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An integral curve follows a vector field by solving the displayed ordinary differential equation. For a smooth vector field, integral curves through initial points assemble into its local flow.

With fixed $X,Z$, compare $N(X\beta ,{\sigma }^{2}I)$ and $N(X\beta ,{\sigma }^{2}I+{\tau }^{2}Z{Z}^T)$ by maximizing the ordinary likelihood function in both models. Their likelihood-ratio test statistic is unchanged by $Y\mapsto Xb+cY$, $c>0$, because both maximized log-likelihoods change by the same $-n\log c$. Its null law can therefore be simulated using independent $N(0,I)$ responses. Comparing the observed statistic with simulated statistics by $(1+\#\{T_b\ge T_{obs}\})/(B+1)$ gives a conservative finite-sample Monte Carlo p-value, subject to correctly maximizing both likelihoods.

In the Schrödinger picture, states evolve with the Hamiltonian operator while observables without explicit time dependence are fixed. For time-independent $H$, the corresponding Heisenberg picture operator is $O_H(t)=e^{iHt/\hslash }{O}_S{e}^{-iHt/\hslash }$.

An eigenphase of a unitary operator is a real number $\theta$ modulo one specifying an eigenvalue $e^{2\pi i\theta }$. Every unitary eigenvalue has modulus one. Quantum phase estimation records the eigenphase corresponding to an input eigenvector; a superposition retains the corresponding labels coherently before measurement.

Assuming $V=L$, every uncountable transitive model of ZFC contains all ambient countable ordinals and witnesses their countability internally. Its ordinal height is at least ${\omega }_1$, and absoluteness of constructible levels puts $L_{{\omega }_1}$ inside it. Every countability witness for a countable ordinal can be chosen in $L_{{\omega }_1}$ by hereditarily countable constructible sets appear below omega-one.

For transonic accretion in a dipolar flux tube, the Bernoulli equation at the sonic point contains $c_s^2[1/(\gamma -1)-5/2]$. A positive reservoir energy requires $\gamma <7/5$. Compression in the $r^3$ dipolar flux-tube area raises the specific enthalpy more rapidly than in the $r^2$ geometry of spherical Bondi accretion, whose corresponding upper index is $5/3$.

For $L=\left(\begin{array}{cc}{\lambda }_1& 0\\ 1& {\lambda }_2\end{array}\right)$ with distinct real negative eigenvalues, the matrix exponential is $A=\left(\begin{array}{cc}a& 0\\ b& d\end{array}\right)$, where $a=e^{{\lambda }_{1}t}$, $d=e^{{\lambda }_{2}t}$ and $b=(a-d)/({\lambda }_1-{\lambda }_2)$. The optimal energy amplification of a linear system isIts eigenvectors are nonorthogonal, and immediate energy growth occurs exactly when ${\lambda }_1{\lambda }_2<1/4$. If ${\lambda }_2>{\lambda }_1$, putting $\alpha =({\lambda }_2-{\lambda }_1{)}^{-1}$ gives $G(t)\sim (1+{\alpha }^2)e^{2{\lambda }_{2}t}$. The optimal asymptotic initial direction $(\alpha ,1)$ is an adjoint eigenvector, while the eventual state aligns with the right eigenvector $(0,1)$.

For the matrix exponential $A(t)=e^{Lt}$, maximizing the energy ratio over nonzero initial conditions givesThe Rayleigh quotient achieves its maximum at an eigenvector of $A^{\ast }A$ with largest eigenvalue, equivalently a right singular vector of $A$. Such an optimal disturbance is generally not an eigenvector of $L$.

For $x^{\prime }=Lx$ and $E=\Vert x{\Vert }_2^2$, the Hermitian matrix $H=(L+L^{\ast })/2$ gives $\dot{E}=2x^{\ast }Hx$. Thus some initial condition grows immediately iff the largest eigenvalue of $H$ is positive. All trajectories have nonincreasing energy iff $H$ is a negative semidefinite matrix. This differs from requiring negative real parts for the eigenvalues of $L$: a non-normal matrix can satisfy modal decay while allowing transient growth.

For a continuous Loewner trace generating its hulls, the inverse mapping-out function of a compact H-hull $g_t^{-1}$ extends continuously to the closed complex upper half-plane. The finite trace is a continuous map image of an interval, and the boundary of its unbounded complementary component has local connectedness; the Caratheodory boundary extension theorem gives the extension. In particular $g_t^{-1}(U_t)=\gamma (t)$, while $g_t^{-1}$ maps the open complex upper half-plane into $D_t$. This is the deterministic boundary-extension input of Rohde and Schramm, Theorem 4.1. Under a Conformal Markov property of SLE restart, this relates boundary contacts of the mapped future to contacts with the old hull, without assuming the old trace is simple.

With zero forward differences at the grid boundary, discrete isotropic total variation is the sum of the Euclidean lengths of the two-component forward gradients. Its dual constraint is the product of unit Euclidean balls, and minimizing $\frac{1}{2}\Vert g-D^{\ast }p{\Vert }_2^2$ over that product yields $u_{\ast }=g-D^{\ast }{p}_{\ast }$. The exact difference adjoint operator is essential. A dual proximal gradient method projects $p+\tau D(g-D^{\ast }p)$ pointwise onto unit balls; $0<\tau \le 1/8$ is safe on an unscaled square grid.

Staircasing is the formation of approximately constant plateaux in reconstructed data, even when the original data vary smoothly. The linear growth of the total variation seminorm on a domain favors sparse spatial gradients, so preserving jumps can come at the cost of flattening gradual transitions.

In a Cartesian closed category, exponentiation satisfies $(-{)}^{A\times B}\cong ((-{)}^B{)}^A$. If the two exponential functors have right adjoints, their composite has the composite of those right adjoints in reverse order. Therefore a product of tiny objects is tiny. The terminal object is tiny since exponentiation by it is the identity, covering the empty product.

For a small category $\mathcal{C}$ with binary coproducts, exponentiation by $h_A=\mathcal{C}(A,-)$ in $[\mathcal{C},\mathbf{Set}]$ is naturally $F\mapsto F(-⨿A)$: the exponential at $B$ is $\mathrm{Nat}(h_B\times h_A,F)\cong \mathrm{Nat}(h_{B⨿A},F)\cong F(B⨿A)$ by the Yoneda lemma. This precomposition functor has a right adjoint given by Right Kan extension, so $h_A$ is a tiny object.

Let $\mathcal{C}$ be a small Cauchy-complete category with initial object $0$. The terminal object of $[\mathcal{C},\mathbf{Set}]$ is $h_0$, so $\mathrm{Nat}(P,F)\cong (F^P)(0)$. If $P$ is tiny, exponentiation by $P$ is itself a left adjoint and preserves all colimits. Evaluation at zero also preserves colimits. Consequently $\mathrm{Nat}(P,-)$ preserves coproducts and epimorphisms, since every epimorphism in a set-valued functor category is the coequalizer of its kernel pair. Thus $P$ is an irreducible projective in a set-valued functor category, hence representable.

A sentence with closed-term parameters holds in a term model if and only if it belongs to the complete consistent theory defining that model. Prove this by structural induction: atoms are the definition, Boolean steps use completeness and consistency, and existential sentences use the Henkin witness property. Conversely any existential witness represented by a term gives the existential sentence by logical inference.

A tensor derivation is a real-linear operation preserving every tensor type, obeying the tensor-product Leibniz rule, and commuting with every tensor contraction. A derivation $D$ of smooth functions and a real-linear operator on vector fields satisfying $D(fY)=fDY+(Df)Y$ extend uniquely to a tensor derivation. On a differential one-form it must satisfyThis expression is linear over smooth functions in $Y$. In a local frame $D{e}_a=A^b{}_a{e}_b$, the dual rule is $D{ϵ}^a=-A^a{}_b{ϵ}^b$. Apply the product rule to every coefficient and frame factor to define the extension; these dual signs cancel under contraction. Frame changes agree by differentiating the inverse matrix. Cutoffs prove locality and hence uniqueness from local expansions.

A multilinear operation on vector fields is tensorial when it is linear over smooth functions in every argument. Its value at a point then depends only on the argument vectors at that point. To see this, use a smooth cutoff function to reduce to local fields, expand them in a local frame, and apply linearity over smooth functions to their coefficients. A smoothly valued tensorial operation therefore defines a tensor field. The curvature of an affine connection is tensorial, whereas a covariant derivative differentiates a scalar coefficient in its second argument and is not tensorial there.

For integer paths with increments in $\{-1,0,1\}$, the positive part obeys a telescoping identity with slope $f(x)=1_{\{x>0\}}+\frac{1}{2}{1}_{\{x=0\}}$ and a correction $\frac{1}{2}{1}_{\{S_{t-1}=K\}}(\Delta S_t{)}^2$ at the kink. For a martingale, the slope term is a martingale transform.

Let $\alpha$ be finite and measurable in $\mathcal{T}(\xi )$ for a finite measurable partition $\xi$, and put $h=h_{\mu }(T,\xi )$. Choose $r$ with $H(\alpha \mid {\xi }_0^r)<\epsilon$. For $\gamma ={\alpha }_0^{n-1}$ and $\beta ={\xi }_0^{n+r-1}$, conditional subadditivity gives $H(\gamma \mid \beta )<n\epsilon$. Tail measurability and block conditional entropy given the infinite future give $H(\beta \mid \gamma )\ge (n+r)h$. Thus $H(\gamma )\le H(\beta )-(n+r)h+n\epsilon$. Divide by $n$, then let $n\to \infty$ and $\epsilon \downarrow 0$, proving $h_{\mu }(T,\alpha )=0$. The proof also works for noninvertible transformations.