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A vector $v$ can be included in a diagonal basis for a real quadratic form precisely when $v\ne 0$ and either its quadratic value is nonzero or it lies in the radical of a bilinear form. A nonzero quadratic value gives the direct splitting $\mathbb{R}v\oplus v^{\perp }$ and induction. A nonzero radical vector can be included in a radical basis. Conversely, an isotropic vector in a diagonal basis is orthogonal to every basis vector, and must lie in the radical.

For a function with bounded derivatives through order $k$, the derivative supremum norm is $|u{|}_{k;U}={\sum }_{j=0}^k{\sup }_U|D^{j}u|$, with any equivalent finite-dimensional norm for each derivative tensor. This is a classical $C^k$ norm; it is different from the Lp norm with exponent $k$.

If a nonnegative supermartingale reaches zero at a discrete stopping time, its later nonnegative values have conditional expectation at most zero and must also be zero almost surely.

For a right-continuous nonnegative supermartingale $M$ and $R>0$, the first level-hit time $\sigma$ and the optional stopping theorem give $R\mathbb{P}(\sigma \le T)\le \mathbb{E}[M_{\sigma \wedge T}]\le \mathbb{E}[M_0]$. Let $T\to \infty$ and first use levels $R-\epsilon$, then $\epsilon \downarrow 0$, to include a supremum not attained. For a continuous nonnegative local martingale on a stochastic interval, apply the argument on increasing localized compact subintervals, then pass to their limit. Letting $R\to \infty$ shows that its running supremum is finite almost surely.

The sum of the largest eigenvalues has the semidefinite program representationFor every feasible $Y$ in the fantope, positive semidefinite trace nonnegativity gives $\mathrm{tr}(XY)\le kt+\mathrm{tr}S$. To attain equality, use an orthonormal eigenbasis of $X$ and choose $S=\mathrm{diag}(({\lambda }_i-t{)}_{+})$ in that basis, with the same threshold choice as in the threshold formula for the sum of the largest components. This argument also covers $k=n$, without requiring strict feasibility of the maximization program.

For a real symmetric matrix $X$ and $1\le k\le n$,Indeed $Y=Q{Q}^T$ is a rank-$k$ orthogonal projection matrix in the fantope, and projecting onto the eigenvectors of the largest eigenvalues attains the support function maximum. Equivalently, the largest matrix trace of a compression to a $k$-dimensional vector subspace is the sum of the largest eigenvalues.

For $1\le k\le n$, linear programming duality giveswhere $(\cdot {)}_{+}$ is the positive part of a real-valued function. To see equality directly, order the coordinates $x_{[1]}\ge \cdots \ge x_{[n]}$ and choose $x_{[k+1]}\le t\le x_{[k]}$ for $k<n$, or $t\le x_{[n]}$ for $k=n$. The threshold expression then equals the sum of the largest components. Introducing the variables $s_i$ turns this convex piecewise-linear objective into a linear program.

For a real symmetric matrix $A$, set $v(z)=(z_1^2,\dots ,z_n^2{)}^T$. ThenFor sufficiency, factor $P=B^{T}B$. Its contribution is a sum of squares of linear combinations of $z_i^2$, while the contribution of $N$ is ${\sum }_i{N}_{ii}(z_i^2{)}^2+{\sum }_{i<j}2N_{ij}(z_i{z}_j{)}^2$. For necessity, use a homogeneous sum of squares representation and sign averaging of a sum of squares. Writing each quadratic summand with coefficients $a_{\ell i},b_{\ell ij}$ gives $P={\sum }_{\ell }{a}_{\ell }{a}_{\ell }^T$, $N_{ii}=0$ and $N_{ij}=\frac{1}{2}{\sum }_{\ell }{b}_{\ell ij}^2$ for $i<j$. Comparing coefficients gives $A=P+N$.

This is the basic semidefinite programming certificate of copositivity discussed in Parrilo's paper on matrix copositivity.

A polynomial invariant under changing the sign of each coordinate can have its sum of squares representation averaged over independent Rademacher random variables ${\epsilon }_i\in \{-1,1\}$. Distinct parity patterns of monomials have zero cross terms because $\mathbb{E}[{\prod }_i{\epsilon }_i^{a_i}]$ vanishes when some exponent is odd. For a quadratic homogeneous polynomialthis givesThis identity separates the even monomials from each distinct two-coordinate parity pattern.

If a degree-$2d$ homogeneous polynomial is a sum of squares polynomial, it has a representation as squares of degree-$d$ homogeneous polynomials. In any sum of squares, the highest-degree terms cannot cancel, since their homogeneous part is itself a sum of squares. Likewise the lowest nonzero degree cannot cancel. Thus every nonzero summand has only degree $d$.

For a real-valued loss function depending only on evaluations at $x_1,\dots ,x_n$, and a positive squared-norm penalty, any minimizing tuple in a sum of reproducing-kernel Hilbert spaces has the formwith the same coefficient vector for every component. The tuple must be the unique minimum-norm decomposition of its sum. Apply the representer theorem to that sum and note that the displayed tuple has total squared norm ${\sum }_j{\hat{\alpha }}^T{K}_j\hat{\alpha }={\hat{\alpha }}^T({\sum }_j{K}_j)\hat{\alpha }$, exactly the norm of the sum. This argument is conditional on existence of a minimizer; arbitrary loss functions need not attain their infimum.

An image of $f:A\to B$ is the least subobject $m:I\hookrightarrow B$ through which $f$ factors. In a category with pullback in a category constructions this gives a strong epimorphism–monomorphism factorization $f=me$: pull back any mono in a lifting square for $e$ to $I$; minimality of the image forces the resulting mono to be invertible, producing a diagonal. Conversely, the lifting property of the strong part proves minimality. No stability under pullback is asserted; that is an additional property in a regular category.

For a finitely generated module over a principal ideal domain, its invariant factors are nonzero nonunits $d_i$ in a decomposition $M\cong R^r\oplus {⨁}_{i}R/(d_i)$, ordered by $d_i\mid d_{i+1}$. They are unique up to multiplication by units and describe the torsion submodule. Over $F[t]$, making $t$ act as a linear operator recovers the invariant factors of a linear operator used in its rational canonical form.

Expand the string embedding map along target geodesics as $X={\exp }_{\bar{X}}Y$. The quadratic Euclidean fluctuation action has a covariant derivative term and a curvature term $-R_{acbd}{Y}^a{Y}^b\partial {\bar{X}}^c\partial {\bar{X}}^d$. The logarithmically divergent coincident contraction is proportional to ${\alpha }^{\prime }{g}^{ab}$; its curvature contraction is the Ricci tensor. A compensating metric counterterm gives ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}+O({\alpha }^{\prime 2})$. This is why the target gravitational equation arises from quantum worldsheet Weyl anomaly cancellation, rather than from the classical worldsheet metric variation. Infrared regulation and removal of equation-of-motion terms by field redefinitions specify the usual one-loop scheme.

The string-frame metric is the target metric appearing directly in the kinetic term of the string nonlinear sigma model. With conventional Euclidean dilaton $\phi$, its leading spacetime gravitational action contains $\sqrt{-g^{(s)}}{e}^{-2\phi }(R+4|\nabla \phi {|}^2)$. The dilaton-dependent Ricci coefficient distinguishes it from the Einstein-frame metric. Frame changes are local field redefinitions where the conformal factor is finite and nonzero; the Lorentzian dilaton coupling sign convention fixes which named field is $\phi$.

For small residual phase errors, $S\simeq e^{-{\sigma }_{\varphi }^2}$, where ${\sigma }_{\varphi }=2\pi h_{\mathrm{rms}}/\lambda$. The variance is taken after removing irrelevant constant phase. This approximation to the Strehl ratio is most useful near the diffraction limit of a telescope.

Suppose a nonnegative stopping time $T$ and constants $L>0$, $0<q\le 1$ satisfy $\mathbb{P}(T\le nL+L\mid {\mathcal{F}}_{nL})\ge q$ on $\{T>nL\}$. Take $L$ to be a positive integer in discrete time. ThenOn the surviving event, conditional survival is at most $1-q$. The tower property of conditional expectation therefore givesInduction proves the geometric bound. Partitioning the tail integral into intervals of length $L$ then gives $\mathbb{E}T\le L{\sum }_{n\ge 0}\mathbb{P}(T>nL)\le L/q$; in discrete time, group the tail sum into $L$ successive integer indices instead. This proves finite expected hitting times when every surviving state has a uniform positive chance of escaping within a fixed time. It works in discrete or continuous time without assuming independence of successive survival events.

A bounded stopping time is a stopping time bounded almost surely by a deterministic finite constant. In discrete time, a bound $T\le N$ makes $M_T$ integrable whenever $M_0,\dots ,M_N$ are integrable random variables, and the optional stopping theorem applies to a martingale without additional limiting hypotheses. Almost-sure finiteness alone is weaker than boundedness.

An announcing sequence consists of stopping times increasing to $T$ almost surely, strictly smaller than $T$ on $\{T>0\}$. Such a lifetime can be approached through stopped intervals on which a locally defined stochastic process is an ordinary adapted process. For the usual maximal local solution of a stochastic differential equation, the exit times ${\tau }_n$ from a nested compact exhaustion of the open domain, capped as $T_n={\tau }_n\wedge n$, supply this sequence. The existence of such a sequence is an extra lifetime convention; arbitrary stopping times need not admit one.

Differentiating the Stokeslet gives its rate-of-strain tensor and vorticity, displayed above for $r=|\mathbf{x}|>0$. The pressure is $p=\mathbf{F}\cdot \mathbf{x}/(4\pi r^3)$. Integrating the Newtonian fluid stress tensor over a sphere enclosing the origin gives traction resultant $-\mathbf{F}$, fixing the point-force normalization.