For emitted spectral power per steradian $P_{\nu }\propto {\nu }^{-\alpha }$, the luminosity distance gives $S_{{\nu }_o}=(1+z)P_{(1+z){\nu }_o}/d_L^2$. Photon energy and arrival-rate redshifts supply two factors of $1+z$, while the observed frequency interval is reduced by one. Substitution of the power-law spectrum gives the displayed K correction. This convention uses spectral flux per unit frequency and power per steradian; using per unit wavelength or total spectral luminosity changes the corresponding factors.

The running intersection property says that every variable belongs to a connected subtree of a junction tree. Equivalently, every clique along the path between two cliques contains their intersection. This ensures that eliminating variables on one side of a tree edge leaves a message depending only on its separator with the other side.

Assign each factor exactly once to a containing clique and multiply the factors there into ${\psi }_i$. The displayed sum-product belief propagation message sums out variables of $C_i$ outside the separator $S_{ij}=C_i\cap C_j$. By the running intersection property, eliminating all variables in the subtree on the $i$ side creates precisely this function of the separator. An inward pass computes the normalizing constant at the root; an outward pass produces marginal distributions. For pedigree likelihoods, observed genotypes and penetrance factors simply multiply the relevant potentials.

The junction tree algorithm performs exact inference in a finite probabilistic graphical model. First moralize a Bayesian network if necessary; then apply triangulation of an undirected graph, organize its maximal cliques into a junction tree, and assign each original factor to one containing clique. Inward and outward junction-tree sum-product messages compute the normalizing constant and marginal distributions. Its cost depends exponentially on the largest clique's number of variables, rather than necessarily on the total number of variables.

Choose the water, vessel and immersed paddle as the closed system, with fixed outer volume and negligible endpoint changes in bulk mechanical energy. Between common equilibrium temperatures $T_1$ and $T_2$,The effective heat capacity $C_{\mathrm{eff}}$ includes every component that participates in the common temperature change. For nearly constant $C_{\mathrm{eff}}$, the readout is $C_{\mathrm{eff}}\Delta T$. Heat transferred between water and paddle stays inside this boundary; it must not be added again as external input. The drive supplies shaft work even though the vessel has no expansion work.

For a simplified modern model, let $M$ be the total descending mass, let each of $N$ descents cover height $h$, and let $W_{\mathrm{loss},\mathrm{out}}$ be energy diverted by the external drive. With negligible remaining drive kinetic energy and elastic storage,The signed $Q_{\mathrm{room},\mathrm{out}}=-Q_{\mathrm{room},\mathrm{in}}$ may be negative if the room warms the assembly. This model omits the small endpoint and elastic corrections explicitly considered in the historical analysis. It does not equate all of the weights' lost gravitational potential energy with the water's gained internal energy.

As a proposed modern protocol, measure the falling masses and distances, calibrate the thermometer and $C_{\mathrm{eff}}$ independently, record room temperature and resting drift before and after each run, and wait for fluid motion to decay before the final equilibrium reading. Repeat with different fall budgets and run durations. A stable ratio of corrected input to corrected thermal storage supports the first law of thermodynamics over the tested range. Resting controls only estimate leakage if their thermal conditions represent the driven run; sensor lag, nonuniform temperature, unmeasured drive losses, evaporation or changing heat capacity can spoil the inference. Those are limits of the measurement, rather than evidence that heat was stored as a separate substance.

The gas-only condition $\Delta U=0$ requires negligible heat exchange with the apparatus. If the insulated isolated system instead includes the walls, sensors and water, the first law of thermodynamics fixes their combined energy. Once everything returns to a common final temperature, a constant apparatus heat capacity $C_a$ givesDefine $A=U_g(T_1,V_2)-U_g(T_1,V_1)$, the gas's internal energy increase for the same expansion evaluated at the initial temperature. For a small temperature change, linearizing at the final volume givesHere $C_{V,g}$ is the whole gas sample's heat capacity at constant volume at the reference state. Compared with an isolated gas sample's $-A/C_{V,g}$, the common final signal is reduced by $C_{V,g}/(C_{V,g}+C_a)$. This energy accounting explains why a massive bath can conceal a real-gas effect. Fitzpatrick gives the corresponding apparatus correction for the Van der Waals equation. ^[ref]

For a toy calculation, stipulate $A=5\mathrm{J}$, $C_{V,g}=10\mathrm{J}{\mathrm{K}}^{-1}$ and $C_a=990\mathrm{J}{\mathrm{K}}^{-1}$, all constant in this example. The predicted cooling magnitude changes from $0.5\mathrm{K}$ for the gas alone to $0.005\mathrm{K}$ for the combined apparatus. An illustrative effective detection threshold of $0.01\mathrm{K}$ would miss the latter. These are chosen numbers, not Joule's measurements. In this approximation a null observation only bounds $|A|\lesssim (C_{V,g}+C_a){ϵ}_T$, where ${ϵ}_T$ is the detectable temperature change after allowing for drift and other experimental uncertainties; a display's smallest digit is not automatically this threshold.

The historical limitation was recognized explicitly: Joule and Thomson later estimated that the common apparatus in Joule's experiment would cool by only about $0.003\mathrm{K}$, explaining why the earlier external thermal effect was undetectable. This was a later calculation about the apparatus, not a measured cooling trace. ^[ref]

For real $a\ne 0$ and $|x|<\pi$, integrate $z\sin (xz)/((a^2+z^2)\sin \pi z)$ around a thin positively oriented strip containing the integers. Its integer residues give the series. Closing the two strip sides outward instead gives minus the sum of the residues at $z=\pm ia$, each $\sinh (ax)/(2\sinh (a\pi ))$. The factor $|x|<\pi$ ensures exponential decay on those closures. The bilateral series is interpreted by symmetric partial sums; at $a=0$ its continuous extension is $-x$ with the zero-index summand defined to be zero.

An involution in characteristic two has Jordan blocks of sizes at most two. Its conjugacy type is determined by the rank of $N$, subject to $2\mathrm{rank}N\le \dim V$. GL-conjugacy can split in SL if the centralizer determinant map is not onto; that issue must be checked separately. In $S{L}_3(4)$ a one-dimensional untouched block makes that determinant map onto, yielding one involution class. In $S{L}_4(2)=G{L}_4(2)$ the two possible nonzero ranks give two classes. Their projective quotients have equal order but different involution class counts.

A matrix is defective when an eigenvalue has geometric multiplicity smaller than its algebraic multiplicity. Over a field in which the characteristic polynomial splits, this is precisely the obstruction to an eigenbasis: at least one Jordan block has size greater than one, and generalized eigenvectors are needed to complete a basis. The shear mapping in diagonalizable matrix is the simplest example.

A Jordan curve in the two-sphere is carried to an equator by a homeomorphism of the sphere; equivalently, the closures of its two complementary regions are discs. This strengthens the separation assertion of the Jordan curve theorem and controls the boundary degrees of characteristic maps for cell decompositions of a surface.

A Jordan domain is the bounded complementary component of a closed simple curve in the plane. The Jordan curve theorem separates the plane into this inside component and one unbounded outside component. A conformal bijection from the disc or complex upper half-plane to this simply connected domain extends to a homeomorphism of their closed conformal boundaries by the Caratheodory boundary extension theorem.

For a compact set $J$ homeomorphic to a circle, lifting an argument along its parametrization identifies the quotient by continuous exponentials with $\mathbb{Z}$. By planar logarithm factorization it is also free abelian on the bounded complementary components, so exactly one exists. A proper closed arc has trivial exponential quotient, and therefore connected complement. Given a small open arc of $J$ inside any neighborhood of a point, join points from the two complementary regions by a path avoiding the remaining closed arc. The first and last intersections with $J$ occur in the small arc, proving that the neighborhood meets both regions. Thus $J$ is the common boundary of its unique bounded and unbounded regions.

Draw a $k\times D$ random matrix $G$ whose entries are independent random variables with the standard normal distribution, and setThe factor $1/\sqrt{k}$ makes the expected value of the squared L2 norm equal to the original squared L2 norm. This Gaussian Johnson–Lindenstrauss transform is a linear map, generally not an orthogonal projection: its rows are not required to be orthogonal vectors. ^[ref]

For the fixed finite set $S$, let $0<\delta <1$ be an allowed failure probability, and defineWith $\log$ denoting the natural logarithm, a sufficient integer target dimension isThen one draw of this specified random matrix satisfies all the squared Euclidean distance bounds simultaneously with probability at least $1-\delta$. Since $c(\epsilon )\ge {\epsilon }^2/12$, the simpler sufficient bound $k\ge \lceil 12{\epsilon }^{-2}\log (n(n-1)/\delta )\rceil$ also works. Both are sufficient bounds for this argument; they do not claim the smallest possible target dimension.

This is a probability statement about a map drawn independently of the fixed finite set. It implies existence because the success probability is positive, but permits failed draws. The set cannot be chosen after inspecting the map with this same guarantee. In particular, no single draw preserves every possible finite set in the ambient Euclidean space when $k<D$.

A centrally symmetric convex body is in John position when its John ellipsoid is the Euclidean unit ball.

For a centrally symmetric convex body in John position, there are contact points $u_j\in \partial K\cap S^{d-1}$ and positive weights $c_j$ such that ${\sum }_j{c}_j{u}_j\otimes u_j=I$. Taking the trace gives ${\sum }_j{c}_j=d$. A supporting hyperplane at $u_j$ shows $|⟨u_j,x⟩|\le \Vert x{\Vert }_K$.

Here is the maximum-volume argument for the decomposition. If $I$ were outside the convex cone generated by the contact matrices, a separating hyperplane would give a symmetric matrix $A$ with $\mathrm{tr}A>0$ and $⟨Au,u⟩\le 0$ at every contact. Subtract a small positive multiple of $I$ so these inequalities become strict while the trace remains positive. The support function of $(I+tA)B_2^d$ is $|(I+tA)v|$. It decreases below $1$ near all contact directions for small positive $t$; away from them, the strict gap $h_K(v)>1$ preserves containment. Its volume increases, since $\det (I+tA)=1+t\mathrm{tr}A+O(t^2)$, a contradiction. The cone is closed because its generators have trace $1$ and form a compact set; finite-dimensional convexity gives a finite decomposition.

In inclusive electron-positron annihilation into hadrons, hard gluon radiation produces a quark, an antiquark and a gluon, giving three resolved particle jets. Observation of such events at PETRA in 1979 provided direct evidence for gluons. The institutional account is www.desy.de/news/news_search/index_eng.html?openDirectAnchor=1643&printversion=1 .

Apply Jensen's inequality to the two equally weighted values $a,b$ and the convex function $t\mapsto t^p$. Equality for $a=b>0$ proves that $2^{p-1}$ is the smallest possible constant. For $p>1$, strict convexity makes equality equivalent to $a=b$; for $p=1$, equality holds for every pair.

The Jensen gap is the nonnegative difference $\mathbb{E}[f(X)]-f(\mathbb{E}X)$ in Jensen's inequality, under its integrability assumptions. It quantifies how much applying a convex function before averaging differs from applying it afterward. For $f(x)=x^2$ it is the variance of $X$; for an affine function it is zero. For general convex functions, it depends on both the function and the distribution of $X$, and is not determined by the variance alone.

For a meromorphic function with no zero or pole on $|w|=R$, set $G_R(z,a)=\log |(R^2-\overline{a}z)/(R(z-a))|$. At source points that are neither zeros nor poles, the displayed identity expresses its logarithmic modulus as the Poisson integral of its boundary values, minus its zero Green potentials plus its pole Green potentials. The Poisson kernel is $(R^2-|z{|}^2)/|R{e}^{i\theta }-z{|}^2$ with normalized angular measure. Subtracting the corresponding logarithmic singularities leaves a harmonic function, so the harmonic Poisson formula proves the identity. At zero it reduces to Jensen's formula.

With physical wavelength $2\pi /k$, the Jeans length is $2\pi /k_J$, where the Jeans wavenumber is $\sqrt{4\pi G\rho }/c_s$. Linear pressure support dominates gravity below this length, whereas longer wavelengths can grow. In an expanding background the growth also depends on the Hubble damping term, so the static threshold does not by itself give the time dependence.