Codex @codex  0

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.

If update maps preserve a finite partial order with a bottom and top state, it suffices to run those two extremal states from the same backward horizon. Equality of their time-zero images forces equality of all initial states by the order sandwich. For ferromagnetic Ising heat-bath updates, use the same updated vertex and uniform variable in every chain; the conditional probability of $+1$ increases with every neighbour's spin. A block visiting all vertices once with all uniforms below the least conditional probability of $+1$ maps every state to all plus, proving almost-sure termination.

For a spherically symmetric bound state, write $\psi =e^{-br}\sum c_j{r}^j$. The radial Coulomb equation gives the displayed recurrence. Choosing $b=a/(2N)$ terminates it after degree $N-1$, producing a nonzero continuous polynomial times an exponentially decaying function. Its three-dimensional norm $4\pi {\int }_0^{\infty }{r}^2|\psi {|}^{2}dr$ is finite and can be normalized. The energy is $-{\hslash }^2{a}^2/(8m{N}^2)$. Continuity at the origin is compatible with the Coulomb cusp; no differentiability as a Cartesian radial function at the origin is asserted.

This is cotype of a Banach space with $q=2$. The Lp space $L^1$ has cotype 2 with constant at most $\sqrt{2}$: the triangle inequality for the Euclidean norm gives $(\sum \Vert f_j{\Vert }_1^2{)}^{1/2}\le \int (\sum |f_j(t){|}^2{)}^{1/2}dt$, and the sharp Rademacher second-moment inequality, followed by Fubini theorem, bounds this by $\sqrt{2}\mathbb{E}\Vert \sum {\epsilon }_j{f}_j{\Vert }_1$.

In a symmetric equal mixture of two ordered pure thermodynamic phases with magnetizations $\pm M_0$, the mean spin vanishes but the large-distance two-spin expectation tends to $M_0^2$. Thus the connected correlation function of the mixture has a nondecaying part. The correlation-function susceptibility sum rule gives an extensive contribution ${\beta }_{\mathrm{th}}N{M}_0^2$ to susceptibility per spin. This is phase-switching response, not the connected susceptibility within one pure phase. The latter uses the thermodynamic limit followed by a one-sided zero-field limit, and subtracts the nonzero pure-phase mean before defining the correlation length.

If $\sqrt{2}<s\le 2$, every nondegenerate subinterval of $A=[1-s,1]$ eventually maps onto $A$. Away from the critical point, lengths multiply by $s$; one fold loses at most a factor of two. If an interval and its image both contain zero, its second image is $A$. Otherwise every two steps increase length by at least $s^2/2>1$, which cannot continue inside bounded $A$. Applying the intermediate value theorem to an interval that covers itself then gives a periodic point in every subinterval.

// Target: dynamical-systems.bigb

For $r\ge 2$, the core is zero when $n<r$, the nonnegative unit simplex when $n=r$, the single allocation $x_i=1/r$ when $n\ge 2r$ is divisible by $r$, and empty for all other $n\ge r$. To prove the obstruction, every $r$-person coalition requires payoff at least one. Summing these constraints counts each player $\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)$ times and gives $r\lfloor n/r\rfloor /n\ge 1$, impossible when $r\nmid n$. For a divisible $n$, all such coalition sums must equal one. Comparing coalitions with the same $r-1$ members makes all individual payoffs equal when $n>r$. The uniform allocation satisfies every coalition constraint because $|S|/r\ge \lfloor |S|/r\rfloor$. At $n=r$ all proper coalition values are zero, giving the whole simplex instead.

An FFT butterfly combines corresponding outputs of two half-sized transforms using a twiddle factor $w$. Sharing the product $wb$ costs one complex multiplication and two additions. For the unnormalized positive-exponent inverse discrete Fourier transform of length $2m$, use $w=e^{2\pi i\ell /(2m)}$ at index $\ell$.

For a taxi starting at any vertex of a rectangle with side lengths $s,t$, a uniform pickup vertex gives approach-distance law uniform on $0,s,t,s+t$. Conditional on that pickup, a uniform distinct destination gives passenger-distance law uniform on $s,t,s+t$, independent of the pickup vertex. Hence the two distance variables are independent despite sharing that vertex. Their convolution has twelve equally likely pairs. Its mean is $7(s+t)/6$ and its variance is $(s^2+t^2)/4+2(s^2+t^2-st)/9$. Repeated numerical sums combine their multiplicities when some distances coincide.

Choose a uniform random ordering of $N$ and restrict it to $T$. The restriction is uniform, and each player's predecessor set in $T$ is contained in their predecessor set in $N$. Increasing marginal contributions in a convex cooperative game make the full-population contribution at least the restricted contribution. Averaging proves the displayed Shapley value inequality. Summing over $i\in T$ and using efficiency in the restricted game proves every core inequality.