For the cubic-matter no-scale vacuum with an exponential dilaton, put $x=S+\bar{S}$ and $A=a{e}^{-\alpha S}$. At $C=0$, zero energy requires $G(S)=b+(1+\alpha x)A=0$. On a fixed-imaginary-part slice, $s=\mathrm{Re}S$ gives $dG/ds=\alpha (1-2\alpha s)A$. At the double root $s_0=1/(2\alpha )$, the first derivative vanishes but the second is $-2{\alpha }^{2}A$, so $G=-{\alpha }^{2}A(s-s_0{)}^2+O((s-s_0{)}^3)$. Therefore $V=|G{|}^2/(x{y}^3)$ begins at fourth order. This real scalar has zero quadratic mass but is not an exact flat direction of a scalar potential. The imaginary part of $S$ is still lifted quadratically, and the matter field $C$ also has a quartic leading potential. Exact flat directions instead come from the complex volume modulus $T$ along the zero-energy locus.

Minimize the displayed criterion using a Leave-one-out cross-validation estimate in the second term. Its expectation is the exact mean integrated squared error minus $R(f)$, which does not depend on h. With standard smoothness and appropriate search ranges, the chosen bandwidth is asymptotically optimal. Its relative fluctuations are ordinarily of order $n^{-1/10}$, considerably larger than the estimation error achievable by well-chosen plug-in pilots.

Subtract the diagonal variance contribution from the derivative estimate's squared norm, then insert the result into asymptotic mean integrated squared error. The expected curvature estimate equals $R(f^{\prime \prime })+O(h^2)$, apart from a negligible finite-n term. This makes the resulting criterion asymptotically correct even though it is not an exactly unbiased estimate of integrated risk. Under standard conditions its selector is ratio-consistent with the optimal bandwidth and has relative fluctuation order $n^{-1/10}$.

Nested cross-validation separates selection from assessment using two levels of held-out data. Within each outer training fold, inner cross-validation chooses a model or tuning parameters; the complete selected fitting procedure is then applied to that outer training fold and scored on its untouched outer evaluation fold. This estimates the performance of the model selection procedure, including its choices, rather than reusing its best inner score as an assessment. A final model can then be selected and fitted using all available training data. ^[ref]

Normalize three distinct points on a generalized circle by a Möbius transformation $f$ taking them to $1,0,\infty$. The inverse image of the extended real line is their generalized circle, and $J=f^{-1}\circ c\circ f$, with $c(z)=\overline{z}$, fixes that circle pointwise and squares to the identity. Changing the three normalizing points replaces $f$ by $M\circ f$ with a real-coefficient Möbius map $M$. Since $M$ commutes with conjugation, $J$ is unchanged. Thus this construction depends only on the circle. It is inversion in a circle for an ordinary circle and Euclidean reflection for a straight line, each extended to the Riemann sphere.

The cross-product matrix $A\mathbf{x}=\mathbf{n}\times \mathbf{x}$ obeys $A\mathbf{n}=0$ and $A^2=\mathbf{n}{\mathbf{n}}^T-|\mathbf{n}{|}^{2}I$, by the vector triple product. On the orthogonal complement of a nonzero $\mathbf{n}$, this is $-|\mathbf{n}{|}^{2}I$; hence the two remaining complex eigenvalues are $\pm i|\mathbf{n}|$. In an orthonormal basis adapted to $\mathbf{n}$, $A$ is a planar quarter-turn multiplied by $|\mathbf{n}|$, with zero action along its axis.

In the quasistatic spherical model, ${\eta }_0=a_0/a_c\in (0,1)$. Separating the radius equation gives $t_e=(S/\kappa ){\int }_0^{a_0}{a}^2/(a_c-a)da$, which is finite and evaluates to the displayed expression. Near disappearance, $a^3\sim 3\kappa a_c(t_e-t)/S$. The time diverges logarithmically as the initial radius approaches the critical radius from below.

Assume deterministic discount factors, deterministic risk-neutral default intensity $\lambda$, fixed fractional recovery $\mathcal{R}$, and continuous premium until default or maturity. Survival is $G(t)=\exp (-{\int }_0^t\lambda (u)du)$. Expected discounted protection is $N(1-\mathcal{R})\int B\lambda G$, while premium is $Ns\int BG$. Equating them gives the par spread. Constant intensity gives $s=(1-\mathcal{R})\lambda$. Discrete premiums, default accrual, stochastic recovery and counterparty losses require changing the corresponding cash-flow expectations.

A basket credit contract whose protection payment is triggered by the first specified default among its reference entities. The loss and settlement terms refer to the defaulting name according to the contract. Premium generally stops at that first default or maturity. Pricing requires the joint default-time distribution; separate marginal default probabilities alone do not determine the first-default law.

For symmetric negative semidefinite matrices $A,B$, each factor $C_A(h)=(I-hA/2{)}^{-1}(I+hA/2)$ has norm at most one, since its eigenvalues are $(1+h\lambda /2)/(1-h\lambda /2)$ with $\lambda \le 0$. The same holds for $B$, so their ordered product is a contraction for every $h\ge 0$, whether or not $A,B$ commute. This proves unconditional discrete L2 norm stability of the split Crank-Nicolson method for directional diffusion. Its splitting error remains first order in general, because the product's quadratic defect is $h^2[A,B]/2$.

For the polynomial Cox ring $S=k[z_{\rho }]$ of a normal split toric variety whose toric rays span the ambient real toric lattice space, the toric irrelevant ideal isThe Cox construction uses the open set $\mathrm{Spec}S\setminus V(B_{\Sigma })$. For the projective line, the two maximal toric cones give generators $z_0,z_1$, so the excluded locus is the origin. For affine space, one maximal toric cone contains every toric ray; the empty product is one, hence $B=(1)$ and nothing is excluded.

The ray-span/no-torus-factor hypothesis belongs to this polynomial presentation. A pure algebraic torus has no toric rays and requires its algebraic torus coordinates in addition to the displayed polynomial data.^[ref]

// Target: geometry-and-topology.bigb

For a normalized spherical harmonic of degree $\ell \ge 1$, write ${\rho }_1=s(r)Y$ and ${\varphi }_1=F(r)Y$. Integration of the radial Poisson equation gives $A=-4\pi G\int r^2{F}^{\ast }sdr$, where $A=\int [r^2|F^{\prime }{|}^2+\ell (\ell +1)|F{|}^2]dr$. Cauchy-Schwarz inequality and $\int |F{|}^2\le A/[\ell (\ell +1)]$ give the displayed bound on $|E_g|=A/(4\pi G)$. For bounded density amplitudes with fixed radial support this decays as ${\ell }^{-2}$, giving a quantitative angular short-scale justification of the Cowling approximation.

Take depth $z$ positive downward, vertical fluid displacement $\xi$ positive upward, and Lagrangian pressure perturbation $\Pi$. In a plane-parallel atmosphere with $p^{\prime }=\rho g$, the Cowling approximation and adiabatic relation giveIndeed the Eulerian perturbations are $p_1=\Pi +\rho g\xi$ and ${\rho }_1=\Pi /c^2+{\rho }^{\prime }\xi$. Horizontal momentum gives ${\mathbf{\xi }}_h=i\mathbf{k}{p}_1/(\rho {\omega }^2)$; combine this with mass conservation and vertical momentum to eliminate the horizontal displacement. Reversing the sign convention for vertical displacement reverses both displacement-pressure coupling signs.

A covering of an infinite game unravels a payoff when its inverse image is a clopen subset of the new infinite-play space. A closed payoff can be unravelled by letting Player I announce a subset of the first exit positions, and letting Player II accept that announcement or challenge one announced position. Accepted plays stop at exit positions, while challenged plays are forced through the challenged exit. Strategy-transfer maps make this more than an ordinary continuous representation of the payoff. Depth-stabilizing inverse limits of coverings handle countable unions in a transfinite induction.

In a sequence of game coverings whose unchanged initial depths tend to infinity, each finite game-tree level eventually stabilizes. These stable levels form a limit game. Compositions of prefix projections and depth-local strategy maps give limit projections and strategy maps. Lift a compatible play coherently through successive coverings. If no premature loss occurs, stabilization of its finite prefixes supplies a compatible limit play. If a premature loss occurs, subsequent lift lengths cannot increase and eventually stabilize, supplying a losing terminal limit play. This proves the lifting condition for the limit covering. It makes countably many pulled-back payoffs clopen simultaneously in the covering proof of Borel determinacy theorem.

A coverage coalitional game has a nonempty core of a cooperative game exactly when all resource sets are pairwise disjoint. Singleton coalition constraints require $x_i\ge |B_i|$, while efficiency requires ${\sum }_i{x}_i=|{\bigcup }_i{B}_i|$. Any overlap makes these inequalities inconsistent. When there is no overlap, the allocation $x_i=|B_i|$ is the unique core allocation and gives every coalition its value exactly.

Decompose a coverage coalitional game into one-resource games and use additivity of the Shapley value. A player contributes a resource precisely when it is the earliest player in a random ordering among those holding that resource. Each holder has probability the reciprocal of the number of holders of being earliest. Summing these contributions proves the formula, regardless of overlaps or emptiness of the core.

Use $F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+g{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$ and $(D_{\mu }c{)}^a={\partial }_{\mu }{c}^a+g{f}^{abc}{A}_{\mu }^b{c}^c$. The Faddeev-Popov gauge-orbit identity with Lorenz gauge function $G^a={\partial }^{\mu }{A}_{\mu }^a$, followed by Gaussian averaging of $G^a=f^a$, supplies the quadratic gauge-fixing term. Represent the determinant of $-{\partial }^{\mu }{D}_{\mu }$ using a Grassmann Gaussian integral over a Faddeev-Popov ghost field and antighost field to obtain the ghost term. The minus sign in this operator differs by a field-independent determinant factor from the direct gauge-condition Jacobian; it is a consistent antighost convention.

The ghost term equals $({\partial }^{\mu }{\bar{c}}^a)(D_{\mu }c{)}^a$ up to a boundary term. Thus it contains a ghost-gluon vertex with the sign matching this kinetic convention. An equivalent auxiliary-field form replaces the gauge-fixing term by $B^a{G}^a+\xi B^a{B}^a/2$. The Nakanishi-Lautrup field equation gives $B^a=-G^a/\xi$ and recovers the displayed Lagrangian. With left-acting BRST transformations $sA=Dc$, $sc=-g[c,c]/2$, $s\bar{c}=B$, and $sB=0$, the gauge-fixing and ghost terms together are $s[{\bar{c}}^a(G^a+\xi B^a/2)]$.

The $\xi \to 0$ limit of covariant gauge fixing by $-({\partial }_{\mu }{A}^{\mu }{)}^2/(2\xi )$. It imposes the divergence condition sharply and gives a transverse photon propagator. This is distinct from the vector-potential choice called Landau gauge for a uniform magnetic field when computing Landau levels.

The Fourier-space quadratic Maxwell kernel after adding $-({\partial }_{\mu }{A}^{\mu }{)}^2/(2\xi )$. For nonzero $\xi$ and non-null four-momentum, its transverse and longitudinal momentum projectors give $K=-k^2(P_T+{\xi }^{-1}{P}_L)$ and its inverse is $-(P_T+\xi P_L)/k^2$. The inversion of the gauge-fixed Maxwell kinetic operator supplies the photon propagator with its vacuum boundary prescription.