Codex @codex  0

Give every site a distinct label and interchange labels at the edge marks defining the symmetric simple exclusion process. Each forward or backward label path makes only finitely many jumps in a bounded time interval, since its total rate is $2c$. Occupations are transported along these paths. The backward ancestors of a finite set have the law of the finite-particle symmetric exclusion process, by time reversal of the Poisson marks. This proves the product self-duality of symmetric exclusion.

On a finite graph with symmetric edge-swap rates, the stirring representation of symmetric exclusion transports the occupations by a random permutation. An independent identically distributed Bernoulli distribution occupation field is invariant under every deterministic permutation, and hence under this random permutation when it is independent of the initial field. Therefore the displayed product measure is invariant, including densities zero and one. Equivalently, an edge swap preserves the number of particles and has the same reverse rate, giving detailed balance.

Its adjoint is $i{f}^{\prime }$ on all of $H^1(0,1)$, with no endpoint conditions. Every complex $\lambda$ is an adjoint eigenvalue, with eigenfunction $e^{-i\lambda x}$. Thus every complex $\lambda$ obstructs surjectivity of $\lambda I-A_0$. The original operator has no eigenvalues because its zero initial value kills any solution of $i{f}^{\prime }=\lambda f$.

// Target: analysis.bigb

For a symmetric finite game, pure-action expected payoffs $e(i,p)$ against independent identical opposing mixed strategies are continuous. The displayed map takes the probability simplex continuously into itself. At a Brouwer fixed-point theorem fixed point, put $r_i=[e(i,p)-e(p){]}_{+}$ and $S={\sum }_i{r}_i$. The fixed-point equations give $r_i=p_{i}S$. If $S>0$, every supported action has strictly positive excess payoff, contradicting ${\sum }_i{p}_i(e(i,p)-e(p))=0$. Hence $S=0$ and every supported action is a best response, proving a symmetric Nash equilibrium.

For functions in the form domain, this symmetric bilinear energy equals $-⟨Lf,g⟩$ when $f$ is in the generator domain. Its finite-time approximation is $(2t{)}^{-1}\int (f(x)-f(y))(g(x)-g(y))\mu (dx)P_t(x,dy)$. The quadratic energy ${\mathcal{E}}_{\mu }(f)={\mathcal{E}}_{\mu }(f,f)$ may take the extended value infinity.

// Target: analysis.bigb

This is the critical infection rate for the contact process started from one infected site to avoid its absorbing empty state forever. It concerns infection somewhere in the graph at all times, rather than repeated infection of a fixed site. On a transitive graph it is independent of the starting site. Infection-arrow inclusion couples different rates monotonically.

Local survival means repeated infection of a fixed vertex at arbitrarily large times in the contact process. It implies global survival, but infection can survive by moving away from every fixed vertex on an infinite graph. On a transitive graph the threshold is independent of the chosen vertex, and ${\lambda }_1\le {\lambda }_2$, with ${\lambda }_1$ the global survival threshold of the contact process.

For a smooth tangent vector field $v$ on a compact orientable surface with smooth boundary and outward conormal of a surface boundary, the integral of its surface divergence is its boundary flux. Extend the unit normal vector to a unit field $m$. The curl of a cross product gives $m\cdot \nabla \times (m\times v)=\nabla \cdot v-m_i{m}_j{\partial }_j{v}_i$ on the surface: $m\cdot v=0$ there and $m\cdot {\partial }_{j}m=0$ by unit length. Apply Stokes theorem and use $(m\times v)\cdot d\mathbf{r}=v\cdot (d\mathbf{r}\times m)$ to obtain the formula.

For a Borel probability measure on a Polish space and a continuous map to another Polish space, the image measure has the displayed support. Every neighborhood of an image of a support point has an open inverse image of positive measure. Conversely, the closed image closure has full image measure because the original support has full measure. This transfers a support description for a driving rough path to the solution of a rough differential equation.

In flat four-dimensional N=1 supersymmetry, let ${\bar{D}}^2={\bar{D}}_{\dot{\alpha }}{\bar{D}}^{\dot{\alpha }}$, contracting the two spinor indices with the antisymmetric epsilon invariant tensor. The supercovariant derivative algebra with left derivatives has $\{{\bar{D}}_{\dot{\alpha }},{\bar{D}}_{\dot{\beta }}\}=0$. There are only two dotted components, so every product of three barred derivatives repeats one component and vanishes. Therefore ${\bar{D}}_{\dot{\alpha }}{\bar{D}}^{2}X=0$ for any superfield $X$: ${\bar{D}}^{2}X$ is a chiral superfield. This map is called chiral projection, although ${\bar{D}}^2$ is not an idempotent projector.

If $\Phi$ is a chiral superfield, holomorphic closure of chiral superfields also makes $W^{\prime }(\Phi )$ chiral. Thus an equation ${\bar{D}}^2{\Phi }^{†}=4W^{\prime }(\Phi )$ is consistent with the chiral constraint: applying any ${\bar{D}}_{\dot{\alpha }}$ gives zero on both sides. Both sides are scalar superfields, and the squared supersymmetric covariant derivative commutes with the supercharges, so the equation is Lorentz invariant and supersymmetry covariant.

A solid can locally exceed its solidus because a thin solute-diffusion layer changes composition faster than thermal conduction changes temperature. Near a melting phase boundary, a depleted solid can join the equilibrium solidus at the interface while the adjacent, more concentrated solid remains above its own lower solidus. The assumed single sharp interface is then thermodynamically unstable to additional melting and may be replaced by a mushy layer.

The axis of a bipolar solar active region is statistically tilted relative to the local east-west direction, with its following polarity typically farther from the equator and a mean tilt that grows with latitude. Scatter is substantial. The separation in latitude of the two polarities allows their surface dispersal to generate a net poloidal field through the Babcock-Leighton mechanism.

Suppose a fixed pair of Gassmann equivalent finite covers of a compact two-dimensional orbifold has isometric limiting hyperbolic metrics. Choose nearby orbifold-smooth nowhere locally homogeneous metrics and rescale them to the displayed distinct areas. Scaling by a positive constant multiplies area by that constant in dimension two, preserves local rigidity, and preserves convergence when the constants tend to one. Sunada theorem supplies equal spectra within each pair. Local rigidity forces any quotient isometry to descend to the identity on the base; nonconjugate covering subgroups therefore exclude within-pair isometry. Distinct areas exclude all cross-pair isometries and spectra, because the leading heat trace coefficient is area divided by $4\pi t$. Fixed markings of the two isometric limiting covers give convergence to the same metric. The existence of those isometric limits is an additional geometric input, not a consequence of Gassmann equivalence alone.

For a finite isometric group action on a closed Riemannian surface, choose a disk in the free locus whose translates are disjoint. Modify the Riemannian metric there, and identically on its translates, by small supported changes with three distinguished high Gaussian curvature maxima. Give their heights distinct values occurring nowhere else, arrange them as a tight noncollinear cluster, and keep separate clusters far apart. Such markers can be made by local conformal bump functions controlling curvature and its derivatives. Every quotient isometry must send an entire labelled cluster to another. After identifying the marked disks with the original one, an isometry fixes its three noncollinear points and is therefore the identity locally. If $p_i:M\to U_i\backslash M$ are free finite quotients, any quotient isometry $F$ consequently makes $F\circ p_1=p_2\circ t$ on an open disk for some element $t$ of the original group. Two local isometries agreeing on an open set agree on the connected surface, by propagation along geodesics. The equality therefore holds globally and implies $t{U}_1{t}^{-1}\subseteq U_2$, hence equality for equal subgroup orders. Nonconjugate equal-order subgroups thus have nonisometric quotients for this marked invariant metric; constant curvature is not required.

Let a finite group $T$ be generated by elements of exact orders $p,q$ whose product has exact order $r$, with $p^{-1}+q^{-1}+r^{-1}<1$. The epimorphism from the hyperbolic triangle group sending its generators to these elements has torsion-free kernel. Its quotient of the hyperbolic plane is a closed surface with a $T$ action. A subgroup acts freely precisely when it meets no conjugate of a nonidentity power of the three elliptic generators. Free almost conjugate subgroups give isospectral manifolds by the Sunada theorem.

Take a hyperbolic torus with a single cone point of angle $2\pi /7$. Its orbifold fundamental group is $⟨a,d\mid [d,a{]}^7=1⟩$. An epimorphism to $\mathrm{PSL}(3,2)$ whose commutator has exact order $7$ has torsion-free kernel. The point and line stabilizers have index $7$ and order $24$, hence avoid all nontrivial cone stabilizers. Their quotient covers are smooth surfaces with Euler characteristic $7(-6/7)=-6$, so have genus $4$. Their equal coset permutation characters give equal heat traces by the Sunada theorem. Nonisometry still requires a geometric distinction, such as the intersection test for nonisometric finite covers.

Let two finite locally isometric covers carry uniquely length-identified primitive closed geodesics ${\alpha }_i,{\beta }_i$. If their geometric intersection numbers differ, the covering surfaces are not isometric, because an isometry preserves both lengths and intersections. For base simple curves intersecting once, lift components correspond to cycles of their monodromy permutations. Their intersection number is the number of sheets common to the two cycles. This distinguishes covers even when every individual permutation has the same cycle type in both actions, as in Gassmann equivalence.

A finite isometry group decomposes $L^2(N)$ into its irreducible representation factors $V_{\rho }$ and multiplicity Hilbert spaces ${\mathcal{H}}_{\rho }$. The Laplacian commutes with the group, so it acts as the identity on each representation factor. The averaging projection gives $\dim V_{\rho }^{U_i}=|U_i{|}^{-1}{\sum }_{u\in U_i}{\chi }_{\rho }(u)$. For Gassmann equivalent subgroups these dimensions agree. Choosing unitary identifications of the invariant finite-dimensional factors therefore intertwines the quotient Laplacians, even when a noncompact covering manifold has no finite heat trace.

For a finite subgroup $U$ of a group $T$ of Riemannian isometries of a closed manifold $N$, assume $U$ acts freely and define $I_t(g)={\int }_N{H}_N(t,x,gx)dV(x)$. Invariance of the Riemannian heat kernel and change of variables show that $I_t$ is a class function on $T$. For $U\le T$, the heat kernel on a finite isometric quotient and integration over a fundamental domain give the displayed heat trace formula. Grouping by conjugacy classes proves equality for Gassmann equivalent subgroups. More generally the same argument allows fixed points of $T$ outside the freely acting subgroups; only the quotient under consideration must be smooth.

A matrix minor is the determinant of a square submatrix obtained by selecting rows and columns of a matrix. Over a field, the matrix rank is less than $r$ exactly when every $r$-by-$r$ matrix minor vanishes. In a matrix of regular functions, these equations define a closed rank-defect locus, explaining the determinantal variety construction.