the output is a regular value. For $b=0$ its input has matrix rank one, so the output is a critical value. For $b<0$ the fiber of a function is the empty set, and the output is a regular value despite being outside the image of a function.

In general, for a smooth map $f:X\to Y$, write $C_f$ for its set of critical points of a smooth map. Its set of critical values is $f(C_f)\subseteq Y$. Equivalently, $q\in Y$ is a critical value if some $p\in f^{-1}(q)$ has a differential that is not surjective. A regular value requires a surjective differential at every input in that fiber of a function. The universal condition holds vacuously for an empty fiber of a function: every output outside $f(X)$ is a regular value. These quantifiers are explicit in Loring W. Tu, An Introduction to Manifolds, Definition 8.22 and its two following remarks.

The distinction is useful when solving equations. For a smooth map $f:X^m\to Y^n$ between smooth manifolds without boundary and a regular value $q$ with nonempty fiber of a function, the regular level set theorem makes $f^{-1}(q)$ an embedded submanifold of dimension $m-n$, with tangent spaceThe implicit function theorem explains the freedom count: locally, $n$ independent equations fix $n$ coordinates and leave $m-n$ free. The rank-nullity theorem gives the same count for the displayed kernel of a linear map. See Eckhard Meinrenken, Differentiable Manifolds, Theorems 1.11 and 1.15 for the local projection description and its consequence for level sets.

For example, $H:{\mathbb{R}}^2\to \mathbb{R}$, $H(x,y)=x^2+y^2$, has gradient $(2x,2y)$ and just one critical point, the origin, with critical value zero. For $c>0$, its fiber of a function is the circle of radius $\sqrt{c}$, a one-dimensional embedded submanifold; negative $c$ are regular values with empty fibers. At $c=0$, the fiber of a function is a single point instead of a circle. A critical value does not force its fiber of a function to be nonsmooth: that singleton is itself a zero-dimensional embedded submanifold, but the regular dimension conclusion $2-1=1$ does not apply.

Nor does a single regular input certify a regular value. The scalar smooth function $h(t)=t^2(t-1)$ has $h^{-1}(0)=\{0,1\}$ and derivative $h^{\prime }(t)=3t^2-2t$. The input zero is a critical point, while the input one is regular because $h^{\prime }(1)=1$. Their shared output zero is nevertheless a critical value. Conversely, different critical points of a smooth map can share one critical value: $f(t)=(t^2-1{)}^2$ sends both critical inputs $-1$ and $1$ to zero. Counting inputs and counting critical outputs are different tasks.

Finally, for $G(t)=(t,t^2):\mathbb{R}\to {\mathbb{R}}^2$, every output on the parabola is a critical value, and every output off it is a regular value with empty fiber of a function. There is no nonempty regular fiber of a function here, so the negative expression $1-2$ never becomes a claimed manifold dimension. Both diagrams show exact mathematical examples, not measured data.

On the augmented precedence graph, the linear programming problemhas dual $\max {\sum }_{i\to j}{\tau }_i{f}_{ij}$, with $f\ge 0$ a unit source-to-sink flow. Equivalently, the dual is the negative of the optimal uncapacitated minimum-cost flow value with arc costs $-{\tau }_i$. The flow selects a longest path, while the time variables provide a matching weak duality bound.

// Target: mathematical-optimization.bigb

Task float, also called total float or task slack, measures an activity's available start-time delay relative to its earliest start, while allowing a feasible schedule that meets the fixed project finish. For earliest start $E_i$, latest start $L_i$, and fixed duration ${\tau }_i$, it isIn the six-activity project scheduling example with start zero and finish 10, B has task float 2 and E has task float 6. B can move from the interval $[3,5]$ to $[5,7]$, and E from $[0,3]$ to $[6,9]$, without changing D's or F's earliest starts. A, C, D, and F have zero task float. With the finish fixed to the optimal duration and the basic critical path method assumptions, zero task float identifies exactly the activities lying on at least one critical graph path. See Hendrickson, §10.4, Activity Float and Schedules.

Task float is not generally an independent allowance that can be spent separately on every activity. For a small example, let two duration-1 activities X then Y form one chain, running in parallel with a separate duration-4 activity. The optimal finish is 4. X has earliest/latest starts 0/2 and Y has earliest/latest starts 1/3, so each has task float 2. Delaying X to 2 forces Y to start at 3, using Y's allowance too. Choosing X at its latest start 2 and Y at its earliest start 1 violates precedence, although both chosen times lie inside their individual ranges. The optimization constraints still couple the decisions. Here B and E can both use their full task floats together because they lie on separate branches; that freedom must be checked from the directed acyclic graph.

// Target: mathematical-optimization.bigb

For a monotone inviscid parallel shear flow, let a regular neutral Rayleigh equation for inviscid shear flow mode have $U(0)=c_0$, $U^{\prime \prime }(0)=0$, $d=U^{\prime }(0)>0$, and $p={\varphi }_0(0)\ne 0$. A perturbation $c=c_0+\epsilon c_1$ with $\mathrm{Im}{c}_1>0$ resolves the singularity on the inner variable $\eta =y/\epsilon$. The first outer correction has ${\varphi }_1=a+\beta y\log |y|+b_{\pm }y+\cdots$, with $\beta =c_1{U}^{\prime \prime \prime }(0)p/d^2$. The inner particular solution contains $(d\eta -c_1)\log (d\eta -c_1)$. Its complex logarithm has argument tending to zero at positive infinity and to $-\pi$ at negative infinity, giving the displayed matching jump.

A credit rating is an ordinal assessment of an issuer's or obligation's creditworthiness. Agency scales group credits into broad investment-grade and speculative-grade classes, with default categories and rating transitions. A rating is not itself a calibrated default probability, and identical ratings do not establish independence or constant intensities of defaults.

A yield difference between a credit-risky instrument and a chosen benchmark, with maturity and other terms matched as appropriate. The benchmark and quotation convention must be specified. The spread may compensate for default losses, default-risk premiums, liquidity and other effects; it is not generally identical to a physical default probability or loss rate.

The risk that a trading counterparty defaults while owing a positive replacement value or settlement amount. It differs from the reference borrower's default exposure in a credit default swap. A credit hedge may fail precisely when its protection seller is distressed. Collateral, netting and margining change exposures and loss amounts; they do not turn every mismatched hedge into a risk-free claim.

In the basic structural model, the firm's asset value $V$ follows geometric Brownian motion with constant volatility and a single debt obligation of face value $F$ matures at $T$. Default at maturity occurs when $V_T<F$. Equity is a call on firm assets; debt is the risk-free face claim minus a put on those assets, since $\min (V_T,F)=F-(F-V_T{)}^{+}$. This links debt valuation to asset value, volatility and capital structure. The basic model does not permit default before maturity; richer structural models add a default barrier or more detailed obligations.

Loss given default is the loss conditional on a default, measured as an amount or as a fraction of exposure under the chosen convention. In an independent marked default model the loss amounts are independent marks attached to the events of a Poisson process. Their raw second moment, not just their variance, controls the aggregate compound Poisson distribution variance.

Bayesian credibility estimates a risk's conditional expectation by its Bayesian posterior expected value under squared-error loss. When that posterior mean is an affine combination of exposure-weighted individual experience and a prior mean, the coefficient of the experience is its credibility factor. Exact Bayesian credibility is distinct from merely selecting the best affine estimator when the true posterior mean is nonlinear.

For a Hermitian fermion bilinear, an antiunitary operator implementing CPT conjugates numerical coefficients and exchanges particle and antiparticle fields at $-x$. Reordering the fermions then gives $\Theta :\overline{\psi }\Gamma \psi (x):{\Theta }^{-1}=:\overline{\psi }{\gamma }_5\Gamma {\gamma }_5\psi (-x):$ when ${\Gamma }^{†}={\gamma }^0\Gamma {\gamma }^0$. Thus scalar, pseudoscalar and tensor bilinears are even, while vector and axial-vector bilinears are odd. Each derivative contributes another minus sign. The graded reordering is essential; treating Dirac fields as commuting quantities gives the wrong transformation.

Under CPT, a Hermitian local tensor monomial picks up a minus sign for each Lorentz vector index, including derivative indices. A Lorentz scalar contracts all such indices, so the total sign is positive; charged terms are exchanged with their Hermitian conjugates. Therefore a local Hermitian Lagrangian density obeys $\Theta \mathcal{L}(x){\Theta }^{-1}=\mathcal{L}(-x)$, leaving its action invariant. This field-level argument illustrates the CPT theorem for ordinary scalar, spinor and vector theories; the axiomatic theorem also uses positive energy and microcausality.

In a CPT-invariant vacuum, a subatomic particle and its antiparticle have equal masses and total decay widths, equal spin magnitudes and opposite conserved additive charges. CPT exchanges initial and final states in a scattering experiment; it relates a process to the reversed process with antiparticles and reversed spins. It does not force equality of every pair of conjugate partial decay widths when CP violation is present.

A weak phase is a phase in a coupling or decay amplitude that reverses sign in the CP symmetry conjugate process. A physically relevant difference must survive allowed field rephasings; a complex coupling alone does not prove CP violation.

A direct CP asymmetry compares decay widths for a process and its CP symmetry conjugate. Let two decay amplitudes contribute with real nonnegative magnitudes $a_1,a_2$, absorptive phases ${\delta }_i$ and weak phases ${\varphi }_i$. Then $\mathcal{A}=a_1{e}^{i({\delta }_1+{\varphi }_1)}+a_2{e}^{i({\delta }_2+{\varphi }_2)}$ and $\overline{\mathcal{A}}=a_1{e}^{i({\delta }_1-{\varphi }_1)}+a_2{e}^{i({\delta }_2-{\varphi }_2)}$. Expanding the absolute squares givesThus two nonzero contributions and both kinds of relative phase are required in this example. Equal total decay widths under CPT symmetry constrain the sum over channels and do not forbid individual channel asymmetries.

The vector current is odd and the axial current is even under charge conjugation. Under parity symmetry in quantum field theory, the first is a vector and the second a pseudovector. Thus their difference obeys $j_L^{\mu }(x)\mapsto -P^{\mu }{}_{\nu }{j}_L^{\nu }(x_P)$ under CP, with $P=\mathrm{diag}(1,-1,-1,-1)$. If a neutral vector has $V^{\mu }(x)\mapsto \xi P^{\mu }{}_{\nu }{V}^{\nu }(x_P)$ under CP, the interaction transforms by the factor $-\xi$. It is invariant for $\xi =-1$. Being a real field restricts its intrinsic phases but does not alone fix this sign. The conventional C-odd neutral gauge-vector assignment makes real diagonal Z boson vector and axial couplings CP invariant, despite their separate C and P violations.

For a time step $h>0$, choose the undiscounted stock multipliers and money-market multiplier asFor sufficiently small $h$, $D_h<R_h<U_h$ and the risk-neutral up probability is $q_h=(R_h-D_h)/(U_h-D_h)$. A one-step log-stock increment takes values $\pm \sigma \sqrt{h}$; expanding $q_h$ gives mean $(\rho -{\sigma }^2/2)h+O(h^2)$ and variance ${\sigma }^{2}h+O(h^2)$. The Donsker invariance principle, in its triangular-array form, gives the risk-neutral log-price limitThus prices converge to the geometric Brownian motion used in the Black-Scholes model.^[ref]

The parent model's discounted multipliers are instead $u_h=U_h/R_h$ and $d_h=D_h/R_h$. Their log-price drift loses the $\rho$ term, as required for the discounted price to be a martingale. Convergence of claim prices also needs the payoff functional to be continuous at almost every limiting path, together with Uniform integrability when payoffs are unbounded. A process limit alone does not justify taking expectations of every payoff.

A likelihood-based analogue of the coefficient of determination compares the fitted and intercept-only log-likelihoods ${\ell }_1$ and ${\ell }_0$:For a Cox proportional-hazards model, these are maximized partial likelihood logarithms with and without the covariates. The expression measures improvement in fit, rather than the fraction of variation in event times explained. Its attainable maximum can be below one. It does not test the proportional hazards assumption or establish predictive calibration.

A homotopy into the base of a covering space has a unique continuous lift once its initial map is continuously lifted. Local inverse sheets construct the lift along each time interval, and compactness of that interval permits a finite subdivision locally around each point of the homotopy's parameter space. Uniqueness on overlapping intervals follows from path lifting theorem uniqueness. In a homotopy of paths with endpoints fixed, a chosen lifted endpoint remains fixed because it varies continuously in a discrete fiber.

An oriented coordinate neighborhood of a smooth manifold gives two sheets distinguished by the sign of its orientation. On an overlap, the sheets agree where the two local orientation forms differ by a positive function. These open agreement sets define a two-sheeted smooth covering space, whose projection is locally a diffeomorphism.