With three evenly matched players, the previous winner plays the previous spectator. A pair of successive distinct winners $ij$ moves to $jj$ or $jk$ with equal probabilities, where $k$ is the remaining player. The transient states form two directed 3-cycles and each step has absorption Probability one half. Starting with a game between two specified players, their overall winning probabilities are each $5/14$, while the original spectator wins with Probability $2/7$.

For a path with $T+1$ vertices and edge weight $1/2$, the eigenvalues are $1-\cos [\pi k/(T+1)]$, $k=0,\dots ,T$. The corresponding coefficients are proportional to $\cos [\pi k(t+1/2)/(T+1)]$. The zero mode is uniform, and the spectral gap is at least $2/(T+1{)}^2$. The endpoint difference equations use reflecting ghost values, not periodic ones.

For $M(s)=(1-s)D+sE$ in the unary history subspace, the first Gershgorin disc lies in $[0,s]$ and every other disc has real part at least $1-s$. For $s<1/2$, the first disc is isolated and contains the unique ground state energy. Thus the spectral gap is at least $1-2s$, in particular at least $1/3$ for $s\le 1/3$.

The Frobenius norm is the finite-matrix Hilbert-Schmidt norm. It equals $\sqrt{\mathrm{tr}(A^{\ast }A)}$, is unchanged by unitary multiplication on either side, and is the square root of the sum of squared singular values. For unit vectors $u,v$, $\Vert u{u}^{\ast }-v{v}^{\ast }{\Vert }_F^2=2(1-|u^{\ast }v{|}^2)$, a useful sign-invariant measure of distance between rank-one orthogonal projection matrices.

If $\rho$ is a density operator with decreasing eigenvalues ${\lambda }_j$ and $T$ has matrix rank at most $k$, let $\Pi$ project onto the image of $T$. The Cauchy-Schwarz inequality for the Hilbert-Schmidt inner product givesThe Hermitian effect variational principle bounds $\mathrm{Tr}(\rho \Pi )$ by ${\sum }_{j<k}{\lambda }_j$, proving the claimed estimate without assuming $\rho$ is invertible.

An $SU(2)$ gauge theory with a real adjoint scalar and a vacuum $\varphi =v{e}_3$ breaks to $U(1)$. The Higgs mechanism gives two charged vectors mass $gv$, leaves one massless neutral vector, and gives the radial scalar mass $\sqrt{2\lambda }v$ for the quartic potential. Unlike the Standard Model electroweak interaction, this model has no additional hypercharge field and no massive neutral Z boson.

A Yang-Mills theory with gauge group $SO(3)$ and a real three-component scalar field in its defining vector representation. With a symmetry-breaking scalar potential, a nonzero vector vacuum leaves its $SO(2)$ stabilizer subgroup unbroken. The two transverse internal directions produce massive gauge bosons.

The hereditarily small set structure $H_{{\omega }_2}$ fails the Axiom of power set at ${\omega }_1$. Every subset of ${\omega }_1$ belongs to $H_{{\omega }_2}$, but their full power set has cardinality at least ${\aleph }_2$ by the Cantor theorem, so cannot itself be hereditarily smaller than ${\aleph }_2$.

The pitch is the axial displacement per complete turn. For a circular helix with $z=a\varphi +b$, the signed pitch for increasing angle is $2\pi a$, and its magnitude is $2\pi |a|$. Pitch is independent of the cylinder radius.

In four-dimensional supersymmetry, a finite-helicity massless supermultiplet has helicity interval of width $\mathcal{N}/2$. If every state obeys $|h|\le s$, that interval fits inside one of width $2s$, giving $\mathcal{N}\le 4s$. The bounds $4$ for $s=1$ and $8$ for $s=2$ are attained by the four-dimensional N=4 super Yang-Mills theory vector multiplet and the four-dimensional N=8 supergravity supergravity multiplet. Spin bounds alone do not prove that an arbitrary action is renormalizable.

A bipartite spin rotation rotates the spin on one sublattice by $\pi$, converting an alternating classical Néel state to an all-up reference state. Rotation about $x$ sends $S^z\mapsto -S^z$ and $S^{\pm }\mapsto S^{\mp }$ on that sublattice. For an isotropic nearest-neighbor Heisenberg antiferromagnet, a bond becomes $-S_i^z{S}_j^z+(S_i^{+}{S}_j^{+}+S_i^{-}{S}_j^{-})/2$. The transformation is unitary and preserves the spin commutation relations; it changes the reference frame, not the model's spectrum.

Concavification replaces a nonconcave incentive utility function by its least concave majorant. In a complete market with an atomless state-price density, the optimizing payoff may avoid all intervals where the majorant strictly exceeds the original utility, making the relaxed optimum exactly attainable. At a pricing atom corresponding to a linear segment, a lottery between contact points gives the relaxed value at unchanged cost only if the admissible model permits that randomization and the resulting payoff is attainable. For example, an atomless underlying space can allow splitting a positive-measure price level into subsets of the required probabilities. A price level that is a single indivisible outcome cannot be split this way. Without available randomization or an equivalent attainable allocation, the concavified problem can have a relaxation gap.

The rational space of level-one cusp forms has a full lattice of forms with integral Fourier coefficients. It has finite rank by the valence formula for the modular group, and is preserved by the Fourier coefficients of a composite-index Hecke operator formula. Thus every Hecke operator has a monic integral characteristic polynomial on this lattice, and its eigenvalues are algebraic integers. The common eigenvalue field is totally real by the Petersson inner product self-adjointness.

For a normalized level-one weight-$k$ cuspidal Hecke eigenform, the local recurrence gives ${\sum }_{r\ge 0}{a}_{p^r}{X}^r=(1-a_{p}X+p^{k-1}{X}^2{)}^{-1}$. Multiplicativity therefore gives $L(f,s)={\prod }_p(1-a_p{p}^{-s}+p^{k-1-2s}{)}^{-1}$ in a right half-plane. Each degree-two local factor packages all the prime-power Fourier coefficients.

In the one-factor Heath-Jarrow-Morton model, put $b_t^T={\int }_t^T\sigma (t,u)du$. The Itô formula gives $dP/P=r_{t}dt-b_t^{T}d{W}_t$. Thus multiplying by the discount factor yields the displayed stochastic exponential. Bounded forward volatility on a finite maturity horizon implies the Novikov condition, so the discounted bond is a true martingale.

For $L>0$ and $t>0$, the Sturm-Liouville eigenfunction expansion for $u_t=u_{xx}$ on $(0,L)$ with homogeneous Neumann boundary conditions givesThe constant eigenfunction mode preserves the spatial integral. This is the $\beta =0$ limit of the weighted Neumann heat kernel with constant drift, and its boundary inputs have the signs in the Neumann boundary-forcing formula.

For standard Brownian motion in ${\mathbb{R}}^d$, $p_t(y-x)$ is the transition density from $x$ to $y$. It is the heat kernel for generator $\frac{1}{2}\Delta$. For every unit vector $e$, $\int |{\partial }_e{p}_t(x)|dx=\sqrt{2/\pi }/\sqrt{t}$, which bounds the change in this density under a spatial translation and proves the Gaussian heat-kernel proof of the harmonic Liouville theorem.

For the heat equation with zero Dirichlet boundary conditions on $(0,L)$,It is also obtained by the method of images, taking the difference between Gaussian sources at $y+2jL$ and $-y+2jL$. The eigenfunction expansion is useful at long times; the image expansion is useful at short times.

For $x>0$ and the heat equation on a half-line, the zero-initial-data solution with prescribed Dirichlet boundary condition $g$ is ${\int }_0^{t}P(x,t-s)g(s)ds$, whereIts mass concentrates at $t=0$ as $x\downarrow 0$, recovering the boundary trace.

The normalized Gaussian mass of an interval isIt is the convolution of an interval indicator with the heat kernel, and has an error function representation. As the variance tends to zero it converges to the interval indicator away from its endpoints, with value one half at an endpoint. This is a concrete approximate identity.