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For zero scalar potential, the static field equations are $D_i{D}_i\Phi =0$ and $D_i{F}_{ij}=[\Phi ,D_j\Phi ]$. If $B_i=-D_i\Phi$, the gauge-theory Bianchi identity gives the first equation. For the second,using $F_{ik}={ϵ}_{ik\ell }{B}_{\ell }$ and ${ϵ}_{ijk}{ϵ}_{ik\ell }=-2{\delta }_{j\ell }$. Thus the first-order equations imply the full second-order equations locally, independently of the radial ansatz.

For the trace inner product and adjoint covariant derivative, compatibility gives $\Delta |\Phi {|}^2=2|D\Phi {|}^2+2⟨\Phi ,D_i{D}_i\Phi ⟩$. The gauge-theory Bianchi identity and $B_i=\pm D_i\Phi$ imply $D_i{D}_i\Phi =0$. Therefore $\Delta |\Phi {|}^2=2|D\Phi {|}^2$, equal to the energy density $|B{|}^2+|D\Phi {|}^2$ on a Bogomolny-Prasad-Sommerfield monopole. If the energy density is instead defined with an overall factor $1/2$, this Laplacian is twice that density.

The sign $\sigma =\pm 1$ distinguishes an inverted quartic potential from a double-well potential. The origin has determinant $\lambda$ and trace $\kappa$. Nonzero equilibria obey $q^2=\lambda /\sigma$, with determinant $-2\lambda$ and trace $\kappa -2\lambda$. Its energy $p^2/2+\lambda q^2/2-\sigma q^4/4$ has derivative $(\kappa -2\sigma q^2)p^2$. The origin's Hopf bifurcation is supercritical for $\sigma =1$ and subcritical for $\sigma =-1$. In the latter case the nonzero equilibria have supercritical Hopf bifurcations at $\kappa =2\lambda <0$.

// Target: dynamical-systems.bigb

For $\lambda >0$, the equilibrium points are $x=\pm \sqrt{\lambda }$, $y=0$. The positive branch is a saddle; the negative branch has a subcritical Hopf bifurcation at $\mu =\sqrt{\lambda }$. A homoclinic orbit occurs at $\mu =(5/7)\sqrt{\lambda }+O(\lambda )$ near the double-zero point. Between this homoclinic curve and the Hopf curve, an unstable periodic orbit surrounds a stable equilibrium point. The factor $5/7$ follows from a Melnikov energy-balance method, using the integrals of $v^2$ and $u{v}^2$ along the leading conservative saddle loop.

// Destination: dynamical-systems.bigb

For $\lambda >0$, the equilibrium points are the origin and $(s,s^2)$ with $s=(1\pm \sqrt{1+4\mu })/2$. The nonzero branches have Jacobian trace $s-\lambda$ and determinant $\lambda s(2s-1)$. They have a saddle-node bifurcation at $\mu =-1/4$, $s=1/2$; the origin exchanges stability with the lower branch at the transcritical bifurcation $\mu =0$. A Hopf bifurcation occurs on the upper branch at $\mu =\lambda (\lambda -1)$ for $\lambda >1/2$. The point $(\mu ,\lambda )=(-1/4,1/2)$ has a double zero eigenvalue and a nontrivial Jordan block. Its generic unfolding includes a saddle-loop curve and the intervening stable periodic orbit region.

For finite sums of compactly supported convolution squares, $\hat{p},\hat{q}\ge 0$. Insert the Bochner theorem representation of $q$ into $p\ast q$ and apply the Fubini theorem: its representing measure is $\hat{p}{\mu }_q$. Reversing $p,q$ and using uniqueness gives the displayed identity. It makes the local quotients ${\mu }_p/\hat{p}$ consistent wherever $\hat{p}>0$.

A positive-definite function $r$ on the discrete group $\mathbb{Z}$ is the sequence of Fourier coefficients of a unique finite positive Borel measure on the circle, with total mass $r(0)$. Positivity makes the finite sums ${\sum }_{|n|<N}(1-|n|/N)r(n)e^{-2\pi int}$ nonnegative. Their measures converge weakly to the required measure; uniqueness follows from density of trigonometric polynomials in continuous functions.

An ordinary time integral of random variables taken in the L2 space of the underlying probability space. Under strong measurability and ${\int }_a^b(\mathbb{E}|X_t{|}^2{)}^{1/2}dt<\infty$, it exists as a Bochner integral; for a mean-square continuous process it is also the limit of Riemann sums in that space. If the process is centered and the covariance is integrable, Fubini's theorem givesFor a Gaussian process, the integral is a Gaussian random variable, by approximating it in mean square by finite linear combinations of jointly Gaussian random variables.

For a transitive finite permutation group, suppose each point stabilizer has orbits of a common size $r$ on the other points. If a nontrivial block system exists, all two-point stabilizers are trivial. Indeed for block size $b$, $r\mid n-1$ gives $\gcd (r,b)=1$. An orbit of a block not containing $\alpha$ has size at most $br$ and divisible by $br$, so its b point-orbits are disjoint. A stabilizer of $\alpha$ and a point of that block fixes the whole block. Interchanging the two points also fixes the block of $\alpha$. Equal two-point-stabilizer orders then show that each two-point stabilizer fixes every point, and faithfulness makes it trivial.

A block subspace is the closed linear span of an infinite block sequence. Passing to a block subspace preserves the order of supports and permits successive-vector constructions.

Let $Z=⟨z⟩\le Z(G)$ have order $p$ and set $t=z-1$. The nilpotent ideal $tkG$ is generated by a central element. If an idempotent lifts a central idempotent of the quotient, its off-diagonal corner satisfies $eA(1-e)\subseteq teA(1-e)$ and therefore is zero. Thus central block idempotents lift centrally and uniquely. A principal indecomposable module is free over $kZ$, and the $p$ factors of its $t$-filtration are all its reduction modulo $t$. Consequently, under the natural indexing of simple modules, $C_G=p{C}_{G/Z}$.

For a complete discrete valuation ring $\mathcal{O}$ with residue field $k$, class sums give bases of $Z(\mathcal{O}G)$ and $Z(kG)$, so reduction of the center of an associative algebra is surjective. Idempotent lifting in this complete commutative algebra gives a bijection of primitive central idempotents, hence of blocks of a group algebra. Corresponding blocks of a group algebra have the same defect groups of a block: relative traces reduce, and a lifted trace whose reduction is the block identity is a unit in its complete block center.

The principal block of a group algebra is the unique block of a group algebra containing the trivial representation. Its primitive central idempotent is the unique one with augmentation $1$. Its defect groups of a block are the Sylow subgroups for the residue characteristic.

Let $N◃G$, let $b$ be a block of a group algebra of $kN$, and let $I$ be its inertia subgroup. A block of $kG$ covering $b$ is a full matrix algebra of degree $[G:I]$ over its corresponding block of $kI$. When $I=N$, the corner is just $kNb$. Let $e_B$ be the global block idempotent and put $c=e_{B}b$. With coset representatives $g_i$, the elements $g_{i}c{g}_j^{-1}$ are matrix units in that global block because different conjugates of $b$ are orthogonal. When $I=N$, one has $c=b$.

The center of an indecomposable finite-dimensional block of a group algebra is a commutative Artinian local ring. Its quotient by its Jacobson radical is a field, giving the block's central character. Over a splitting field for finite group representations the residue field is the coefficient field; without this assumption one retains the residue field rather than pretending every central character is scalar-valued over $k$.

For shapes of the same size, equal residue-content multiplicities are equivalent to equal cores of a partition at modulus $p$. Removing a rim $p$-hook subtracts one of each residue. With the number of beta numbers divisible by $p$, the packed-runner charges are $q_r=c_r-c_{r+1}$, so the multiplicities recover the core. Symmetric polynomials in Young–Jucys–Murphy elements act by their evaluations on these residues, and generate the modular center. Equality of its characters is precisely the block-equivalence criterion for simple modules.

Suppose $(u_n)$ is a normalized block basic sequence in the l-p sequence space, $1\le p<\infty$, and ${\sum }_n\Vert x_n-u_n{\Vert }_p<1$. Choose norm-one linear functionals ${\varphi }_n$ supported on the respective blocks with ${\varphi }_n(u_n)=1$. The bounded linear operator $A=I+{\sum }_n{\varphi }_n(\cdot )(x_n-u_n)$ is invertible by the Neumann series. The block projection $Qz={\sum }_n{\varphi }_n(z)u_n$ has norm at most one, and $AQ{A}^{-1}$ projects onto the closed span of $(x_n)$. Thus that span is a complemented subspace isomorphic to the l-p sequence space.

The rotation-invariant average of pure qubit states is uniform area measure on the Bloch sphere, induced by Haar measure on unitary transformations. For a uniformly distributed unit Bloch vector $r$, rotational symmetry gives $\mathbb{E}{r}_i=0$ and $\mathbb{E}{r}_i{r}_j={\delta }_{ij}/3$. Uniform polar angle without the factor $\sin \theta$ is a different distribution.

Write each pure density matrix as $(I+r\cdot \sigma )/2$. The trace identities of the Pauli matrices, $\mathrm{Tr}{\sigma }_i=0$ and $\mathrm{Tr}({\sigma }_i{\sigma }_j)=2{\delta }_{ij}$, make the trace of their product equal to the displayed overlap. Antipodal unit vectors represent orthogonal states; vectors at right angles correspond to squared overlap one half. For mixed states the same trace formula gives Hilbert–Schmidt overlap, not in general quantum fidelity.

For a qubit density matrix $\rho =(I+n\cdot \sigma )/2$ with nonzero Bloch vector, $\hat{n}=n/\Vert n\Vert$ and the multiplication rule for the Pauli matrices give these orthogonal rank-one spectral projectors, with eigenvalues $(1\pm \Vert n\Vert )/2$. Their Bloch vectors are opposite unit vectors $\pm \hat{n}$. At $n=0$ the state is maximally mixed and the direction is undefined; any orthonormal eigenbasis gives an antipodal pair, but no pair is distinguished by the state.