Use $-g{\varphi }^3/3!$ and $D=6-2ϵ$ with $m>0$. In an unshifted vacuum, the one-loop mean field has pole $v_{\mathrm{div}}=-g{m}^2/[4(4\pi {)}^3ϵ]$. Attaching it to a cubic vertex gives the connected but one-particle-reducible two-point insertion $(-ig)v_{\mathrm{div}}=i{g}^2{m}^2/[4(4\pi {)}^3ϵ]$. It supplements the proper bubble unless a tadpole subtraction sets the renormalized mean field to zero. A disconnected product of two one-point functions is a separate two-loop term and is not a propagator insertion.

// Target: relativistic-quantum-field.bigb

For every $c>0$, all sufficiently large finite integer sets $A$ with at least $c|A{|}^2$ ordered three-term arithmetic progressions contain a nonconstant four-term arithmetic progression. Cauchy-Schwarz inequality gives $E(A)\ge c^2|A{|}^3$. The Balog-Szemerédi-Gowers theorem supplies a large small-doubling subset. A bounded-modulus Freiman 2-isomorphism gives a positive-density cyclic model; apply the assumed four-term case of Szemerédi's theorem to representatives, then lift the two consecutive second-difference relations.

Around each of $n$ points draw its at most $t$ distinct-distance circles. On every circle containing at least three points, join cyclically consecutive points by arcs. The resulting loopless multigraph has at least $n(n-1)-2nt$ edges and a drawing with $O(n^2{t}^2)$ crossings. A pair joined by several arcs has every corresponding centre on its perpendicular bisector, which is the key to bounding multiplicity.

For a length-$n$, dimension-$k$ cyclic code with generator polynomial $g$ of degree $n-k$, let the message polynomial satisfy $\mathrm{deg}M<k$. Subtracting the remainder upon polynomial division by $g$ makes the displayed word divisible by $g$. The remainder has degree less than $n-k$, so the highest $k$ coordinates retain the coefficients of $M$. This gives systematic encoding without losing any message information.

A Riemann problem for a one-dimensional system of conservation laws ${\partial }_{t}U+{\partial }_{x}f(U)=0$ has two constant initial states, $U_L$ for $x<0$ and $U_R$ for $x>0$. The data and equation are invariant under $(x,t)\mapsto (\lambda x,\lambda t)$. When the physically admissible entropy solution is unique, this invariance makes it a self-similar solution depending on $x/t$. For ideal-gas compressible flow, the characteristic speeds are $u-c,u,u+c$. The two acoustic families carry shock waves or rarefaction waves, while the middle family carries a contact discontinuity. Sufficiently strong separating velocities can instead create a vacuum region.

For constant $P=w\rho$, the perfect fluid stress tensor and inverse perturbed spatial metric give $T^{ij}=a^{-2}w\bar{\rho }[(1+\delta ){\delta }^{ij}-h^{ij}]$ at first order. Normalizing the velocity makes $u^0=a^{-1}$ at this order. Stress-energy conservation yields the synchronous perfect-fluid density equation and Cosmological Euler equation in synchronous gauge; the pressure projection includes $v_i{\bar{P}}^{\prime }$, which changes the velocity damping from $\mathcal{H}$ to $(1-3w)\mathcal{H}$.

// Target: cosmology.bigb

For the FRW metric in conformal time, a displacement ${\xi }^{\mu }=(\alpha ,{\xi }^i)$ changes a raw metric perturbation by $q_{\mu \nu }\mapsto q_{\mu \nu }-{\mathcal{L}}_{\xi }{\bar{g}}_{\mu \nu }$. The displayed first-order time equations set the lapse and shift perturbations to zero. They can be integrated locally for smooth perturbations, demonstrating accessibility of synchronous gauge in cosmology. Their homogeneous solutions leave residual spatial functions, so the gauge is not fully fixed. A globally regular synchronous chart need not survive caustics of its freely falling congruence.

Fix the background signature $+---$ and define a positive spatial metric perturbation by $\delta g_{ij}=+a^2{h}_{ij}$. Thus the synchronous line element isFor a nonzero Fourier mode, take the passive displacement $\tilde{\tau }=\tau +T{e}^{i\mathbf{k}\cdot \mathbf{x}}$ and ${\tilde{x}}^i=x^i+i{\hat{k}}^{i}L{e}^{i\mathbf{k}\cdot \mathbf{x}}$. The transformed scalar shear is $h_S-2kL$, so setting it to zero gives $L=h_S/(2k)$. Removing the mixed metric term gives $T=L^{\prime }/k=h_S^{\prime }/(2k^2)$.

Define the resulting Newtonian gauge line element as $a^2[(1+2\Phi )d{\tau }^2-(1-2\Psi ){\delta }_{ij}d{x}^{i}d{x}^j]$. Its potentials areThese signs follow directly from the linear metric gauge-transformation law. Positive $\delta g_{ij}$ here corresponds to a minus sign for $h_{ij}$ inside the spatial parentheses. The synchronous-to-Newtonian scalar Fourier transformation uses a different explicit background/spatial-amplitude convention. Its displacement signs must be translated along with that convention. The present definitions also fix the signs in the photon redshift identity with positive spatial perturbations.

Quantal neurotransmitter release is release of neurotransmitter in discrete packets associated with individual synaptic vesicles. Packet count and postsynaptic packet response are different quantities; see the binomial distribution example under synaptic vesicle.

A learning rule can constrain the sum or norm of incoming synaptic weights to prevent unbounded Hebbian learning. For ${\sum }_i{w}_{ai}=S$, subtracting the mean unconstrained update from each update preserves the sum. Bounds and renormalization may also be needed to keep excitatory weights nonnegative. Normalization of learning weights is distinct from divisive gain control in instantaneous neural responses.

If there is a symplectic embeddingthen $R\le r$. Symplectic volume alone does not imply this radius obstruction.

For almost complex manifolds $(\Sigma ,j)$ and $(X,J)$, a smooth map $u:\Sigma \to X$ is J-holomorphic when $du\circ j=J\circ du$, equivalently ${\bar{\partial }}_{J}u=0$.

For a finite-dimensional vector space over any field with a nondegenerate alternating bilinear form $B$, the symplectic group over a field consists of invertible linear maps preserving $B$. In a symplectic basis its matrices satisfy $g^{\mathsf{T}}Jg=J$. Matching two symplectic bases gives a unique such linear map, and a symplectic basis exists by successively splitting off paired vectors. This includes arbitrary-characteristic and infinite-field versions, beyond the real Lie group and the symplectic group over a finite field.

Suppose diffeomorphisms preserving capacities of solid ellipsoids converge uniformly on compact sets to a diffeomorphism $\varphi$. For $0<a<1$, uniform closeness and degree of a continuous mapping give ${\varphi }_j(aE)\subseteq \varphi (E)\subseteq {\varphi }_j(a^{-1}E)$. The capacity axioms squeeze $c(\varphi (E))$ between $a^{2}c(E)$ and $a^{-2}c(E)$. Letting $a$ tend to one proves equality. Rescaling a smooth $\varphi$ near a point and repeating the argument also proves capacity preservation by its derivative.

An invertible real linear map preserving symplectic capacities of all solid ellipsoids is symplectic or anti-symplectic. For a pair with $\omega (u,v)=1$, if $0<|\omega (T^{T}u,T^{T}v)|<1$, complete this pair and its normalized transformed pair to symplectic bases. The resulting transpose matrix sends a unit ball into a symplectic cylinder of radius less than one, contradicting the capacity axioms after iterating the resulting linear map: its first coordinate pair contracts geometrically, while the image ellipsoid retains positive capacity. Perturbation handles a zero pairing; using $T^{-1}$ handles absolute value greater than one. Thus the absolute bilinear pairings agree. Their squares agree as polynomials, and the polynomial ring being an integral domain forces one global sign.

A locally uniform limit of symplectomorphisms that is a smooth diffeomorphism is symplectic. By capacity preservation under uniform limits and linear capacity rigidity, its derivative is symplectic or anti-symplectic. Repeat this argument after taking the product of every map with the identity on a symplectic plane. An anti-symplectic derivative on the original factor and a symplectic derivative on the added factor would give neither sign on the product, a contradiction. The argument uses symplectic capacities in the original and stabilized dimensions.

A normalized symplectic capacity has the displayed unit-ball and unit-cylinder values. Dilation gives $\pi R^2$ for both radius-$R$ domains. Monotonicity then implies the sharp Gromov non-squeezing theorem.

An opposite-vertex line in the full octahedral symmetry group is an unoriented coordinate axis. Its stabilizer subgroup has the block form $\mathrm{diag}(A,\epsilon )$, where $A$ is a signed two-dimensional permutation matrix and $\epsilon =\pm 1$. The eight possibilities for $A$ form the dihedral group of a square, of order eight; the independent axial sign gives a cyclic group of order two. Thus the stabilizer is their direct product, of order sixteen. It is larger than the stabilizer of an individual vertex, because exchanging the two endpoints still fixes the line.

The full symmetry group of a cube acts as all signed permutation matrices. The stabilizer subgroups of $(1,0,0)$, $(1,1,0)$ and $(1,1,1)$ have orders eight, four and six, respectively, and their fixed-point subspaces of a group action are the lines through those vectors. The three group orbits have sizes six, twelve and eight. They therefore give distinct axial types for the equivariant branching lemma. More generally the fixed-space dimension of a vector stabilizer subgroup equals the number of distinct nonzero absolute coordinate values, so these are precisely its one-dimensional types.

// Destination: dynamical-systems.bigb

For vertices $(\pm 1,\pm 1,\pm 1)$, the symmetry group of a cube consists of signed permutation matrices. A matrix preserving the unoriented line through $(1,1,1)$ must have all three signs equal, so it has the form $\epsilon P$, where $P$ is a permutation matrix and $\epsilon \in \{1,-1\}$. Central inversion commutes with every $P$ and is not itself a permutation matrix. Consequently $(P,\epsilon )\mapsto \epsilon P$ identifies this twelve-element stabilizer subgroup with the direct product of groups $S_3\times C_2$. A body diagonal is different from a face-normal coordinate axis, whose line stabilizer has order sixteen.