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The orthogonal transformation ${\psi }_{\pm }=({\varphi }_1\pm {\varphi }_2)/\sqrt{2}$ takes $g({\varphi }_1^4+{\varphi }_2^4+6{\varphi }_1^2{\varphi }_2^2)$ to $2g({\psi }_{+}^4+{\psi }_{-}^4)$. These are the same two decoupled Ising models expressed in rotated fields.

For the cotangent bundle projection $\pi :T^{\ast }X\to X$, the canonical differential one-form is ${\lambda }_{(x,\xi )}(v)=\xi (d\pi (v))$. In coordinates it is ${\sum }_j{\xi }_{j}d{x}_j$. The canonical symplectic form is $-d\lambda$ in the position-first convention ${\sum }_{j}d{x}_j\wedge d{\xi }_j$.

For $|N^2|\ll {\Omega }^2$, this leading amplitude-growth rate is largest at thermal diffusion rate of a Fourier mode $\beta =\Omega$, giving ${\sigma }_{\max }=-N^2/(4\Omega )$ when $N^2<0$.

A continuous local martingale $M$ with $M_0=0$ and deterministically bounded quadratic variation on every finite horizon is a square-integrable martingale, with $\mathbb{E}{M}_t^2=\mathbb{E}[M{]}_t$. Localization, the Doob L2 maximal inequality, and the resulting integrable bound on the squared supremum justify removing the localizing sequence.

The quotient of a horizontal strip by a horizontal translation. The model $(\mathbb{R}/\mathbb{Z})\times (0,M)$ has conformal modulus of an annulus $M$. Averaging horizontal loop lengths and applying Cauchy-Schwarz inequality proves that the core-loop extremal length equals $1/M$.

The conditional entropy is the integral of the conditional information function. Its chain rule expresses the entropy of a join of measurable partitions as unconditional entropy plus conditional entropy. Conditioning reduces entropy, also for countable partitions of finite entropy.

For increasing sigma-algebras ${\mathcal{G}}_n$, put $f_n=I_{\mu }(\xi \mid {\mathcal{G}}_n)$ and $F={\sup }_n{f}_n$. For each atom $A\in \xi$ and $t\ge 0$,Together with the bound by $\mu (A)$ and the tail integral formula for moments, this yields $\int Fd\mu \le H_{\mu }(\xi )+1$. The inequality follows by stopping the conditional-expectation martingale $\mathbb{E}[1_A\mid {\mathcal{G}}_n]$ the first time it falls below $e^{-t}$.

If D-separation and conditional independence are equivalent for a Directed acyclic graph, its moral graph is its conditional independence graph. After conditioning on all other vertices, an active path can only be a direct edge or a two-edge collider through a common child: two adjacent interior vertices cannot both be colliders. These are exactly the moral edges.

For Gaussian data, regress each variable on all remaining variables using Lasso. Nonzero coefficients estimate its neighbours in the conditional independence graph. Symmetrize the separate regressions using either the union or the intersection of their selected directed neighbourhoods.

For real $A_{\mu }$ in positive-definite Euclidean signature, $\partial \cdot A$ is real and $A_{\mu }{A}_{\mu }\ge 0$. Therefore $\partial \cdot A+i{A}^2=0$ forces $A^2=0$, and hence $A=0$. Its only real gauge orbit is the zero-field orbit; a field with nonzero electromagnetic curvature cannot be gauge-transformed to zero. Consequently this complex equation is not an admissible gauge fixing for general Hermitian Euclidean U(1) gauge symmetry configurations. A formal complexified treatment needs a specified contour and cannot be justified by a real delta-functional insertion.

A massless complex bosonic charged scalar field with $D_{\mu }={\partial }_{\mu }+i{A}_{\mu }$ contributes $[\det (-D^2){]}^{-1}$ to a Gaussian background-field integral. A Faddeev-Popov ghost field pair contributes $\det {\mathcal{O}}_A$. On the formal complex quadratic Abelian gauge condition, ${\mathcal{O}}_A=-D^2$, so the two determinants cancel and their effective-action contributions sum to zero. Use matching regulators, boundary conditions and removed zero modes. This is a formal cancellation of closed loops; the reality obstruction for a complex Euclidean gauge condition precludes deducing that physical scalar quantum electrodynamics has no quantum effects.

A polydisc is a product of open discs in $\mathbb{C}$. It is a contractible complex manifold, with vanishing positive-degree de Rham cohomology and vanishing Dolbeault cohomology in positive antiholomorphic degree.

The structure sheaf assigns to an open set its ring of holomorphic functions. Its invertible sections form the sheaf ${\mathcal{O}}_M^{\ast }$ of nowhere-zero holomorphic functions.

Expand both sides using the definition of a commutator: the middle terms $BAC$ cancel, leaving $ABC-BCA$. Thus commutation with a fixed operator is a derivation of an algebra. In particular, $[A,[B,C]]=[[A,B],C]+[B,[A,C]]$, which gives a short way to derive a Lie algebra representation from generator actions on an underlying algebra.

An instantaneous kinetic suppression estimate replaces sound speed by the velocity dispersion of cold dark matter. Nonrelativistic particles in thermal contact with photons have $\sigma \propto a^{-1/2}$; after kinetic decoupling, their momenta redshift and $\sigma \propto a^{-1}$. A background dynamical estimate using total density is then constant during radiation domination and decreases as $a^{-1/2}$ during matter domination. Using only the species' own matter density instead makes the self-gravitating collisionless Jeans length decrease as $a^{-1/2}$ throughout the decoupled nonrelativistic era. Neither instantaneous estimate is the accumulated distance traveled by collisionless free streaming.

An associated point of a coherent sheaf is a point whose corresponding prime is the annihilator of an element of the module on an affine chart. There are finitely many on a Noetherian scheme. A section of a line bundle avoiding them acts injectively on the sheaf, even when the scheme is nonreduced. Over an infinite field a sufficiently ample linear system supplies such a section.

The support dimension is the dimension of a scheme of the closed support of a coherent sheaf. Tensoring with a line bundle leaves its support unchanged. Quotienting by a section that avoids its associated points reduces the support dimension by at least one.

For $X=\left(\begin{array}{cc}a& b\\ c& -a\end{array}\right)$, the image of its matrix exponential is compact precisely when $a^2+bc<0$ or $X=0$. If $a^2+bc=-{\omega }^2<0$, then $e^{tX}=\cos (\omega t)I_2+\sin (\omega t)X/\omega$ and its image is conjugate to the circle group. A nonzero nilpotent generator or a generator with real nonzero eigenvalues produces an unbounded image.

For a finite-dimensional memoryless quantum channel,The Holevo bound controls information obtainable from an output ensemble, while collective decoding makes rates below the corresponding coding expression achievable. The Holevo capacity needs regularization because input states can be entangled across channel uses. Watrous's channel-coding chapter states and develops the classical-capacity theorem.

For random proper simple curves from $0$ to infinity in the complex upper half-plane, the probabilities of avoiding every compact H-hull whose closure excludes zero determine the unparameterized trace law. Every point off a simple chordal curve can be connected to a real boundary interval away from zero within one side of its complement. A countable collection of thin rational polygonal tubes attached to that interval therefore recovers the complement from its avoidance indicators. Joint avoidance of finitely many tubes reduces to avoidance of their filled union. If the union blocks every path from zero to infinity the joint event is empty; otherwise filling pockets does not change the avoidance event. The inclusion-exclusion principle gives all finite indicator distributions. These determine the random complement and hence the trace. Simplicity and properness are hypotheses; the assertion is not a general determination theorem for arbitrary random closed sets.