For the Gaussian Brownian drift filter $Y_t=\alpha t+W_t$, with independent prior $\alpha \sim N(m_0,{\tau }_0^{-1})$, the observation law has densityrelative to driftless Wiener measure. The law having fixed observation drift $k$ has density $e^{k{Y}_t-k^{2}t/2}$ relative to the same reference. Their ratio is therefore a positive mean-one density on every finite horizon. In the observation filtration it satisfies $d{D}_t=-D_t(m_t-k)d{\hat{W}}_t$, where $m_t$ is the posterior mean and $\hat{W}$ the innovation process. This density-ratio argument establishes a true martingale without an unsupported global Novikov condition for the unbounded posterior mean.

Initial displacement specifies the configuration of a wave equation at the starting time. For a second-order evolution equation it is paired with the initial velocity. Specifying only the boundary forcing does not determine these initial data.

Initial velocity specifies the time derivative of the displacement field at the starting time. A second-order wave equation requires it independently of initial displacement. Both vanish for a causal scattered field before an incident pulse reaches its scatterer.

For a linear differential equation written $Ly=f$, the prescribed right-hand side $f$ is the forcing term. It may represent an external input or source, and can be constant or time dependent. Setting $f=0$ gives the associated homogeneous linear differential equation.

Choose the four unit Bloch vectors at the vertices of a regular tetrahedron, so ${\sum }_i{\mathbf{n}}_i=0$ and ${\sum }_i{\mathbf{n}}_i{\mathbf{n}}_i^T=4I/3$. The displayed four positive rank-one effects sum to $I$. On input $(I+\mathbf{r}\cdot \mathbf{\sigma })/2$, their probabilities are $p_i=(1+\mathbf{r}\cdot {\mathbf{n}}_i)/4$, giving the reconstruction $\mathbf{r}=3{\sum }_i{p}_i{\mathbf{n}}_i$. This realizes the smallest fixed qubit informationally complete POVM.

In Information Theory and Statistics: A Tutorial by Imre Csiszár and Paul C. Shields (2004, Historical Notes, §3, printed p. 522), the authors connect systematic studies of information projection with earlier relative entropy minimization principles and geometric work, including Imre Csiszár's I-Divergence Geometry of Probability Distributions and Minimization Problems by Imre Csiszár (1975). This is the tutorial's historical account, rather than a claim of one isolated invention. Its finite-alphabet treatment makes the projection identities available without the additional machinery needed for general probability spaces.

Let $p_i>0$ be a probability mass function on a finite alphabet with numerical outcomes $a_i$. Prescribe a mean $m$ strictly between the smallest and largest outcomes. Exponential tilting giveswith $\theta$ chosen so that ${\sum }_i{q}_i^{\ast }{a}_i=m$. For any other feasible law $r$, direct substitution yieldsRelative entropy nonnegativity therefore proves optimality and uniqueness. At an extreme prescribed mean, feasible laws may have zero probabilities; a finite tilting parameter need not realize them, although a limit can.

Under the finite-alphabet, positive-reference, closed-convex set assumptions above, every feasible law $r\in \mathcal{C}$ satisfiesThus a feasible law's excess relative entropy above the optimum is at least its relative entropy from the optimizer. The information projection supplies a quantitative comparison with every other feasible law, not merely the location of a minimum.^[ref]

For affine expected value constraints, the result becomes a Pythagorean identity for relative entropy. In relative entropy minimization under a mean constraint, the logarithm of $q_i^{\ast }/p_i$ is an affine function of the outcome $a_i$. Any feasible $r$ has the same normalization and mean as $q^{\ast }$, making the residual in the three-point identity for relative entropy zero.^[ref]

For a uniform distribution on a finite set of $m$ outcomes, direct expansion of relative entropy giveswhere $H_e$ is information entropy in nats. Therefore minimizing relative entropy from this particular reference is equivalent to maximizing information entropy subject to the same constraint. This statement depends on the uniform reference; an arbitrary reference contributes its own weights.

In the three-outcome example, $H_e(q^{\ast })=\ln 3-D(q^{\ast }\Vert p)\approx 0.901235$ nats. The mean fixes one aggregate quantity, and information projection chooses how to distribute the remaining uncertainty. This extends maximum entropy on a finite alphabet, whose unconstrained maximizer is the uniform law itself.

Let the possible numerical outcomes be $-1,0,1$, and give the reference law $p$ uniform distribution on a finite set: $p=(1/3,1/3,1/3)$. We want a law $q$ with expected value $1/2$. Write the three probabilities in outcome order as $(q_{-},q_0,q_{+})$. The mean constraint is $q_{+}-q_{-}=1/2$, while normalization requires $q_{-}+q_0+q_{+}=1$.

These conditions leave a whole line of possible probability distributions:For example, $(0,1/2,1/2)$ and $(1/4,0,3/4)$ both have the required mean. The constraint alone does not choose between them. Information projection selects the point on this line with the smallest relative entropy from the reference.

Exponential tilting gives weights proportional to $e^{-\theta },1,e^{\theta }$. Set $t=e^{\theta }>0$ and multiply these weights by $t$ to obtain $q^{\ast }=(1,t,t^2)/(1+t+t^2)$. Enforcing the mean givesThe positive root is $t=(1+\sqrt{13})/2$, henceThe most favored outcome becomes $1$, but all three remain possible. This is an exact constructed example with a stipulated reference law and constraint, not a fitted distribution from measurements.

For positive slope parameter $A$, the exponential-potential field increases during cosmic inflation. The quantity $aH$ grows by $e^{(1-{ϵ}_H)N}$ with constant ${ϵ}_H=A^2/(16\pi )$ and $N=8\pi ({\varphi }_R-{\varphi }_i)/(A{m}_{\mathrm{pl}})$. Starting from order-one curvature and demanding order-one or smaller present curvature gives the displayed upper bound on the initial field. A specified tighter final curvature tolerance adds one-half its logarithm to the numerator. Thus sufficient cosmic inflation sets an upper, not a lower, initial-field bound for this sign of the potential slope. Instantaneous reheating and subsequent entropy conservation determine $a_R{H}_R/a_0{H}_0$.

For $|q|>1$, define $S_q(z)={\prod }_{k\ge 0}(1-q^{-2k-1}z)(1-q^{-2k-1}{z}^{-1})$ on ${\mathbb{C}}^{\ast }$. The geometric decay gives locally uniform absolute tail convergence there. Its simple zeros are $q^{2k+1}$ and $q^{-2k-1}$ for $k\ge 0$, and reindexing gives $S_q(qz)=-z{S}_q(q^{-1}z)$. The variable zero is outside its domain, and the poles introduced by each reciprocal factor cannot be interpreted as a value of the product at zero.

// Destination: analysis.bigb

For a stress-free, fixed-temperature layer, the infinite-Prandtl number growth rate of the fundamental vertical normal mode is ${\lambda }_{\infty }=R{k}^2/(k^2+{\pi }^2{)}^2-(k^2+{\pi }^2)$. For large Rayleigh number, its maximum has $k^2={\pi }^2-8{\pi }^6/R+O(R^{-2})$ and ${\lambda }_{\max }=R/(4{\pi }^2)-2{\pi }^2+O(R^{-1})$. The infinite-Prandtl number limit must be taken first, or inertia must remain small on the resulting growth-time scale. This optimum differs from the free-slip convection neutral curve minimum, which selects onset rather than fastest growth far above it.

For an axisymmetric normal mode in a rigid rotating sphere, substitute $W=z(A{R}^2+B{z}^2+C)$ into the pressure equation and its impermeability condition. With $\beta =1-4{\Omega }^2/{\sigma }^2$, the interior equation gives $2A+3\beta B=0$. The boundary polynomial gives $3\beta B=(2+\beta )A$ and $\beta C=-(2+\beta )A{r}_0^2$. A nontrivial oscillatory solution has $\beta =-4$, hence the displayed frequency and $B=A/6$, $C=-A{r}_0^2/2$. It satisfies both equations identically.

In the large-diffusivity regime, equate diffusion rate $\alpha /r^2$ with inertial eddy rate ${ϵ}^{1/3}{r}^{-2/3}$. This gives the Corrsin scalar microscale, which exceeds the Kolmogorov length scale when $\alpha \gg \nu$.

The congruence axiom says that an affordable alternative indirectly revealed at least as desirable as a chosen alternative must also be chosen. For a chosen bundle $x$ and an affordable unchosen bundle $y$, the axiom rules out $yRx$; because $xRy$, every strict-part-preserving completion ranks $x$ strictly above $y$.

In the genuine indexed point-set model, spectrification $L$ is the left adjoint to the forgetful functor from indexed spectra to indexed prespectra. An indexed spectrum has adjoint structure maps $E(V)\to {\Omega }^{W\ominus V}E(W)$ that are homeomorphisms. Applying $L$ to a cellular indexed prespectrum supplies the associated genuine topological spectrum.

For compact $X$ over the space of linear isometries $U\to U\prime$, choose a subordinate flag for a family of linear isometries $V_i,W_i$. The vector bundle ${\xi }_i$ has fiber $W_i\ominus f_x(V_i)$. Form $P(W_i)=\mathrm{Th}({\xi }_i)\wedge E(V_i)$. Its structure maps use the orthogonal decompositionand the structure map of $E$. The twisted half-smash product is the spectrification $LP$, after extending from the cofinal flag. Unlike an arbitrary replacement by $X_{+}\wedge E$, this definition retains the specified isometry family at the point-set level.

A rotation through $\pi$ about the third coordinate axis preserves the strain components $e_{11},e_{22},e_{33},e_{12}$ and reverses $e_{13},e_{23}$. A symmetry-invariant elastic stiffness tensor cannot couple the two parity spaces. Its independent symmetric blocks therefore contain $4\cdot 5/2+2\cdot 3/2=13$ coefficients. A twofold axis alone does not impose invariance under every rotation about that axis.

Draw two different labels uniformly from $\{1,2,3,4\}$ using sampling without replacement. Each position has a uniform distribution on a finite set, but $\mathbb{P}(X_1=X_2)=0$. Two independent random variables with those same marginal distributions would match with probability $1/4$. Equal marginal laws therefore do not make this sample an IID sample. Mark some of those four labels as successes. Their count in the two draws has a hypergeometric distribution; independent Bernoulli trials with a common success probability give a binomial distribution.