For $y^{\prime }+a(x)y=H(x)$ with continuous $a$, the integrating factor $\mu =\exp (\int a)$ gives the displayed solution. It is continuous at the step and solves the differential equation classically on each side; the derivative jumps by one. The value assigned to the Heaviside step function at the single point zero does not affect the integral. A discontinuity in $y$ would instead produce an unintended delta term in its distributional derivative.

The sine and cosine integrals accumulate $\sin t/t$ and $\cos t/t$, with different normalizations at zero and infinity. They arise in oscillatory integrals, Fourier transforms of singular kernels and Fourier series of logarithmic profiles.

The integral remainder in Taylor theorem for $\log (1-t)$ gives the exact finite identityFor $0\le x\le t<1$, $(t-x)/(1-x)\le t$, so the nonnegative remainder is at most $t^{N+1}/(1-t)$ and tends to zero. At $t=1/2$ it is at most $2^{-N}$, proving both convergence to $\log 2$ and an explicit error bound without assuming the logarithmic power series beforehand.

For any algebraic field extension $K/\mathbb{Q}$, including an infinite one, its ring ${\mathcal{O}}_K$ of elements integral over $\mathbb{Z}$ is an integral extension of $\mathbb{Z}$. The Lying-over theorem, going-up theorem and incomparability theorem for integral extensions give $\dim {\mathcal{O}}_K=\dim \mathbb{Z}=1$. The argument does not require that ${\mathcal{O}}_K$ be Noetherian or finitely generated over $\mathbb{Z}$.

A volume average is the normalized integral average of a field over a three-dimensional region of finite positive volume. For a fixed region, differentiating in time commutes with this average under the usual regularity conditions. The divergence theorem converts the average of a divergence into its net boundary flux divided by volume.

For an almost complex structure, closure of $T^{1,0}$ under the Lie bracket is equivalent to the absence of a $(0,2)$ component in $d\alpha$ for every $(1,0)$-form $\alpha$. Conjugation exchanges the bracket condition with closure of $T^{0,1}$. For sections $U,V$ of $T^{0,1}$, the exterior derivative identity gives $d\alpha (U,V)=-\alpha ([U,V])$. Its vanishing for all $(1,0)$-forms is exactly $[U,V]\in T^{0,1}$.

On a finite simply connected planar graph, fix the height of the exterior face to zero. Every divergence-free integer bond current is uniquely the oriented difference of the two neighboring face heights. The positive graph Laplacian $L$ on the remaining faces gives its energy. Applying the Poisson summation formula to these heights and completing the square givesThe Gaussian prefactor $Z_{\mathrm{sw}}$ equals the spin wave determinant of the original graph, up to angle zero-mode normalization. One way to verify this is that the matrix-tree theorem equates the spanning-tree counts of a planar graph and its dual. Boundaries can admit nonzero total vortex charge; neutrality is an additional periodic or finite-energy condition.

Fix a trivialization at infinity and divide smooth finite-action instantons only by gauge transformations tending to the identity there. Global gauge orientations remain genuine parameters. For regular $SU(2)$ instantons of charge magnitude $k$, the real dimension is $8k$ by the ADHM construction. Forgetting the framing divides by the three-dimensional global group action of $SU(2)$, giving $8k-3$ on the irreducible locus. For one instanton the framed parameters are four center coordinates, one positive size, and three orientation coordinates.

The scale $\rho >0$ in a BPST instanton changes the width of its action density while leaving total action unchanged. This reflects conformal invariance of four-dimensional classical Yang-Mills action. As $\rho \to 0$, gauge curvature concentrates at the center and the smooth family approaches a bubbling boundary, not another smooth zero-size instanton.

Under the instantaneous recycling approximation, forming mass $dM$ locks $ds=(1-\beta )dM$ into stars. If a fraction $f$ of returned material remains in the interstellar medium, the retained original metal mass is $\beta fZdM$, and retained newly made metal mass is $fyds$, where $y$ is the stellar yield per locked mass. Subtracting the metal mass $ZdM$ initially consumed gives the displayed equation. Pristine galactic gas inflow adds gas but no metals; enriched inflow adds its metal mass separately. The retained fraction applies to both old and newly made metals in this idealization.

Cumulative mixing efficiency is the ratio of total irreversible density-mixing conversion to total irreversible mechanical-energy loss over an interval. It is a loss-rate-weighted average of instantaneous mixing efficiency, not an unweighted time average. Stored kinetic energy, changes in available potential energy and boundary work must be accounted for before replacing its denominator by integrated forcing.

The actual derivative of ${\theta }_K(\gamma (t)(x-X(t)))$ is $[\dot{\gamma }(x-X)-\gamma \dot{X}]{\theta }_K^{\prime }$. The omitted contribution is first order in acceleration. Its cross term with the translation velocity integrates to zero because $\int y({\theta }_K^{\prime }(y){)}^{2}dy=0$. The square adds ${\dot{\gamma }}^2\int y^2({\theta }_K^{\prime }{)}^{2}dy/(2{\gamma }^3)={\pi }^2{\dot{\gamma }}^2/(3{\gamma }^3)$ to the spatially integrated density, since ${\theta }_K^{\prime }=2\mathrm{sech}y$ and $\int y^2({\theta }_K^{\prime }{)}^{2}dy=2{\pi }^2/3$. Thus the instantaneous-boost effective action for a sine-Gordon kink is an adiabatic truncation rather than the exact action restricted to that accelerating configuration-space profile.

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]